PIGTIKAL (puzzles in geometry that I know and love)

PIGTIKAL (puzzles in geometry that I kno w and lo v e) An ton P etrunin Asso ciation for Mathematical Researc h Monographs V olume 2 An ton P etrunin Departmen t of Mathematics The P ennsylv ania State Universit y Univ ersity P ark, P A 16802 USA AMR Monographs Editorial Board: George Andrews Colin Adams Eric F riedlander Rob ert Ghrist Jo el Hass Robion Kirby Alex Kon toro vich Sergei T abachnik ov Cop yright © 2022 by An ton Petrunin This work is licensed under a CC BY-SA 4.0 license T o view a c op y of this license, visit: https://creativecommons.org/licenses/by-sa/4.0/legalcode Second printing: 2025 Asso ciation for Mathematical Research Da vis, CA; Jenkinto wn P A Co ver design: Rob ert Ghrist Con ten ts 1 Curv es 5 2 Surfaces 21 3 Comparison geometry 35 4 Curv ature-free diﬀerential geometry 62 5 Metric geometry 78 6 A ctions and co verings 96 7 T op ology 110 8 Piecewise linear geometry 120 9 Discrete geometry 131 Bibliograph y 145 Preface This collection is ab out ideas, and it is not ab out theory . An idea might feel more comfortable in a suitable theory , but it has its own life and history , and it can sp eak for itself. I am collecting these problems for fun, but they migh t be used to impro ve the problem-solving skills in geometry . Ev ery problem has a short elegan t solution — this giv es a hint which was not av ailable when the problem was disco vered. Ho w to read it. Op en a random chapter; make sure you lik e the prac- tice problem — if yes try to solve a random problem in the chapter. A semisolution is giv en at the end of the c hapter, but think b efore reading, otherwise, it will not help. Some problems are marked b y ◦ , ∗ , + or  . ◦ — easy problem; ∗ — the solution requires at least tw o ideas; + — the solution requires knowledge of a theorem;  — there are interesting solutions based on diﬀeren t ideas. A ckno wledgments. I wan t to thank every one who help ed me; here is an incomplete list: Stephanie Alexander, Ilya Alexeev, Miroslav Baˇ c´ ak, Christopher Crok e, Bogdan Georgiev, Sergei Gelfand, Mohammad Ghomi, Jouni Luukk ainen, Alexander Lytchak, Andrei Malyutin, Rostislav Mat- v eyev, Dmitri Pano v, P eter P etersen, Idzhad Sabitov, Thomas Sharp e, Serge T abachnik ov, and Sergio Zamora Barrera. This collection is partly inspired by c onnoisseur’s c ol le ction of puzzles of P eter Winkler [1]. Many problems w ere suggested on MathOv erﬂow [2]. This work was partially supported b y the following grants: NSF grants DMS 0103957, 0406482, 0905138, 1309340, 2005279, Simons F oundation gran ts 245094 and 584781, and Minobrnauki grant 075-15-2022-289. Chapter 1 Curv es Recall that a curve is a contin uous map from a real interv al in to a space (for example, Euclidean plane) and a close d curve is a contin uous map deﬁned on a circle. If the map is injective then the curve is called simple . W e assume that the reader is familiar with related deﬁnitions including length of curv e and its curv ature. The necessary material is cov ered in the ﬁrst couple of lectures of a standard in tro duction to diﬀerential geometry , [see Part I in 3, Chapter 1 in 4, or §26–27 in 5]. W e giv e a practice problem with a solution — after that, you are on y our o wn. Spiral The follo wing problem states that if you drive on the plane and turn the steering wheel to the righ t all the time, then y ou will not b e able to come bac k to the same place.  L et γ b e a smo oth r e gular plane curve with strictly monotonic curva- tur e. Show that γ has no self-interse ctions. Semisolution. The trick is to sho w that the os- culating circles of γ are nested. Without loss of generalit y , w e ma y assume that the curve is parametrized by its length and its curv ature decreases. Let z ( t ) b e the center of the osculating circle at γ ( t ) and r ( t ) its radius. Note that z ( t ) = γ ( t ) + γ ′′ ( t ) | γ ′′ ( t ) | 2 , r ( t ) = 1 | γ ′′ ( t ) | . 5 6 CHAPTER 1. CUR VES Straigh tforward calculations show that | z ′ ( t ) | = r ′ ( t ) . Note that the curve z ( t ) has no straight arcs; therefore ( ∗ ) | z ( t 1 ) − z ( t 0 ) | < r ( t 1 ) − r ( t 0 ) . if t 1 > t 0 . Denote b y D t the osculating disk of γ at γ ( t ) ; it has a center at z ( t ) and radius r ( t ) . By ( ∗ ) , D t 1 lies in the interior of D t 0 for any t 1 > t 0 . Hence the result follows. This problem was considered by Peter T ait [6] and later rediscov ered b y A dolf Kneser [7]. The osculating circles of the curve giv e a p eculiar decomp osition of an ann ulus in to circles; it has the follo wing property: if a smo oth function is constant on each osculating circle it must b e constant in the annulus [see Lecture 10 in 8]. The same idea leads to a solution of the following problem:  L et γ b e a smo oth r e gular plane curve with strictly monotonic curva- tur e. Show that no line c an b e tangent to γ at two distinct p oints. It is instructive to chec k that the 3-dimensional analog does not hold; that is, there are self-in tersecting smooth regular space curv es with strictly monotonic curv ature. Note that if the curve γ ( t ) is deﬁned for t ∈ [0 , ∞ ) and its curv ature tends to ∞ as t → ∞ , then the problem implies the existence of the limit of γ ( t ) as t → ∞ . The latter result could b e considered as a contin uous analog of the Leibniz test for alternating series. F Mo on in a puddle  A smo oth close d simple plane curve with cur- vatur e less than 1 at every p oint b ounds ﬁgur e F . Pr ove that F c ontains a unit disk. Wire in a tin  L et α b e a close d smo oth curve immerse d in a unit disk. Pr ove that the aver age absolute curvatur e of α is at le ast 1 , with e quality if and only if α is the unit cir cle p ossibly tr averse d mor e than onc e. Curv e on a sphere  Show that if a close d curve on the unit spher e interse cts every e quator then its length is at le ast 2 · π . 7 Ov al in an o v al  Consider two close d smo oth strictly c onvex planar curves, one inside the other. Show that ther e is a chor d of the outer curve that is tangent to the inner curve at the midp oint of the chor d. Capture a sphere in a knot ∗ The follo wing form ulation uses the notion of smo oth isotop y of knots, that is, a one-parameter family of embeddings f t : S 1 → R 3 , t ∈ [0 , 1] suc h that the map [0 , 1] × S 1 → R 3 is smo oth.  Show that one c annot c aptur e a spher e in a knot. Mor e pr e cisely, let B b e the close d unit b al l in R 3 and f : S 1 → R 3 \ B a knot. Show that ther e is a smo oth isotopy f t : S 1 → R 3 \ B , t ∈ [0 , 1] such that f 0 = f , the length of f t is non-incr e asing with r esp e ct to t and f 1 ( S 1 ) c an b e sep ar ate d fr om B by a plane. Link ed circles  Supp ose that two linke d simple close d curves in R 3 lie at a distanc e at le ast 1 fr om e ach other. Show that the length of e ach curve is at le ast 2 · π . Surrounded area  Consider two simple close d plane curves γ 1 , γ 2 : S 1 → R 2 . Assume | γ 1 ( v ) − γ 1 ( w ) | ⩽ | γ 2 ( v ) − γ 2 ( w ) | for any v , w ∈ S 1 . Show that the ar e a surr ounde d by γ 1 do es not exc e e d the ar e a surr ounde d by γ 2 . Cro ok ed circle  Construct a b ounde d set in R 2 home omorphic to an op en disk such that its b oundary c ontains no simple curves. 8 CHAPTER 1. CUR VES Rectiﬁable curv e F or the following problem we need the notion of Hausdorﬀ me asur e . Cho ose a compact set X ⊂ R 2 and α > 0 . Giv en δ > 0 , set h ( δ ) = inf ( X i (diam X i ) α ) , where the greatest lo wer b ound is taken ov er all ﬁnite cov erings { X i } of X such that diam X i < δ for eac h i . Note that the function δ 7→ h ( δ ) is not decreasing in δ . In particular, h ( δ ) → H α ( X ) as δ → 0 for some (possibly inﬁnite) v alue H α ( X ) . This v alue H α ( X ) is called the α -dimensional Hausdorﬀ measure of X .  L et X ⊂ R 2 b e a c omp act c onne cte d set with ﬁnite 1-dimensional Hausdorﬀ me asur e. Show that ther e is a r e ctiﬁable curve p assing thru al l the p oints in X . Shortcut ∗  L et X ⊂ R 2 b e a c omp act c onne cte d set. Show t hat any two p oints x, y ∈ X c an b e c onne cte d by a p ath α such that the c omplement α \ X has arbitr arily smal l length. Note that it migh t b e impossible to connect x and y b y a path in X . In fact, there are connected sets in the plane (for example, the pseudo-arc) that contain no curves. Straigh t set A set X in the plane is called δ -straigh t if, for any disc D ( x, r ) with radius r > 0 and center at a point x ∈ X , the intersection X ∩ D ( x, r ) is δ · r -close in the sense of Hausdorﬀ to a diameter of D ( x, r ) .  Show that for any ε > 0 ther e is δ > 0 such that any δ -str aight close d plane set X is a curve that admits a lo c al ly (1 ± ε ) -bi-H¨ older p ar ametrization. That is, X is c onne cte d and for any x ∈ X ther e is a curve α : [0 , 1] → R 2 that c overs a neighb orho o d of x in X , and the ine quality c · | t 1 − t 0 | 1+ ε ⩽ | α ( t 0 ) − α ( t 1 ) | ⩽ C · | t 1 − t 0 | 1 − ε holds for any t 0 and t 1 and p ositive c onstants c and C , Before trying to solve the problem, it migh t b e useful to look at the follo wing example. Consider the set X k formed b y the origin and the tw o 9 logarithmic spirals ρ = ± e kφ in the polar coordinates ( ρ, ϕ ) . Observe that if k is large, then X k is δ -straight. Another example of a δ -straight set is shown; it is constructed in a w ay similar to the Koch snowﬂak e curv e, where we used a very obtuse isosceles triangle instead of equilateral. The Hausdorﬀ dimension of this example is larger than 1; in particular, it sho ws that one can not exp ect to hav e a Lipschitz parametrization of X . T ypical con v ex curv es F ormally w e do not need it in the problem, but it is worth noting that the curv ature of a conv ex curve is deﬁned almost everywhere; it follo ws from the fact that monotonic functions are diﬀerentiable almost ev erywhere.  Show that most of the c onvex close d curves in the plane have vanishing curvatur e at every p oint wher e it is deﬁne d. W e need to explain the meaning of the word “most” in the form ulation; it uses Hausdorﬀ distanc e and G-delta sets . The Hausdorﬀ distance | A − B | H b et ween tw o closed b ounded sets A and B in the plane is deﬁned by | A − B | H = sup x ∈ R 2 {| dist A ( x ) − dist B ( x ) |} , where dist A ( x ) denotes the smallest distance from A to x . Equiv alen tly , | A − B | H can b e deﬁned as the greatest low er b ound of the p ositiv e n umbers r such that the r -neighborho o d of A contains B and the r - neigh b orho o d of B contains A . It is straigh tforward to sho w that the Hausdorﬀ distance deﬁnes a metric on the space of all closed plane curves. The obtained metric space is lo cally compact. The latter follows from the sele ction the or em [see §18 in 9], which states that closed subsets of a ﬁxed closed b ounded set in the plane form a compact set with resp ect to the Hausdorﬀ metric. A G-delta set in a metric space X is deﬁned as a coun table in tersection of open sets. According to the Bair e c ate gory the or em , in lo cally compact metric spaces X , the intersection of a countable collection of op en dense sets has to b e dense. (The same holds if X is complete, but we will not need it.) In particular, in X , the in tersection of a ﬁnite or countable collection of G-delta dense sets is also a G-delta dense set. It means that eac h G- 10 CHAPTER 1. CUR VES delta dense set con tains most of X . This is the meaning of the word most used in the problem. Semisolutions Mo on in a puddle. In the pro of we will use the cut lo cus of F with resp ect to its b oundary (also known as me dial axis .); it will b e further denoted by T . The cut lo cus can b e deﬁned as the closure of the set of p oin ts x ∈ F for whic h there exist tw o or more p oin ts in ∂ F minimizing the distance to x . z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z T ∂ F F or each p oin t x ∈ T , consider the subset X ⊂ ∂ F where the minimal distance to x is attained. If X is not connected then w e sa y that x is a cut p oint ; equiv alen tly it means that for any suﬃciently small neigh b orho o d U ∋ x , the complemen t U \ T is disconnected. If X is connected then w e say that x is a fo c al p oint ; equiv alen tly it means that the osculating circle to ∂ F at an y p oin t of X is centered at x . The trick is to show that T con tains a fo cal p oin t, say z . Since ∂ F has curv ature of at most 1 , the radius of any osculating circle is at least 1 . Hence the distance from ∂ F to z is at least 1, and the statement will follow. After a small perturbation of ∂ F , w e may assume that T is a graph em b edded in F with a ﬁnite num b er of edges. Note that T is a deformation retract of F . The retraction F → T can b e obtained the following wa y: (1) given a p oint x ∈ F \ T , consider the (necessarily unique) p oin t ˆ x ∈ ∂ F that minimizes the distance | x − ˆ x | and (2) mov e x along the extension of the line segment [ ˆ xx ] b ehind x until it hits T . In particular, T is a tree. Therefore T has an end vertex, sa y z . The p oin t z is fo cal since there are arbitrarily small neigh b orho o ds U of z such that the complements U \ T are connected. W e prov ed a slightly stronger statement; namely , ther e ar e at le ast two p oints on ∂ F at which osculating cir cles lie in F . Note that these p oints are vertic es of ∂ F ; that is, they are critical p oints of its curv ature. Note further that inv ersion resp ects osculating circles. That is, if γ is an osculating circle of curve α at t 0 , γ ′ is the inv ersion of γ , and α ′ is the in version of α , then γ ′ is an osculating circle of curve α ′ at t 0 . Therefore applying an inv ersion about a circle with the center in F , we also get a pair of osculating circles of ∂ F which surround F . This w ay w e obtain 11 4 osculating circles that lie on one side of ∂ F . The latter statement is a generalization of the four-vertex theorem [10]. The case of con vex curves of this problem app ears in the b o ok of Wil- helm Blaschk e [see §24 in 9]. In full generalit y , the problem w as discussed b y Vladimir Ionin and German P estov [11]. A solution via curve shorten- ing ﬂow of a weak er statemen t was giv en b y K onstantin Pankrashkin [12]. The statemen t still holds if the curve fails to b e smo oth at one point [10]. A spherical v ersion of the later statemen t was used by Dmitri Pano v and me [13]. The statemen t admits a generalization to curves that b ound a disc F in a surface with nonp ositiv e curv ature. The latter can b e used to prov e the following problem which was suggested by Dmitri Burago.  L et γ b e a close d sp ac e curve with curvatur e at most 1. Show that γ c annot b e ﬁl le d by a disc with ar e a less than π . As you can see from the following problem, the 3-dimensional analog of this statement do es not hold.  Construct a smo oth emb e dding S 2  → R 3 with al l the princip al curva- tur es b etwe en − 1 and 1 such that it do es not surr ound a b al l of r adius 1. Suc h an example can b e obtained by fattening a non trivial contractible 2-complex in R 3 [Bing’s house constructed in 14 will do the job]. This problem is discussed b y Abram F et and Vladimir Lagunov [15–17] and it w as generalized to Riemannian manifolds with b oundary b y Stephanie Alexander and Richard Bishop [18]. A similar argument shows that for any Riemannian metric g on the 2-sphere S 2 and any p oint p ∈ ( S 2 , g ) there is a minimizing geo desic [ pq ] with conjugate ends. On the other hand, for ( S 3 , g ) this is not true. Moreo ver, there is a metric g on S 3 with sectional curv ature bounded ab o ve b y arbitrarily small ε > 0 and diam( S 3 , g ) ⩽ 1 . In particular, ( S 3 , g ) has no minimizing geo desic with conjugate ends. An example was originally constructed b y Mikhael Gromov [19]; a simpliﬁcation was given b y P eter Buser and Detlef Gromoll [20]. Wire in a tin. T o solve this problem, you should imagine that you tra vel on a train along the curve α ( t ) and watc h the p osition of the cen ter of the disk in the frame of your train car. Denote by  the length of α . Equip the plane with complex c oordinates so that 0 is the cen ter of the unit disk. W e can assume that α is equipp ed with an  -p erio dic parametrization by arc length. Consider the curve β ( t ) = t − α ( t ) α ′ ( t ) . Observe that β ( t +  ) = β ( t ) +  12 CHAPTER 1. CUR VES for any t . In particular ( ∗ ) length( β | [0 ,ℓ ] ) ⩾ | β (  ) − β (0) | = . Also | β ′ ( t ) | = | α ( t ) · α ′′ ( t ) α ′ ( t ) 2 | ⩽ ⩽ | α ′′ ( t ) | . Note that | α ′′ ( t ) | is the absolute curv ature of α at t . Therefore, the result follo ws from ( ∗ ) . The statement was originally prov ed by Istv´ an F´ ary in [21]; sev eral diﬀeren t pro ofs are discussed b y Serge T abachnik ov [see 22 and also 19.5 in 8]. If instead of the disk we hav e a region b ounded by a closed conv ex curv e γ , then it is still true that the av erage curv ature of α is at least as big as the a verage curv ature of γ . The pro of was given b y Jeﬀrey Lagarias and Thomas Richardson [see 23 and also 24]. Our solution can be adapted to the u nit ball of arbitrary dimension. F urther, the same argument together with Liouville’s theorem (geo desic ﬂo w preserv es the phase volume) implies that a closed smo oth submanifold in a unit ball has av erage normal curv atures at least 1. Let us formulate t wo more related problems.  L et M b e a close d smo oth n -dimensional submanifold in the unit b al l in R q . Denote by H p the me an curvatur e ve ctor of M at p . Show that the aver age value of | H p | is at le ast n .  Supp ose T is a smo othly emb e dde d 2-torus in the unit b al l in R q . Show that the aver age value of squar e d normal curvatur es of T is at le ast 3 2 . See also the problem “Small-twist embedding” on page 43. Curv e on a sphere. Let us present t wo solutions. W e assume that α is a closed curve in S 2 of length 2 ·  that intersects each e quator. A solution with Cr ofton ’s formula. Given a unit vector u , denote by e u the equator with the pole at u . Let k ( u ) b e the num b er of in tersections of α and e u . Note that for almost all u ∈ S 2 , the v alue k ( u ) is even or inﬁnite. Since each equator intersects α , we get k ( u ) ⩾ 2 for almost all u . Then we get 2 ·  = 1 4 · w u ∈ S 2 k ( u ) ⩾ ⩾ 1 2 · area S 2 = = 2 · π . 13 The ﬁrst iden tit y ab o ve is called Cr ofton ’s formula . T o pro ve this form ula, start with the case when the curv e is formed b y one geo desic segmen t, summing up we get it for broken lines, and by approximation it holds for all curves. A solution by symmetry. Let ˇ α b e a sub-arc of α of length  , with end- p oin ts p and q . Let z b e the midpoint of a minimizing geodesic [ pq ] in S 2 . Let r b e a p oin t of in tersection of α with the equator with p ole at z . Without loss of generality , w e ma y assume that r ∈ ˇ α . The arc ˇ α together with its reﬂection with respect to the p oint z forms a closed curv e of length 2 ·  passing thru b oth r and its antipo dal p oint r ∗ . Therefore  = length ˇ α ⩾ | r − r ∗ | S 2 = π . Here | r − r ∗ | S 2 denotes the angle metric in the sphere S 2 . Diﬀeren t solutions of this problem are discussed in a short note by Rob ert F oote [25]; the second pro of is due to Stephanie Alexander. The problem w as suggested by Nikolai Nadirashvili. It is nearly equiv alent to F enchel’s theorem:  Show that total curvatur e of any close d smo oth r e gular sp ac e curve is at le ast 2 · π . Let us also mention the problem of Karol Borsuk that was solved by John Milnor and Istv´ an F´ ary; it states that an y embedded circle of total curv ature less than 4 · π is unknotted. Six pro ofs of this statement are surv eyed b y Stephan Stadler and the author [26]. Ov al in an o v al. Cho ose a c hord that divides the area of the bigger o v al in the minimal (or maximal) ratio. If the c hord is not divided into equal parts, then you can rotate it sligh tly to decrease the ratio. Hence the problem follows. r l u x u A lternative solution. Giv en a u nit v ector u , denote by x u the p oint on the inner curv e with the outer normal v ector u . Dra w a chord of the outer curve that is tan- gen t to the inner curve at x u ; denote by r = r ( u ) and l = l ( u ) the lengths of the segmen ts of this chord to the right and to the left of x u , resp ectiv ely . Arguing by contradiction, assume that r ( u )  = l ( u ) for all u ∈ S 1 . Since the functions r and l are con tinuous, we can assume that ( ∗ ) r ( u ) < l ( u ) for all u ∈ S 1 . 14 CHAPTER 1. CUR VES Pro ve that each of the following tw o integrals 1 2 · w u ∈ S 1 r 2 ( u ) and 1 2 · w u ∈ S 1 l 2 ( u ) giv e the area b et ween the curves. In particular, the in tegrals are equal. The latter contradicts ( ∗ ) . This is a problem of Serge T abac hniko v [27]. A closely related e qual tangents pr oblem is discussed b y the same author in [28]. Capture a sphere in a knot. W e can assume that the knot lies on the sphere ∂ B . Cho ose a M¨ obius transformation m : S 2 → S 2 close to the iden tity and not a rotation. Note that m is a conformal map; that is, there is a function u deﬁned on S 2 as u ( x ) = lim y ,z → x | m ( y ) − m ( z ) | | y − z | . (The function u is called the c onformal factor of m .) Applying the area formula for m , we get 1 area S 2 · w u ∈ S 2 u 2 = 1 . By Buny ak ovsky inequalit y , 1 area S 2 · w u ∈ S 2 u < 1 . It follows that after a suitable rotation of S 2 , the map m decreases the length of the knot. Iterating this construction we get a sequence of knots f n : S 1 → S 2 with length decreasing and tending to zero. Passing to the limit as m → id , w e get a contin uous one-parameter family of M¨ obius transformations which shorten the length of the knot. Therefore it drifts the knot to a single hemisphere and allows the ball to escap e. This is a problem b y Zarathustra Brady , the giv en solution is based on the idea of Da vid Eppstein [29]. A solution to the follo wing problem is based on the same idea.  Show that a spher e c annot b e c aptur e d in a link with 3 c omp onents, but c an b e c aptur e d in a link with 4 c omp onents. It seems unknown which conv ex b o dies can b e captured b y a knot. The following problem is quite tricky already . 15  Show that one c an c aptur e some c onvex b o dies in a knot. The following problem of Abram Besico vitch is closely related [30]; it can b e solv ed using spherical Crofton’s formula.  Show the total length of strings in a net that c an hold a unit spher e has to b e lar ger than 3 · π . Link ed circles. Denote the linked circles b y α and β . Cho ose a p oint x ∈ α . Note that there is a p oint y ∈ α such that the line segment [ xy ] in tersects β , say at z . Indeed, if this is not the case, then a homothet y with center x to α w ould shrink it to x without crossing β . The latter con tradicts that α and β are linked. x y z β α Let α ∗ b e the image of α under the central pro jec- tion onto the unit sphere around z . Since | α ( t ) − z | ⩾ 1 for any t , we hav e that length α ⩾ length α ∗ . Observ e that α ∗ is a closed spherical curv e that contains tw o an tip odal p oin ts, one corresp onds to x and the other to y . It follo ws that length α ∗ ⩾ 2 · π . Hence the result follows. This problem w as prop osed by F rederic k Gehring [see 7.22 in 31]; solutions and generalizations are survey ed in [32]. The presented solution is attributed to Marvin Ortel in [33] and it is close to the solution of Mic hael Edelstein and Biny amin Sch warz [34]. Surrounded area. The tric k is to use the Kirszbraun theorem: A ny L -Lipschitz map f : Q → R n deﬁne d on a subset Q ⊂ R m c an b e extende d to an L -Lipschitz map ¯ f : R m → R n . This theorem app ears in the thesis of Mo j ˙ zesz Kirszbraun [35]; it was redisco vered later by F rederick V alentine [36]. An interesting survey is giv en b y Ludwig Danzer, Branko Gr ¨ un baum and Victor Klee [37]. Let C 1 and C 2 b e the compact regions b ounded by γ 1 and γ 2 resp ec- tiv ely . By the Kirszbraun theorem, there is a 1-Lipsc hitz map f : R 2 → R 2 suc h that f ( γ 2 ( v )) = f ( γ 1 ( v )) for any v ∈ S 1 . Note that f ( C 2 ) ⊃ C 1 . Hence the statement follows. Cro ok ed circle. A contin uous function f : [0 , 1] → [0 , 1] will b e called ε -cro ok ed if f (0) = 0 , f (1) = 1 and for any segment [ a, b ] ⊂ [0 , 1] one can c ho ose a ⩽ x ⩽ y ⩽ b such that | f ( y ) − f ( a ) | ⩽ ε and | f ( x ) − f ( b ) | ⩽ ε. 16 CHAPTER 1. CUR VES A sequence of 1 n -cro ok ed maps can b e constructed recursiv ely . Figure out the construction by lo oking at the following diagram. ε = 1 2 ε = 1 3 ε = 1 4 ε = 1 5 No w, start with the unit circle, γ 0 ( t ) = (cos 2 π t, sin 2 π t ) . Choose a sequence of p ositiv e num b ers ε n con verging to zero very fast. Construct recursiv ely a sequence of simple closed curves γ n : [0 , 1] → R 2 suc h that γ n +1 runs outside of the disk b ounded by γ n and | γ n +1 ( t ) − γ n ◦ f n ( t ) | < ε n , for an ε n -cro ok ed function f n . (It is hard to draw γ n +1 ; it runs like crazy bac k-and-forth almost along γ n .) Denote b y D the union of all disks b ounded by the curves γ n . Clearly D is homeomorphic to an op en disk. F or the right choice of the sequence ε n , the set D is bounded. By construction, the b oundary of D con tains no simple curves. In fact, the only curves in the b oundary of the constructed set are constan t. Compare to the problem Simple p ath on page 113. The pro of uses the so-called pseudo-ar c constructed by Bronis l a w Knaster [38]. The pro of resembles the construction of Can tor’s set. Here are a few similar problems:  Construct thr e e distinct op en sets in R with the same b oundary.  Construct thr e e op en disks in R 2 having the same b oundary. These disks are called lakes of W ada ; it is describ ed by Kunizˆ o Y one- y ama [39].  Construct a Cantor set in R 3 with a non-simply-c onne cte d c omplement. This example is called Antoine’s ne cklac e [40].  Construct an op en set in R 3 with a fundamental gr oup isomorphic to the additive gr oup of r ational numb ers. More adv anced examples include the Whitehe ad manifold , Do gb one sp ac e , and Casson hand le ; see also the problem “Conic neighborho o d” on page 112. Rectiﬁable curve. The 1-dimensional Hausdorﬀ measure will b e de- noted by H 1 . 17 Set L = H 1 ( K ) . Without loss of generality , we may assume that K has diameter 1 . Since K is connected, we get ( ∗ ) H 1 ( B ( x, ε ) ∩ K ) ⩾ ε for any x ∈ K and 0 < ε < 1 2 . Let x 1 , . . . , x n b e a maximal set of p oints in K with | x i − x j | ⩾ ε for all i  = j . F rom ( ∗ ) w e hav e n ⩽ 2 · L/ε . Note that there is a tree T ε with vertices x 1 , . . . , x n and straight edges with lengths at most 2 · ε each. Therefore the total length of T ε is b elo w 2 · n · ε ⩽ 4 · L . By construction, T ε is ε -close to K in the Hausdorﬀ metric. Clearly , there is a closed curve γ ε whose image is T ε and its length is t wice the total length of T ε ; that is, length γ ε ⩽ 8 · L. P assing to a subsequential limit of γ ε as ε → 0 , we get the needed curv e. In terms of measure, the optimal bound is 2 · L ; if in addition the diameter D is kno wn then it is 2 · L − D . The problem is due to Samuel Eilen b erg and Orville Harrold [41]; it also app ears in the b o ok of Kenneth F alconer [see Exercise 3.5 in 42]. Shortcut. Cho ose ε > 0 . Let us construct a set X ′ ⊂ X and a collection of paths α 0 , . . . , α n suc h that (i) the total length of α i \ X is at most ε , (ii) the set X ′ is a union of a ﬁnite collection of closed connected sets X 0 , . . . , X n , (iii) diameter of each X i is at most ε , (iv) x ∈ α 0 , y ∈ α n , and (v) the union X ′ ∪ α 0 ∪ · · · ∪ α n is connected Imagine that the construction is giv en. Let us show that the statement can b e prov ed by applying this construction recursively for a sequence of ε n that conv erges to zero very fast. Indeed we can apply the construction to eac h of the subsets X n and tak e as X ′′ the union of all closed subsets provided by the construction. This wa y we obtain a nested sequence of closed sets X ⊃ X ′ ⊃ X ′′ ⊃ . . . whic h break into a ﬁnite union of closed connected subsets of arbitrary small diameter and a countable collection of arcs with total length at most ε 1 + ε 2 + . . . outside of X . Consider the Cantor set Y = X ∩ X ′ ∩ X ′′ ∩ . . . 18 CHAPTER 1. CUR VES Note that there is a simple curv e from x to y that runs in the constructed arcs and Y . The total length of the constructed curves outside of X can not exceed the sum ε 1 + ε 2 + . . . ; whence the result. It remains to do the construction. Let us co v er X by a grid of ε 2 -squares Q 1 , . . . , Q k . Denote by ∆ the union of all the sides of the squares. By the regularity of length, we may cov er ∆ ∩ X b y a ﬁnite collection of arcs with total length arbitrarily close to the length of ∆ ∩ X . Denote these arcs by α 0 , . . . , α n . Without loss of generality , we ma y assume that x ∈ α 0 and y ∈ α n . Consider a ﬁnite graph with v ertices lab eled b y α 0 , . . . , α n ; tw o ver- tices α i and α j are adjacent if there is a connected set Θ ⊂ X ∩ Q k for some k such that Θ intersects α i and α j . Note that the graph is connected. Therefore we may choose a path from α 0 to α n in the graph. The path corresp onds to a sequence of arcs α i and a sequence of Θ - sets. The Θ -sets that correspond to the edges in the path can b e taken as X i in the construction. This solution was found by T aras Banakh [43]; it was used b y Stephan Stadler and the author [44]. The pro of works only in dimension tw o and w e are not a ware of a generalization to higher dimensions. Namely , the follo wing question is op en:  L et x, y b e two p oints in a c omp act c onne cte d subset X ⊂ R 3 . Is it always p ossible to c onne ct x and y by a p ath α such that the c omplement α \ X has arbitr arily smal l length? Straigh t set. A set L in the plane will b e called r -reachable if it is formed by a collection of disjoint prop er smo oth curves with r -tubular neigh b orho o d; that is, the closest-point pro jection to L is uniquely deﬁned in the r -neighborho o d of L . Note that in this case, L has curv ature at most 1 r at any p oint. F urther, δ i will denote a p ositiv e v alue that dep ends on δ suc h that δ i → 0 as δ → 0 . In fact, every statement b elo w holds for δ i = 239 · δ for an y i . W e assume that δ and, therefore, eac h δ i are small. Fix an δ -straigh t set X . Note that for any r > 0 , there is a r -reachable set L that is δ 1 · r -close to X in the sense of Hausdorﬀ. Indeed, using δ - straigh tness one may approximate X by a p olygonal line that has sides ab out r , v ery obtuse angles, and that do es not come close to a ﬁxed p oint t wice. Smo othing its corners pro duces L . Let L n b e a sequence of such approximations for r n = 1 2 n . Note that L n − 1 lies in a δ 2 · r n -neigh b orho o d of L n . In particular, the closest p oin t pro jection f n : L n − 1 → L n is uniquely deﬁned. Moreov er, it is straigh tforward to chec k that f n is (1 ∓ δ 3 ) -bi-Lipsc hitz. 19 L n − 1 L n Giv en x 1 ∈ L 1 , consider the sequence x n ∈ L n deﬁned by x n +1 = = f n ( x n ) . F rom ab ov e, we hav e that | x n +1 − x n | < δ 4 · r n , | x n +1 − y n +1 | ≷ (1 ∓ δ 4 ) · | x n − y n | . ( ∗ ) In particular, for any x 1 ∈ L 1 , the sequence ( x n ) conv erges in itself. Denote its limit by F ( x 1 ) ; evidently , F : L 1 → X is on to. It follo w s in particular, that any pair of p oints in X on distance at most 1 lie in one connected comp onen t. Since the δ -straightness does not dep end on the scale, it implies that X is connected. Note that ( ∗ ) implies the following tw o pairs of inequalities | x n +1 − y n +1 | ⩽ (1 + δ 5 ) · | x n − y n | , | x n +1 − y n +1 | ⩽ | x n − y n | + δ 5 · r n , and | x n +1 − y n +1 | ⩾ (1 − δ 5 ) · | x n − y n | , | x n +1 − y n +1 | ⩾ | x n − y n | − δ 5 · r n . T o pro ve that F is bi-H¨ older, apply recursively the ﬁrst inequality in each pair until | x n − y n | > r n and after the second one. This puzzle is a baby case of the so-called R eifenb er g’s lemma . It was in tro duced by Ernst Reifenberg [45] and became a useful to ol in metric geometry; in particular, it is used to study the limit spaces with low er Ricci curv ature b ound [46, 47]. T ypical conv ex curves. Denote by C the space of all closed conv ex curv es in the plane equipp ed with the Hausdorﬀ metric. Recall that C is lo cally compact. In particular, by the Baire theorem, a countable in tersection of everywhere dense op en sets is everywhere dense. Note that if a curv e γ ∈ C has non-zero second deriv ative at a p oint p , then γ lies b et ween t wo nested circles tangent to each other at p . Fix these t wo circles. It is straigh tforward to chec k that there is ε > 0 such that the Hausdorﬀ distance from an y conv ex curv e γ squeezed b et ween the circles to any conv ex n -gon is at least ε n 100 . 20 CHAPTER 1. CUR VES γ Cho ose a countable dense set of conv ex p olygons p 1 , p 2 , . . . in C . Denote by n i the n umber of sides in p i . F or any p ositive inte- ger k , consider the set Ω k ⊂ C deﬁned by Ω k = n ξ ∈ C    min i {| ξ − p i | H } < 1 k · n 100 i o , where |∗ − ∗| H denotes the Hausdorﬀ dis- tance F rom the abov e, w e get that γ / ∈ Ω k for some k . Note that Ω k is op en and everywhere dense in C . Therefore Ω = \ k Ω k is a G-delta dense set. Hence the statement follows. This problem states that typical conv ex curves hav e an unexp ected prop ert y . This is common — it is hard to see the typical ob jects and these ob jects often ha v e surprising prop erties. F or example, it was prov ed by Bernd Kirchheim, Emanuele Spadaro, and L´ aszl´ o Sz´ ek elyhidi, that typic al 1-Lipschitz maps fr om the plane to itself pr eserve the length of al l curves [48]. The same w ay , one could sho w that the b oundaries of typic al op en sets in the plane c ontain no nontrivial curves , but the construction of a concrete example is not trivial [see “Crooked circle”, page 7]. More problems of that t yp e are survey ed b y T udor Zamﬁrescu [49]. Chapter 2 Surfaces W e assume that the reader is familiar with smooth surfaces and the re- lated deﬁnitions including intrinsic metric, geo desics, con vex and saddle surfaces as well as diﬀerent types of curv ature. An introductory course in diﬀeren tial geometry should cov er all necessary background material [see 4, 3, or §28–29 in 5]. ∆ Σ Π Con vex hat  Supp ose that a plane Π cuts fr om a smo oth close d c onvex surfac e Σ a disk ∆ . Assume that the r eﬂe ction of ∆ with r esp e ct to Π lies inside of Σ . Show that ∆ is con vex with r esp e ct to the intrinsic metric of Σ ; that is, if b oth ends of a minimizing ge o desic in Σ lie in ∆ , then the entir e ge o desic lies in ∆ . Semisolution. Assume the contrary , then there is a minimizing geodesic γ ⊂ ∆ with ends p and q in ∆ . Without loss of generality , we may as sume that only one arc of γ lies outside of ∆ . Re ﬂection of this arc with resp ect to Π together with the remaining part of γ forms another curve ˆ γ from p to q ; it runs partly along Σ and partly outside Σ , but do es not get inside Σ . Note that length ˆ γ = length γ . Denote b y ¯ γ the closest p oint pro jection of ˆ γ on Σ . Since Σ is conv ex, the closest p oin t pro jection decreases the length. Therefore the curve ¯ γ lies in Σ , it has the same ends as γ , and length ¯ γ < length γ . This means that γ is not length-minimizing — a contradiction. 21 22 CHAPTER 2. SURF A CES Σ p q γ ( t ) p t In volute of geo desic  L et Σ b e a smo oth close d strictly c onvex surfac e in R 3 and γ : [0 ,  ] → Σ a unit-sp e e d minimizing ge o desic. Set p = γ (0) , q = γ (  ) , and p t = γ ( t ) − t · γ ′ ( t ) , wher e γ ′ ( t ) denotes the velo city ve ctor of γ at t . Show that for any t ∈ (0 ,  ) , one cannot see q fr om p t ; that is, the line se gment [ p t q ] interse cts Σ at a p oint distinct fr om q . Simple geo desic  L et Σ b e a c omplete unb ounde d c onvex surfac e in R 3 . Show that ther e is a two-side d inﬁnite ge o desic in Σ with no self-interse ctions. Let us review a couple of statements ab out Gauss curv ature which migh t help to solve the problem [see §28 in 5, for more details]. If Σ is a con vex surface in R 3 then its Gauss curv ature is nonnegative. Assume that a simply-connected region Ω in the surface Σ is b ounded b y a closed broken ge odesic γ . Denote by κ (Ω) the integral of the Gauss curv ature o ver Ω . F or any p oin t p ∈ Σ consider the outer unit normal vector n ( p ) ∈ S 2 . Then κ (Ω) = area[ n (Ω)] and by the Gauss–Bonnet formula κ (Ω) = 2 · π − σ ( γ ) , where σ ( γ ) denotes the sum of the signed exterior angles of γ . In partic- ular, | σ ( γ ) | ⩽ 2 · π . Geo desics for birds The total curvatur e of a space curve γ is deﬁned as the integral of its cur- v ature. That is, if a curve γ : [ a, b ] → R 3 has unit sp eed parametrization, then its total curv ature equals b w a | γ ′′ ( t ) | · dt, 23 the vector γ ′′ ( t ) is called the curvatur e ve ctor and its length | γ ′′ ( t ) | is the curvatur e of γ at time t . The ab o ve deﬁnition mak es sense for C 1 , 1 smo oth curv es, that is, in the case when γ ′ ( t ) is lo cally Lipschitz; in this case the curv ature | γ ′′ ( t ) | is deﬁned almost everywhere. The ge o desics in the following problem are deﬁned as the curv es locally minimizing the length; that is, any suﬃciently short arc of the curve con taining a given v alue of the parameter is length-minimizing.  L et f : R 2 → R b e a smo oth  -Lipschitz function. L et W ⊂ R 3 b e the epigr aph of f ; that is, W =  ( x, y , z ) ∈ R 3   z ⩾ f ( x, y )  . Equip W with the induc e d intrinsic metric. Show that any ge o desic in W has total curvatur e at most 2 ·  . A ctually , geo desics in W are C 1 , 1 -smo oth; in particular, the formula for the total curv ature mentioned ab o ve makes sense. This is an easy exercise in real analysis which can b e also taken for gran ted. Immersed surface  L et Σ b e a smo oth c onne cte d immerse d surfac e in R 3 with strictly p ositive Gauss curvatur e and nonempty b oundary ∂ Σ . Assume that the b oundary ∂ Σ lies in a plane Π and Σ lies entir ely on one side of Π . Pr ove that Σ is an emb e dde d disk. P erio dic asymptote  L et Σ b e a close d smo oth surfac e with non-p ositive curvatur e at every p oint and γ a ge o desic in Σ . Assume that γ is not p erio dic and the curvatur e of Σ vanishes at every p oint of γ . Show that γ do es not have a p erio dic asymptote; that is, ther e is no p erio dic ge o desic δ such that the distanc e fr om γ ( t ) to δ c onver ges to 0 as t → ∞ . Saddle surface Recall that a smo oth surface Σ in R 3 is sadd le at a p oint p if its principal curv atures at p hav e opp osite signs. W e say that Σ is sadd le if it is saddle at all p oin ts.  L et Σ b e a sadd le surfac e in R 3 home omorphic to a disk. Assume that the ortho gonal pr oje ction to the ( x, y ) -plane maps the b oundary of Σ inje ctively to a c onvex close d curve. Show that this pr oje ction is inje ctive on Σ . 24 CHAPTER 2. SURF A CES In p articular, Σ is the gr aph z = f ( x, y ) of a function f deﬁne d on a c onvex domain in the ( x, y ) -plane. Asymptotic line The saddle surfaces are deﬁned in the previous problem. Recall that a smo oth curve γ on a smo oth surface Σ ⊂ R 3 is called asymptotic if γ ′′ ( t ) is tangent to the surface at γ ( t ) for any t .  L et Σ ⊂ R 3 b e the gr aph z = f ( x, y ) of a smo oth function f and γ a close d smo oth asymptotic line in Σ . Assume that Σ is sadd le in a neighb orho o d of γ . Show that the pr oje ction of γ to the ( x, y ) -plane c annot b e star-shap e d; that is, ther e is no p oint p in the plane such that e ach half-line fr om p interse cts the pr oje ction at exactly one p oint. Minimal surface Recall that a smo oth surface in R 3 is called minimal if its mean curv ature v anishes at all p oin ts. The me an curvatur e at e ac h p oint is deﬁned as the sum of the principal curv atures at that p oint.  L et Σ b e a minimal surfac e in R 3 having its b oundary on a unit spher e. Assume that Σ p asses thru the c enter of the spher e. Show that the ar e a of Σ is at le ast π . Round gutter ∗ A round gutter is the surface shown on the picture. More precisely , consider the torus T ; a surface generated b y revolving a circle in R 3 around an axis coplanar with the circle. Let γ ⊂ T b e one of the circles in T that lo cally separates p ositive and negative curv ature on T ; a plane containing γ is tangent to T at all p oints of γ . Then a neigh b orho o d of γ in T is called a r ound gutter and the circle γ is called its main latitude .  L et Ω ⊂ R 3 b e a r ound gutter with the main latitude γ . Assume that ι : Ω → R 3 is a smo oth length-pr eserving emb e dding that is suﬃciently close to the identity. Show that γ and ι ( γ ) ar e c ongruent; that is, ther e is an isometric motion of R 3 sending γ to ι ( γ ) Non-con tractible geo desics  Give an example of a non-ﬂat metric on the 2 -torus such that no close d ge o desic is c ontr actible. 25 T w o disks  L et Σ 1 and Σ 2 b e two smo othly emb e dde d op en disks in R 3 that have a c ommon close d smo oth curve γ . Show that ther e is a p air of p oints p 1 ∈ Σ 1 and p 2 ∈ Σ 2 with p ar al lel tangent planes. Second deriv ativ e b ounds ﬁrst ◦  L et f b e a smo oth function on the L ob achevsky plane with b ounde d Hessian. Show that f is Lipschitz. Semisolutions In volute of geo desic. Let W b e the closed un b ounded set formed by Σ and its exterior p oin ts. Cho ose t ∈ (0 ,  ) ; denote by γ t the concatenation of the line segmen t [ p t γ ( t )] and the arc γ | [ t,ℓ ] . The k ey step is to show the follo wing: ( ∗ ) The curve γ t is a minimizing ge o desic in the intrinsic metric induc e d on W . T ry to prov e it b efore reading further. Let Π t b e the tangen t plane to Σ at γ ( t ) . Consider the curv e α ( t ) = p t . Note that α ( t ) ∈ Π t , α ′ ( t ) ⊥ Π t , and α ′ ( t ) p oints to the side of Π t opp osite to Σ . It follows that for any x ∈ Σ the function t 7→ | x − p t | and, therefore, t 7→ | x − p t | W are non-decreasing; here | x − p t | W stands for the intrinsic distance from x to p t in W . Σ W Π p q γ ( t ) p t α ′ ( t ) On the other hand, by construction | q − p t | W ⩽ | q − p | Σ ; therefore, from the ab ov e | q − p t | W = | q − p | Σ for any t . Hence ( ∗ ) follows. No w assume that q is visible from p t for some t ; that is, the line segmen t [ q p t ] in tersects Σ only at q . F rom the ab ov e, γ t coincides with the line segment [ q p t ] . On the other hand, γ t con tains γ ( t ) ∈ Σ — a con tradiction. 26 CHAPTER 2. SURF A CES This problem is based on an observ ation used by Anatoliy Milk a in the pro of of the following generalization of the comparison theorem for con vex surfaces [50, Theorem 2].  L et W b e the close d unb ounde d set forme d by a (not ne c essarily smo oth) close d c onvex surfac e Σ and its exterior p oints. Supp ose γ 1 : [0 ,  1 ] → Σ and γ 2 : [0 ,  2 ] → Σ ar e unit-sp e e d minimizing ge o desics that start at the same p oint p . Set x i = γ i (  i ) and y i = p +  i · γ ′ i (0) . Show that | x 1 − x 2 | W ⩽ | y 1 − y 2 | W , wher e | − | W stands for the intrinsic distanc e in W . Simple geo desic. Look at tw o combina- torial t yp es of a self-intersection shown in the diagram. One of them can and the other cannot app ear as self-intersections of a geodesic on an un b ounded con vex sur- face. T ry to determine which is which b e- fore reading further. Let γ b e a tw o-sided inﬁnite geodes ic in Σ . The following is the k ey statemen t in the pro of. ( ∗ ) The ge o desic γ c ontains at most one simple lo op. T o prov e ( ∗ ) , we use the following observ ation. ( ∗∗ ) The inte gr al curvatur e ω of Σ c annot exc e e d 2 · π . Indeed, since Σ is un b ounded and con v e x, it surrounds a half-line. Consider a co ordinate system with this half-line as the p ositive half of its z -axis. In these co ordinates, the surface Σ is describ ed as a graph z = f ( x, y ) of a con v ex function f . In particular, all outer normal vectors to Σ hav e a negativ e z -co ordinate; in other w ords, they point to the southern hemisphere. Therefore the area of the spherical image of Σ is at most 2 · π . The area of this image is the in tegral of the Gauss curv ature o ver Σ . Hence ( ∗∗ ) follows. F rom the Gauss–Bonnet formula, we get the follo wing conclusion. If ϕ is the angle at the base of a simple geo desic lo op then the integral curv ature surrounded by the loop equals π + ϕ . In particular, ( ∗∗ ) implies that ϕ ⩽ π ; in other words, there are no c onc ave lo ops . No w assume that ( ∗ ) do es not hold, so that a geo desic has tw o simple lo ops. Note that the disks bounded by the lo ops hav e to o verlap, otherwise the curv ature of Σ would exceed 2 · π . But if they ov erlap, then it is easy 27 to show that the curve also contains a concav e lo op, which contradicts the ab o ve observ ation. 1 If a geo desic γ has a self-intersection, then it con tains a simple loop. F rom ( ∗ ) , there is only one such lo op; it cuts a disk from Σ and go es around it either clo c kwise or counterclockwise. This w ay w e divide all the self-in tersecting geo desics in to tw o sets which we will call clo ckwise and c ounter clo ckwise . Note that the geo desic t 7→ γ ( t ) is clo ckwise if and only if t 7→ γ ( − t ) is coun terclo ckwise. The sets of clo ckwise and coun terclo c kwise are op en and the space of all geodesics is connected. It follo ws that there are geo desics that aren’t clo c kwise nor coun terclo c kwise. Those geodesics ha ve no self-intersections. Note that the proof implies that a t wo-sided inﬁnite geo desic can b e found among geo desics containing a given p oint in Σ . The problem is due to Stephan Cohn-V ossen [Satz 9 in 51]; general- izations were obtained by Vladimir Streltsov and Alexandr Alexandrov [52] and by Victor Bangert [53]. The following problem is of the same style [54].  L et γ b e a close d ge o desic on a close d surfac e of p ositive curvatur e Σ . Show that γ c annot lo ok like one of the curves on the diagr am. In other wor ds, γ c annot sub divide Σ into (1) one hexagon, one triangle, and thr e e mono gons, and (2) one p entagon, one quadr angle, and thr e e mono gons. Geo desics for birds. Cho ose a unit-sp eed geo desic in W , say γ : t 7→ ( x ( t ) , y ( t ) , z ( t )) . W e can assume that γ is deﬁned on a closed interv al [ a, b ] . The key step is to show the following: ( ∗ ) The function t 7→ z is c onc ave. P arametrize the plane curve t 7→ ( x ( t ) , y ( t )) b y the arc length s and reparametrize γ b y s . 1 This observ ation implies that the right picture on the ab ov e diagram cannot b e realized by a geo desic. 28 CHAPTER 2. SURF A CES s 0 s z Note that the function s 7→ z is conca ve. In- deed, supp ose s 7→ z is not conca ve around s 0 . Then one could shorten γ b y increasing its z comp onen t in a small interv al around s 0 while k eeping its endp oints ﬁxed. After the deforma- tion, the curve still lies in W . The latter contra- dicts that γ is lo cally length-minimizing. Finally , note that the concavit y of s 7→ z is equiv alen t to the concavit y of t 7→ z . Hence ( ∗ ) follows. Since f is smo oth, the curve γ ( t ) is C 1 , 1 ; that is, its ﬁrst deriv ativ e γ ′ ( t ) is a w ell-deﬁned Lipschitz function. It follo ws that its second deriv ativ e γ ′′ ( t ) is deﬁned almost everywhere. Since z ( t ) is concav e, we ha ve z ′′ ( t ) ⩽ 0 . Since f is  -Lipschitz, z ( t ) is ℓ √ 1+ ℓ 2 -Lipsc hitz. It follo ws that b w a | z ′′ ( t ) | · dt ⩽ 2 · ℓ √ 1+ ℓ 2 . The curv ature vector γ ′′ ( t ) is p erp endicular to the surface. Since the surface has slop e at most  , we get | γ ′′ ( t ) | ⩽ | z ′′ ( t ) | · p 1 +  2 . Hence b w a | γ ′′ ( t ) | · dt ⩽ 2 · . □ The statement holds for general  -Lischitz functions, not necessarily smo oth. The giv en b ound is optimal, the equalit y is attained by a tw o-side inﬁnite geo desic on the graph of f ( x, y ) = −  · p x 2 + y 2 . The problem is due to David Berg [55], the same b ound for conv ex  -Lipsc hitz surfaces was prov ed earlier by Vladimir Usov [56]. The obser- v ation ( ∗ ) is called Lib erman ’s lemma ; it was used earlier to bound the total curv ature of a geo desic on a conv ex surface [57]. 2 This lemma is often useful when working with geo desics on general conv ex surfaces. Immersed surface. Let  be a linear function that v anishes on Π and is p ositiv e on Σ . W e will apply a Morse-type argument for the restriction of  to Σ . 2 It is a part of the thesis of Joseph Lib erman, defended a couple of months b efore he died in WWI I. 29 Let z 0 b e a maximum of  on Σ ; set s 0 =  ( z 0 ) . Given s < s 0 , denote b y Σ s the connected comp onen t of z 0 in Σ ∩  − 1 ([ s, s 0 ]) . Note that ⋄ Σ s is an embedded disk, and ⋄ ∂ Σ s is a conv ex plane curve for all s suﬃciently close to s 0 . Consider the set A ⊂ [0 , s 0 ) such that for any a ∈ A these tw o condi- tions hold for any s ∈ [ a, s 0 ) . Observe that A is open and closed in [0 , s 0 ) . Whence A = [0 , s 0 ) ; in particular, these conditions hold for s = 0 . Since Σ is connected, Σ 0 = Σ . Hence the result follows. This problem is discussed in the lectures of Mikhael Gromov [see § 1 2 in 58]. P erio dic asymptote. Arguing by contradiction, assume that there is a geo desic γ on the surface Σ with a p erio dic asymptote δ . P assing to a ﬁnite co ver of Σ , w e can ensure that the asymptote has no self-intersections. In this case, the restriction γ | [ a, ∞ ) has no self- in tersections if a is suﬃciently large. Cut Σ along γ ([ a, ∞ )) and then cut from the obtained surface an inﬁnite triangle △ . The triangle has tw o sides formed by b oth sides of cuts along γ ; let us denote these sides of △ b y γ − and γ + . Note that ( ∗ ) area △ < area Σ < ∞ , and b oth sides γ ± are inﬁnite minimizing geo desics in △ . Consider the Busemann function f for γ + [deﬁned on page 38]; denote b y  ( t ) the length of the lev el curv e f − 1 ( t ) . Let − κ ( t ) b e the total cur- v ature of the sup-level set f − 1 ([ t, ∞ )) . F rom the Gauss–Bonnet formula, ( ∗∗ )  ′ ( t ) = κ ( t ) . The level curve f − 1 ( t ) can b e parametrized by a unit-sp eed curv e, say θ t : [0 ,  ( t )] → △ . By the coarea form ula w e hav e κ ′ ( t ) = − ℓ ( t ) w 0 K θ t ( τ ) · dτ , where K x denotes the Gauss curv ature of Σ at the p oint x . Since K θ t (0) = = K θ t ( ℓ t ) = 0 and the surface is smo oth, there is a constant C such that | K θ t ( τ ) | ⩽ C ·  ( t ) 2 for all t , τ . Therefore ( * * * ) κ ′ ( t ) ⩽ C ·  ( t ) 3 T ogether, ( ∗∗ ) and ( * * * ) imply that there is ε > 0 such that  ( t ) ⩾ ε t − a 30 CHAPTER 2. SURF A CES for suﬃciently large t . By the coarea formula we get area △ = ∞ w a  ( t ) · dt = ∞ ; the latter contradicts ( ∗ ) . I learned the problem from Dmitri Burago and Sergei Iv anov, it origi- nated from a discussion with Keith Burns, Mic hael Brin, and Y ako v P esin. Here is a motiv ation. Let Σ b e a closed surface with non-p ositive curv ature that is not ﬂat. The space Γ of all unit-sp eed geo desics γ : R → → Σ can b e identiﬁed with the unit tangent bundle UΣ . In particular Γ comes with a natural choice of measure. Denote by Γ 0 ⊂ Γ the set of geo desics that run in the set of zero curv ature all the time. It is exp ected that Γ 0 has a v anishing measure. In all kno wn examples Γ 0 con tains only p eriodic geo desics in only ﬁnitely many homotopy classes [see also 59]. Saddle surface. Denote by Σ ◦ the interior of Σ . Choose a plane Π . Note that the intersection Π ∩ Σ ◦ lo cally lo oks either like a curve or like t wo curv es intersecting transv ersally; in the latter case, Π is tangent to Σ ◦ at the intersection p oint. F urther, note that Π ∩ Σ ◦ has no cycle. Otherwise, Σ would fail to b e saddle at the p oint of the disk surrounded by that cycle maximizing the distance to Π . If Σ is not a graph then there is a p oint p ∈ Σ with a vertical tangent plane; denote this plane by Π . The in tersection Π ∩ Σ has a cross-p oin t at p . Since the b oundary of Σ pro jects injectively to a closed con vex curve in ( x, y ) -plane, the in tersection of Π ∩ ∂ Σ has at most 2 p oints — these are the only endp oints of Π ∩ Σ . It follows that the connected comp onent of p in Π ∩ Σ is a tree with a v ertex of degree 4 at p and at most t wo end-p oints — a contradiction. The describ ed idea can b e used to prov e the result of Ric hard Schoen and Shing-T ung Y au [60] which gives a suﬃcien t condition for a harmonic map b etw een surfaces to b e a diﬀeomorphism. Unlik e the original pro of, it requires no calculations. The pro of abov e is based on the observ ation that for an y saddle surface Σ and plane Π , each connected comp onen t of Σ \ Π is either unbounded or intersects the b oundary curve. This observ ation plays a central role in the pro of of Sergei Bernstein [61] of the following problem:  Show that a smo oth b ounde d function f : R 2 → R c annot have a strictly sadd le gr aph. One could go further and deﬁne a gener alize d sadd le surfac e as an ar- bitrary (non-necessarily smo oth) surface satisfying the observ ation ab o ve. 31 The geometry of these surfaces is far from being understoo d, Sam uil Shefel has a num b er of b eautiful results ab out them, [see 62, 63, and the refer- ences therein]. The statement of the problem holds for these generalized saddle surfaces, but the pro of is tricky [64]. Asymptotic line. Denote by Π t the tangent plane to Σ at γ ( t ) and by  t the tangent line of γ at time t . Since γ is asymptotic, the plane Π t rotates around  t as t changes. Since Σ is saddle, the sp eed of rotation cannot v anish. 3 Note that Π t is a graph of a linear function, say h t , deﬁned on the ( x, y ) -plane. Denote b y ¯  t the pro jection of  t to the ( x, y ) -plane. The describ ed rotation of Π t can b e expressed algebraically: the deriv ative d dt h t ( w ) v anishes at the point w if and only if w ∈ ¯  t and the deriv ative c hanges sign if w changes the side of ¯  t . Denote by ¯ γ the pro jection of γ to the ( x, y ) -plane. If ¯ γ is star-shap ed with resp ect to a p oint w , then w cannot cross ¯  t . Therefore the function t 7→ h t ( w ) is monotone on S 1 . Observing that this function cannot b e constan t, w e arriv e at a contradiction. This is a stripp ed version of the result of Galina K ov alev a [65], which w as redisco vered b y Dmitri Pano v [66, 67]. Minimal surface. Without loss of generalit y , w e ma y assume that the sphere is centered at the origin of R 3 . Let h be the restriction of the function x 7→ 1 2 · | x | 2 to the surface Σ . Direct calculations sho w that ∆ Σ h = 2 ; here ∆ Σ denoted Laplacian on Σ . Applying the divergence theorem for the gradien t ∇ Σ h in Σ r = Σ ∩ B (0 , r ) , w e get 2 · area Σ r ⩽ r · length[ ∂ Σ r ] . Set a ( r ) = area Σ r . By the coarea formula, a ′ ( r ) ⩾ length[ ∂ Σ r ] for almost all r . Therefore the function f : r 7→ area Σ r r 2 is non-decreasing in the interv al (0 , 1) . Since f ( r ) → π as r → 0 , the result follows. W e describ ed a partial case of the so-called monotonicity formula for minimal surfaces. The same argument sho ws that if 0 is a double point of Σ then area Σ ⩾ ⩾ 2 · π . This observ ation w as used to prov e that the minim al disk b ounded b y a simple closed curve with total curv ature ⩽ 4 · π is necessarily embed- ded. It w as pro ved b y T obias Ekholm, Brian White, and Daniel Wienholtz 3 By the Beltrami–Enneper theorem, if γ has unit sp eed, then the speed of rotation is ± √ − K , where K is the Gauss curv ature which cannot v anish on a saddle surface. 32 CHAPTER 2. SURF A CES [68]; an amusing v ariation of this pro of was obtained b y Stephan Stadler [69]. This result also implies that any embedded circle of total curv ature at most 4 · π is unknot. The latter was prov ed indep endently b y Istv´ an F´ ary [70] and John Milnor [71]. Note that if we assume in addition that the surface is a disk, then the statemen t holds for an y saddle surface. Indeed, denote b y S r the sphere of radius r concentrical with the unit sphere. Then according to the problem “A curve on a sphere” [page 6], length[ ∂ Σ r ] ⩾ 2 · π · r. Then by the coarea formula, we get area Σ ⩾ π . On the other hand, there are saddle surfaces homeomorphic to the cylinder having an arbitrarily small area in the ball. If Σ do es not pass thru the center and we only know the distance, sa y r , from the center to Σ , then the optimal b ound is π · (1 − r 2 ) . This question was op en for ab out 40 years and prov ed by Simon Brendle and P ei-Ken Hung [72]; their pro of is based on a similar idea and is quite elemen tary . Earlier Herb ert Alexander, David Hoﬀman, and Rob ert Os- serman prov ed it for tw o cases: (1) for disks and (2) for arbitrary area minimizing surfaces, any dimension and co dimension [73, 74]. Round gutter. Without loss of generalit y , we can assume that the length of γ is 2 · π and its in trinsic curv ature is 1 at all p oints. Let K b e the conv ex hull of ˆ Ω = ι (Ω) . P art of ˆ Ω touc hes the b oundary of K and the rest lies in the interior of K . Denote b y ˆ γ the curve in ˆ Ω dividing these tw o parts. First note that the Gauss curv ature of ˆ Ω should v anish at the p oin ts of ˆ γ ; in other words, ˆ γ = ι ( γ ) . Indeed, since ˆ γ lies on the conv ex part, the Gauss curv ature at the p oin ts of ˆ γ should b e non-negative. On the other hand, ˆ γ b ounds a ﬂat disk in ∂ K ; therefore its integral in trinsic curv ature should b e 2 · π . If the Gauss curv ature is p ositive at a p oint of ˆ γ , then by the Gauss–Bonnet formula, the total intrinsic curv ature of ˆ γ should b e smaller than 2 · π — a con tradiction. On the other hand, ˆ γ is an asymptotic line. Indeed, if the direction of ˆ γ is not asymptotic at t 0 , then ˆ γ ( t 0 ± ε ) lies the interior of K for some small ε > 0 — a contradiction. Therefore, as the space curve, ˆ γ has to be a closed curv e with constan t curv ature 1 . Any suc h curv e is congruent to a unit circle. It is not known whether ˆ Ω is congruent to Ω or not. The solution presen ted abov e is based on m y answ er to the question of Joseph O’Rourke [75]. Here are some related statements. ⋄ A gutter is second-order rigid; this was pro v ed by Eduard Rembs [see 76 and also page 135 in 77]. 33 ⋄ An y second-order rigid surface do es not admit analytic deformation [pro ved b y Nik olay Eﬁmov; page 121 in 77] and for the surfaces of rev olution, the assumption of analyticit y can b e remo ved [pro ved b y Idzhad Sabitov, see 78]. Non-con tractible geo desics. A torus of revolution is an example. Indeed, it has a family of meridians — a family of circles that form closed geo desics. Note that a geo desic on the torus is either a meridian or it intersects meridians transversally . In the latter case, all the meridians are crossed by the geo desic in the same direction. Note that a contractible curv e has to cross each meridian an equal n umber of times in b oth directions. Therefore no geo desic of the torus is con tractible. I learned this problem from the b o ok of Mikhael Gromov [79], where it is attributed to Y. Colin de V erdi` ere. The same argument can b e used to show that a torus with a geodesic foliation has no contractible closed geo desics. I do not kno w other examples of that type [80]; namely , the follo wing question is op en:  Ar e ther e examples of Riemannian metrics on the 2-torus without close d nul l-homotopic ge o desics and without a ge o desic foliation? T w o disks. Cho ose a contin uous map h : Σ 1 → Σ 2 that is the identit y on γ . Let us prov e that for some p 1 ∈ Σ 1 and p 2 = h ( p 1 ) ∈ Σ 2 , the tangen t planes T p 1 Σ 1 and T p 2 Σ 2 are parallel; this fact is stronger than the one required. Arguing by con tradiction, assume that such a p oint do es not exist. Then for eac h p ∈ Σ 1 there is a unique line  p ∋ p parallel to b oth T p Σ 1 and T h ( p ) Σ 2 . Note that the lines  p form a tangent line distribution ov er Σ 1 and  p is tangent to γ at all p ∈ γ . Let ∆ b e the disk in Σ 1 b ounded by γ . Consider the doubling of ∆ along γ ; it is diﬀeomorphic to S 2 . The line distribution  lifts to a line distribution on the doubling. The latter contradicts the hairy ball theorem. This pro of was suggested nearly simultaneously by Steven Sivek and Damiano T esta [81]. Note that the same pro of w orks when Σ i are orien ted op en surfaces and γ cuts a compact domain in each Σ i . There are examples of three disks Σ 1 , Σ 2 , and Σ 3 with a common closed curve γ suc h that there is no triple of p oin ts p i ∈ Σ i with parallel tangen t planes. Such examples can b e found among ruled surfaces [82]. 34 CHAPTER 2. SURF A CES Second deriv ativ e b ounds ﬁrst. Observe that the gradien t v = ∇ f is almost parallel; that is, there is a constan t C suc h that |∇ u v | ⩽ C for any unit tangent v ector u . In particular, the parallel translation of v ( p ) around a circle has to be close to v ( p ) . If | v ( p ) | is large, the latter con tradicts the Gauss–Bonnet formula. I learned this problem from Christopher Criscitiello [83]. Chapter 3 Comparison geometry In this chapter, we consider Riemannian manifolds with curv ature b ounds. This c hapter is very demanding; we assume that the reader is familiar with the shap e op erator and second fundamental form, equations of Ric- cati and Jacobi, comparison theorems, and Morse theory . The classical b ook [84] co v ers all the necessary material. Geo desic immersion ∗ An isometric immersion ι : N ↬ M from one Riemannian manifold to another is called total ly ge o desic if it maps any geo desic in N to a geo desic in M .  L et M and N b e simply-c onne cte d p ositively-curve d Riemannian man- ifolds and ι : N ↬ M a total ly ge o desic immersion. Assume that dim N > 1 2 · dim M . Pr ove that ι is an emb e dding. Semisolution. Set n = dim N , m = dim M . Cho ose a smo oth increasing strictly concav e function ϕ . Consider the function f = ϕ ◦ dist N , where dist N ( x ) denotes the distance from x ∈ M to N . Note that if f is smo oth at a p oint x ∈ M , then the Hessian of f at x (brieﬂy hess x f ) has at least n + 1 negative eigenv alues. Moreo ver, at any p oin t x / ∈ ι ( N ) the same holds in the barrier sense. That is, there is a smooth function h deﬁned on M such that h ( x ) = f ( x ) , h ( y ) ⩾ f ( y ) for any y and hess x h has at least n + 1 negative eigen v alues. Use that m < 2 · n and the describ ed prop erty to prov e the follo wing analog of Morse lemma for f . 35 36 CHAPTER 3. COMP ARISON GEOMETR Y ( ∗ ) Given x / ∈ ι ( N ) , ther e is a neighb orho o d U ∋ x such that the set U − = { z ∈ U | f ( z ) < f ( x ) } is simply-c onne cte d. Since M is simply-connected, any closed curve in ι ( N ) can b e con- tracted by a disc, say s 0 : D → M . Applying the claim ( ∗ ) , one can construct an f -decreasing homotop y s t that starts at s 0 and ends in ι ( N ) ; that is, there is a h omotop y s t : D → → M , t ∈ [0 , 1] such that s t ( ∂ D ) ⊂ ι ( N ) for any t and s 1 ( D ) ⊂ ι ( N ) . It follo ws that ι ( N ) is simply-connected. Finally , assume that a and b are distinct p oints in N such that ι ( a ) = = ι ( b ) . If γ is a path from a to b in N then the lo op ι ◦ γ is not con tractible in ι ( N ) . Therefore if ι : N → M has a self-intersection, then the image ι ( N ) is not simply-connected. Hence the result follows. The statement was prov ed by F uquan F ang, S ´ ergio Mendon¸ ca, and Xiao c hun Rong [85]. The main idea was discov ered by Burkhard Wilking [86]. Geo desic h yp ersurface The totally geo desic embedding is deﬁned b efore the previous problem.  Assume a c omp act c onne cte d p ositively-curve d manifold M has a to- tal ly ge o desic emb e dde d hyp ersurfac e. Show that either M or its double c overing is home omorphic to the spher e. If con vex, then embedded  L et M b e a c omplete simply-c onne cte d Riemannian manifold with non- p ositive curvatur e and dimension at le ast 3 . Pr ove that any immerse d lo c al ly c onvex c omp act hyp ersurfac e Σ in M is emb e dde d. Let us summarize some statements ab out complete simply-connected Riemannian manifolds with non-p ositive curv ature. By the Cartan–Hadamard theorem, for any p oint p ∈ M the exp o- nen tial map exp p : T p → M is a diﬀeomorphism. In particular, M is diﬀeomorphic to the Euclidean space of the same dimension. Moreo ver, an y geo desic in M is minimizing, and any t wo p oin ts in M are connected b y a unique minimizing geo desic, F urther, M is a CA T(0) space; that is, it satisﬁes a global angle com- parison whic h we are ab out to describ e. Let [ xy z ] b e a triangle in M ; that is, [ xy z ] is formed by three distinct points x, y , z pairwise connected b y geo desics. Consider its mo del triangle [ ˜ x ˜ y ˜ z ] in the Euclidean plane; 37 that is, [ ˜ x ˜ y ˜ z ] has the same side lengths as [ xy z ] . Then each angle in [ xy z ] cannot exceed the corresp onding angle in [ ˜ x ˜ y ˜ z ] . This inequality can b e written as ˜ ∡ ( y x z ) ⩾ ∡ [ y x z ] , where ∡ [ y x z ] denotes the angle of the hinge [ y x z ] formed by tw o geo desics [ y x ] and [ y z ] and ˜ ∡ ( y x z ) denotes the corresp onding angle in the mo del triangle [ ˜ x ˜ y ˜ z ] . F rom this comparison, it follo ws that any connected closed locally con vex sets in M is globally conv ex. In particular, if Σ is embedded then it b ounds a conv ex set. Immersed ball ∗  Pr ove that any immerse d lo c al ly c onvex hyp ersurfac e ι : Σ ↬ M in a c omp act p ositively-curve d manifold M of dimension m ⩾ 3 is the b oundary of an immerse d b al l. That is, ther e is an immersion of a close d b al l f : ¯ B m ↬ M and a diﬀe omorphism h : Σ → ∂ ¯ B m such that ι = f ◦ h . Minimal surface in the sphere A smooth n -dimensional surface Σ in an m -dimensional Riemannian man- ifold M is called minimal if it lo cally minimizes the n -dimensional area; that is, suﬃciently small regions of Σ do not admit area-decreasing defor- mations with a ﬁxed b oundary . The minimal surfaces can b e also deﬁned via the mean curv ature v ector as follows. Let T = T Σ and N = N Σ denote the tangent and the normal bundle respectively . Let s denote the second fundamental form of Σ ; it is a quadratic from on T with v alues in N , see the remark after problem “Hyp ercurv e” b elow. Given an orthonormal basis ( e i ) in T x , set H x = X i s ( e i , e i ) . The vector H x lies in the normal space N x and do es not dep end on the c hoice of orthonormal basis ( e i ) . This vector H x is called the mean cur- v ature v ector at x ∈ Σ . W e say that Σ is minimal if H ≡ 0 .  L et Σ b e a close d n -dimensional minimal surfac e in the unit m - dimensional spher e S m . Pr ove that vol n Σ ⩾ vol n S n . 38 CHAPTER 3. COMP ARISON GEOMETR Y Hyp ercurv e The Riemann curv ature tensor R can b e viewed as an op erator R on the space of tangent bi-vectors V 2 T ; it is uniquely deﬁned by the identit y ⟨ R ( X ∧ Y ) , V ∧ W ⟩ = ⟨ R ( X , Y ) V , W ⟩ . The op erator R : V 2 T → V 2 T is called the curvatur e op er ator and it is said to b e p ositive deﬁnite if ⟨ R ( ϕ ) , ϕ ⟩ > 0 for all non-zero bi-vector ϕ ∈ V 2 T .  L et M m  → R m +2 b e a close d smo oth m -dimensional submanifold and let g b e the induc e d Riemannian metric on M m . Assume that se ctional curvatur e of g is p ositive. Pr ove that the curvatur e op er ator of g is p ositive deﬁnite. The second fundamental form for manifolds of arbitrary co dimension whic h w e are ab out to describ e might help to solve this problem. Let M b e a smo oth submanifold in R m . Given a p oin t p ∈ M , denote b y T p and N p = T ⊥ p the tangent and normal space of M at p . The se c ond fundamental form of M at p is deﬁned by s ( X, Y ) = ( ∇ X Y ) ⊥ , where ( ∇ X Y ) ⊥ a denotes the orthogonal pro jection of co v ariant deriv ativ e ∇ X Y onto the normal bundle. The curv ature tensor of M can b e found from the second fundamental form using the following formula ⟨ R ( X , Y ) V , W ⟩ = ⟨ s ( X , W ) , s ( Y , V ) ⟩ − ⟨ s ( X , V ) , s ( Y , W ) ⟩ , whic h is a direct generalization of the formula for Gauss curv ature of a surface. Horo-sphere W e say that a Riemannian manifold has negatively pinched sectional cur- v ature if its sectional curv atures at all points in all sectional directions lie in an interv al [ − a 2 , − b 2 ] , for ﬁxed constants a > b > 0 . Let M b e a complete R iemannian manifold and γ a ray in M ; that is, γ : [0 , ∞ ) → M is a minimizing unit-sp eed geo desic. The Busemann function bus γ : M → R is deﬁned by bus γ ( p ) = lim t →∞ ( | p − γ ( t ) | M − t ) . By the triangle inequality , the expression under the limit is non-increasing in t ; therefore the limit ab ov e is deﬁned for any p . 39 A hor o-spher e in M is deﬁned as a level set of a Busemann function on M . W e sa y that a complete Riemannian manifold M has p olynomial vol- ume gr owth if, for some (and therefore any) p ∈ M , w e ha ve v ol B ( p, r ) M ⩽ C · ( r k + 1) , where B ( p, r ) M denotes the ball in M and C , k are constan ts.  L et M b e a c omplete simply-c onne cte d manifold with ne gatively pinche d se ctional curvatur e and Σ ⊂ M an hor o-spher e in M . Show that Σ with the induc e d intrinsic metric has p olynomial volume gr owth. Num b er of conjugate p oints Recall that p oin ts p and q on a geo desic γ are called c onjugate if there exists a non-zero Jacobi ﬁeld along γ that v anishes at p and q .  L et s : N → M b e a Riemannian submersion. Supp ose N has non- p ositive se ctional curvatur e. Show that any p oint p in M has at most k = dim N − dim M c onjugate p oints on any ge o desic γ ∋ p . Minimal spheres Recall that tw o subsets A and B in a metric space X are called e quidistant if the distance function dist A : X → R is constan t on B and dist B is constan t on A . The minimal surfaces are deﬁned on page 37.  Show that a 4 -dimensional c omp act p ositively-curve d Riemannian man- ifold c annot c ontain an inﬁnite numb er of mutual ly e quidistant minimal 2-spher es. P ositive curv ature and symmetry +  Assume that S 1 acts isometric al ly on a close d 4 -dimensional Riemann- ian manifold with p ositive se ctional curvatur e. Show that the action has at most 3 isolate d ﬁxe d p oints. The follo wing statement might b e useful. If ( M , g ) is a Riemannian manifold with sectional curv ature ⩾ κ that admits a con tinuous isometric action of a compact group G , then the quotient space A = ( M , g ) /G is an Alexandrov space with curv ature ⩾ κ ; that is, the conclusion of the T op onogov comparison theorem holds in A . F or more on Ale xandro v geometry see [87]. 40 CHAPTER 3. COMP ARISON GEOMETR Y Energy minimizer Let F b e a smo oth map from a closed Riemannian manifold M to a Riemannian manifold N . The energy functional of F is deﬁned by E ( F ) = w x ∈ M | d x F | 2 . W e assume that | d x F | 2 = X i,j a 2 i,j , where ( a i,j ) denote the comp onents of the diﬀerential d x F written in the orthonormal bases of the tangent spaces T x M and T F ( x ) N .  Show that the identity map on R P m is ener gy-minimizing in its ho- motopy class. Her e we assume that R P m is e quipp e d with the c anonic al metric. Curv ature against injectivit y radius +  L et ( M , g ) b e a close d R iemannian m -dimensional manifold. Assume aver age of se ctional curvatur es over al l se ctional dir e ctions of ( M , g ) is 1 . Show that the inje ctivity r adius of ( M , g ) is at most π . Solutions to this and the previous problem use the fact that geo desic ﬂo w on the tangent bundle to a Riemannian manifold preserves the v olume form; this is a corollary of Liouville’s theorem ab out phase volume. Appro ximation of a quotien t  L et ( M , g ) b e a c omp act Riemannian manifold and G a c omp act Lie gr oup acting by isometries on ( M , g ) . Construct a se quenc e of metrics g n on a ﬁxe d manifold N such that ( N , g n ) c onver ges to the quotient sp ac e ( M , g ) /G in the sense of Gr omov–Hausdorﬀ. P olar p oints ♯  L et M b e a c omp act Riemannian manifold with se ctional curvatur e at le ast 1 and dimension at le ast 2 . Pr ove that for any p oint p ∈ M ther e is a p oint p ∗ ∈ M such that | p − x | M + | x − p ∗ | M ⩽ π for any x ∈ M . 41 Isometric section ∗  L et M and W b e c omp act Riemannian manifolds, dim W > dim M , and s : W → M a Riemannian submersion. Assume that W has p ositive se ctional curvatur e. Show that s do es not admit an isometric se ction; that is, ther e is no isometric emb e dding ι : M  → W such that s ◦ ι ( p ) = p for any p ∈ M . W arped pro duct Let ( M , g ) and ( N , h ) b e Riemannian manifolds and f a smo oth p ositive function deﬁned on M . Consider the pro duct manifold W = M × N . Giv en a tangen t v ector X ∈ T ( p,q ) W = T p M × T p N , denote by X M ∈ T M and X N ∈ T N its pro jections. Let us equip W with the Riemannian metric deﬁned by s ( X, Y ) = g ( X M , Y M ) + f 2 · h ( X N , Y N ) . The obtained Riemannian manifold ( W , s ) is called a warp e d pr o duct of M and N with resp ect to f : M → R ; it can b e written as ( W , g ) = ( N , h ) × f ( M , g ) .  L et M b e an oriente d 3-dimensional Riemannian manifold with p ositive sc alar curvatur e and Σ ⊂ M an oriente d smo oth hyp ersurfac e that is ar e a minimizing in its homolo gy class. Show that ther e is a p ositive smo oth function f : Σ → R such that the warp e d pr o duct S 1 × f Σ has p ositive sc alar curvatur e; her e Σ is e quipp e d with the Riemannian metric induc e d fr om M . No appro ximation ♯  Pr ove that if p  = 2 , then R m e quipp e d with the metric induc e d by the  p - norm c annot b e a Gr omov–Hausdorﬀ limit of m -dimensional Riemannian manifolds ( M n , g n ) with Ric g n ⩾ C for a c onstant C . Area of spheres  L et M b e a c omplete non-c omp act Riemannian manifold with non- ne gative Ric ci curvatur e and p ∈ M . Show that ther e is ε > 0 such that area [ ∂ B ( p, r )] > ε for al l suﬃciently lar ge r . 42 CHAPTER 3. COMP ARISON GEOMETR Y Flat co ordinate planes  L et g b e a c omplete Riemannian metric on R 3 such that the c o or dinate planes x = 0 , y = 0 , and z = 0 ar e ﬂat and total ly ge o desic. Assume the se ctional curvatur e of g is either non-ne gative or non-p ositive. Show that in b oth c ases g is ﬂat. T w o-con vexit y ♯ An open subset V with smo oth b oundary in the Euclidean space is called two-c onvex if at most one principal curv ature in the outw ard direction to V is negative. The tw o-conv exity of V is equiv alen t to the following prop ert y . F or an y plane Π and any closed curve γ in the intersection V ∩ Π , if γ is con tactable in V then it is contactable in Π ∩ V .  L et K b e a close d set b ounde d by a smo oth surfac e in R 4 . Assume that K c ontains two c o or dinate planes { ( x, y , 0 , 0) ∈ R 4 } and { (0 , 0 , z , t ) ∈ R 4 } in its interior and also b elongs to the close d 1 -neighb orho o d of these two planes. Show that the c omplement to K is not two-c onvex. Con vex lens +  L et D and D ′ b e two smo oth discs with a c ommon b oundary that b ound a c onvex set (a lens) L in a p ositively-curve d 3-dimensional Riemannian manifold M . Assume that the discs me et at a smal l angle. L D D ′ Show that the inte gr al w D k 1 · k 2 is smal l; her e k 1 and k 2 denote the princip al curvatur es of D . The exp ected solution uses the follo wing relative version of the Bochner form ula. Let M be a Riemannian manifold with b oundary ∂ M . If f : M → R is a smo oth function that v anishes on ∂ M , then w M  | ∆ f | 2 − | hess f | 2 − ⟨ Ric( ∇ f ) , ∇ f ⟩  = w ∂ M H · |∇ f | 2 . Here we denote by Ric the Ricci curv ature of M and by H the mean curv ature of ∂ M , we assume that H ⩾ 0 is the b oundary is con v ex. 43 Small-t wist  Show that any smo oth close d manifold admits an immersion into the unit b al l in a Euclide an sp ac e of suﬃciently lar ge dimension with al l its normal curvatur es less than √ 3 . Semisolutions Geo desic h yp ersurface. Let Σ b e the totally geo desic embedded h yp er- surface in the p ositively-curv ed manifold M . Without loss of generality , w e can assume that Σ is connected. 1 The complement M \ Σ has one or tw o connected comp onents. First let us sho w that if the num b er of connected comp onents is tw o, then M is homeomorphic to a sphere. By cutting M along Σ we get t wo manifolds with geo desic b oundaries. It is suﬃcient to show that eac h of them is homeomorphic to a Euclidean ball. Cho ose one of these manifolds; denote it by N . Denote by f : N → → R the distance functions to the b oundary ∂ N . By the Riccati equation hess f ⩽ 0 at any smo oth p oint, and for an y p oint the same holds in the barrier sense [deﬁned on page 35]. It follows that f is concav e. Cho ose an increasing strictly concav e function ϕ : R → R . Note that ϕ ◦ f is strictly conca v e in the interior of N . Cho ose a compact subset K in the interior of N and smo oth ϕ ◦ f in a neigh b orho o d of K keeping it conca ve. This can b e done by applying the smo othing theorem of Rob ert Greene and Hung-Hsi W u [88, Theorem 2]. After the smo othing, the obtained strictly concav e function, sa y h , has a single critical p oin t which is its maximum. In particular, by Morse lemma, we get that if the set N ′ s = { x ∈ N | h ( x ) ⩾ s } is not empty and lies in K then it is diﬀeomorphic to a Euclidean ball. F or appropriately chosen set K and the smo othing h , the set N ′ s can b e made arbitrarily close to N ; moreov er, its b oundary ∂ N ′ s can b e made C ∞ -close to ∂ N . It follows that N is diﬀeomorphic to a Euclidean ball. This ﬁnishes the pro of of the ﬁrst case. No w assume M \ Σ is connected. In this case, there is a double co vering ˜ M of M that induces a double co vering ˜ Σ of Σ , so ˜ M contains a geo desic h yp ersurface ˜ Σ that divides ˜ M in to tw o connected comp onen ts. F rom the 1 In fact, b y F rank el’s theorem [see page 50] Σ is connected. 44 CHAPTER 3. COMP ARISON GEOMETR Y case which has already b een considered, ˜ M is homeomorphic to a sphere; hence the second case follows. The problem was suggested by Peter Petersen. If con vex, then em b edded. Set m = dim Σ = dim M − 1 . Giv en a p oin t p on Σ , denote by p r the p oin t at distance r from p that lies on the geo desic starting at p in the outer normal direction to Σ . Note that for ﬁxed r ⩾ 0 , the p oints p r sw eep an immersed lo cally con vex h yp ersurface which w e denote b y Σ r . p p r z Σ Σ r Cho ose z ∈ M . Denote by d the maximal distance from z to the p oints in Σ . Note that an y p oint on Σ r lies at a dis- tance at least r − d from z . By comparison, ∡ [ p r z p ] ⩽ arcsin d r . In particular, for large r , each inﬁnite geo desic starting at z in tersects Σ r transv ersally . The space of geo desics starting at z can b e identiﬁed with the sphere S m . There- fore the map that sends a p oin t x ∈ Σ r to a geo desic from z thru x induces a lo cal diﬀeomorphism ϕ r : Σ → S m . Since m ⩾ 2 , the sphere S m is simply-connected. Since Σ is connected, the map ϕ r is a diﬀeomorphism. Therefore Σ r is star-shap ed with a cen ter at z . In particular, Σ r is em b edded. Since Σ r is lo cally conv ex, it b ounds a conv ex region and is embedded. Consider the set W of reals r ⩾ 0 such that Σ r is em b edded and b ounds a conv ex region. Note that W is op en and closed in [0 , ∞ ) . Therefore W = [0 , ∞ ) , and, in particular, Σ 0 = Σ is embedded. The problem is due to Stephanie Alexander [89]. Immersed ball. Equip Σ with the induced in trinsic metric. Denote by κ the low er b ound for principal curv atures of Σ . Note that we can assume that κ > 0 . Cho ose a suﬃciently small ε = ε ( M , κ ) > 0 . Given p ∈ Σ , denote by ∆( p ) the ε -ball in Σ centered at p . Consider the lift ˜ h p : ∆( p ) → T h ( p ) along the exp onen tial map exp h ( p ) : T h ( p ) → M . More precisely: 45 1. Connect each p oint q ∈ ∆( p ) ⊂ Σ to p by a minimizing geo desic path γ q : [0 , 1] → Σ 2. Consider the lifting ˜ γ q in T h ( p ) ; that is, ˜ γ q is the curve suc h that ˜ γ q (0) = 0 and exp h ( p ) ◦ ˜ γ q ( t ) = γ q ( t ) for each t ∈ [0 , 1] . 3. Set ˜ h ( q ) = ˜ γ q (1) . Sho w that all the hypersurfaces ˜ h p (∆( p )) ⊂ T h ( p ) ha ve principal cur- v atures at least κ 2 . Use the same idea as in the solution of “Immersed surface” [page 23] to show that one can ﬁx ε = ε ( M , κ ) > 0 such that the restriction of ˜ h p | ∆( p ) is injective. Conclude that the restriction h | ∆( p ) is injective for an y p ∈ Σ . (Here we use that m ⩾ 3 .) No w consider lo cally equidistant surfaces Σ t in the inw ard direction for small t . The principal curv atures of Σ t remain at least κ in the barrier sense; that is, at an y point p , the surface Σ t can be supported by a smooth surface with principal curv atures at least κ at p . By the same argument as ab o ve, an y ε -ball in Σ t is embedded. W e get a one-parameter family of lo cally conv ex lo cally equidistant surfaces Σ t for t in the maximal interv al [0 , a ] , where the surface Σ a degenerates to a p oint, say p . T o construct the immersion ∂ ¯ B m ↬ M , take the p oint p as the image of the center ¯ B m and take the surfaces Σ t as the restrictions of the em- b edding to the spheres; the existence of the immersion follows from the Morse lemma. As you see from the picture, the analogous state- men t do es not hold in the tw o-dimensional case. The pro of presented ab ov e was indicated in the lectures of Mikhael Gromov [58] and written rigor- ously by Jost Eschen burg [90]. A v ariation of Gromov’s proof was obtained indep enden tly by Ben Andrews [91]. Instead of equidistan t deformation, he uses the so-called in- verse me an curvatur e ﬂow ; this wa y one has to per- form some calculations to show that conv exity survives in the ﬂow, but one do es not hav e to w orry about non-smoothness of the h yp ersurfaces Σ t . Minimal surface in the sphere. Cho ose a geodesic n -dimensional sphere ˜ Σ = S n ⊂ S m . Denote by U r and ˜ U r the closed tubular r -neigh b orho o d of Σ and ˜ Σ in S m resp ectiv ely . Note that ( ∗ ) U π 2 = ˜ U π 2 = S m . 46 CHAPTER 3. COMP ARISON GEOMETR Y Indeed, clearly ˜ U π 2 = S m . If U π 2  = S m , ﬁx x ∈ S m \ U r . Cho ose a closest p oin t y ∈ Σ to x . Since r = | x − y | S m > π 2 the r -sphere S r ⊂ S m with cen ter x is concav e. Note that S r supp orts Σ at y ; in particular, the mean curv ature v ector of Σ at y cannot v anish — a contradiction. By the Riccati equation, H r ( x ) ⩾ ˜ H r for any x ∈ ∂ U r , where H r ( x ) denotes the mean curv ature of ∂ U r at a p oin t x and ˜ H r is the mean curv ature of ∂ ˜ U r , the latter is the same at all p oin ts. Set a ( r ) = v ol m − 1 ∂ U r , ˜ a ( r ) = v ol m − 1 ∂ ˜ U r , v ( r ) = v ol m U r , ˜ v ( r ) = vol m ˜ U r . By the coarea formula, d dr v ( r ) a.e. = = a ( r ) , d dr ˜ v ( r ) = ˜ a ( r ) . Note that d dr a ( r ) ⩽ w x ∈ ∂ U r H r ( x ) ⩽ ⩽ a ( r ) · ˜ H r , and d dr ˜ a ( r ) = ˜ a ( r ) · ˜ H r . It follows that v ′′ ( r ) v ( r ) ⩽ ˜ v ′′ ( r ) ˜ v ( r ) for almost all r . Therefore v ( r ) ⩽ area Σ area ˜ Σ · ˜ v ( r ) for any r > 0 . A ccording to ( ∗ ) , v ( π 2 ) = ˜ v ( π 2 ) = v ol S m . Hence the result follows. 47 This problem is the geometric lemma in the pro of giv en by F rederic k Almgren of his isop erimetric inequalit y [92]. The argumen t is similar to the pro of of isop erimetric inequality for manifolds with p ositive Ricci curv ature giv en b y Mikhael Gromov [93]. Hyp ercurv e. Cho ose p ∈ M . Denote by s the second fundamental form of M at p . Recall that s is a symmetric bilinear form on the tangen t space T p M of M with v alues in the normal space N p M to M , see page 38. By the Gauss formula ⟨ R ( X , Y ) Y , X ⟩ = ⟨ s ( X, X ) , s ( Y , Y ) ⟩ − ⟨ s ( X , Y ) , s ( X, Y ) ⟩ . Since the sectional curv ature of M is p ositive, we get ( ∗ ) ⟨ s ( X, X ) , s ( Y , Y ) ⟩ > 0 for any pair of non-zero vectors X , Y ∈ T p M . The normal space N p M is tw o-dimensional. By ( ∗ ) there is an or- thonormal basis e 1 , e 2 in N p M such that the real-v alued quadratic forms s 1 ( X, X ) = ⟨ s ( X , X ) , e 1 ⟩ , s 2 ( X, X ) = ⟨ s ( X , X ) , e 2 ⟩ are p ositiv e deﬁnite. Note that the curv ature op erators R 1 and R 2 are deﬁned by the for- m ula R i ( X ∧ Y ) , V ∧ W ⟩ = s i ( X, W ) · s i ( Y , V ) − s i ( X, V ) · s i ( Y , W ) are p ositiv e. Finally , note that R = R 1 + R 2 is the curv ature op erator of M at p . The problem is due to Alan W einstein [94]. Note that from [95]/[96] it follo ws that the universal cov ering of M is homeomorphic/diﬀeomorphic to a standard sphere. Horo-sphere. Set m = dim Σ = dim M − 1 . Let bus : M → R b e the Busemann function such that Σ = bus − 1 { 0 } . Set Σ r = bus − 1 { r } , so Σ 0 = Σ . Let us equip each Σ r with the induced Riemannian metric. Note that all Σ r ha ve b ounded curv ature. In particular, the unit balls in Σ r ha ve v olume b ounded ab o ve by a univ ersal constant, say v 0 . Giv en x ∈ Σ , denote b y γ x the unit-speed geodesic suc h that γ x (0) = x and bus( γ x ( t )) = t for an y t . Consider the map ϕ r : Σ → Σ r deﬁned by 48 CHAPTER 3. COMP ARISON GEOMETR Y ϕ r : x 7→ γ x ( r ) . In other words, ϕ r is the closest p oint pro jection from Σ to Σ r . Notice that ϕ r is a bi-Lipschitz map with the Lipschitz constants e a · r and e b · r . In particular, the ball of radius R in Σ is mapp ed by ϕ r to a ball of radius e a · r · R in Σ r . Therefore v ol m B ( x, R ) Σ ⩽ e m · b · r · v ol m B ( ϕ r ( x ) , e a · r · R ) Σ r for all R, r > 0 . T aking R = e − a · r , we get v ol m B ( x, R ) Σ ⩽ v 0 · R m · b a for any R ⩾ 1 . Henc e the statemen t follo ws. The problem was suggested by Vitali Kap ovitc h. There are examples of horo-spheres as abov e with a degree of p olyno- mial gro wth higher than m . F or example, consider the horo-sphere Σ in the complex hyperb olic space of real dimension 4 . Clearly , m = dim Σ = 3 , but the degree of its volume growth is 4 . In this case, Σ is isometric to the Heisenberg group. 2 It is instructiv e to show that any such metric has v olume gro wth of degree 4 . Num b er of conjugate p oin ts. Cho ose a geo desic γ in M and a p oint p ∈ γ . Note that γ can b e lifted to a horizon tal geo desic ¯ γ in N . That is, γ = s ◦ ¯ γ and ¯ γ is p erp endicular to the ﬁb ers of s (in particular, γ and ¯ γ ha ve equal sp eeds). Observ e that each conjugate point of p on γ corresponds to a fo c al p oint on ¯ γ to the ﬁb er F ov er p in N ; that is, ¯ γ lies in a family of geo desics ¯ γ t that are perp endicular to N such that the corresp onding Jacobi ﬁeld along ¯ γ v anish at q . Note that F has dimension k = dim N − dim M . It remains to pro ve that an y smo oth k -dimensional submanifold F in a complete nonp ositively- curv ed manifold N has at most k fo cal p oints on an y geodesic ¯ γ that is p erpendicular to F . The problem is inspired by the pap er of Alexander Lytc hak [97]. Ap- plying it together with the P oincar ´ e recurrence theorem leads to a solution of the following problem.  L et s : N → M b e a Riemannian submersion. Supp ose N has nonp osi- tive se ctional curvatur e and M is c omp act. Show that M has no c onjugate p oints. 2 Heisenb er g group is the group of 3 × 3 upp er triangular matrices of the form   1 a c 0 1 b 0 0 1   under the operation of matrix m ultiplication. 49 In fact, no compact negatively curved manifold N admits a non trivial Riemannian submersion s : N → M [see Theorem F in 98]. Minimal spheres. Assuming the contrary , w e can choose a pair of suﬃcien tly close minimal spheres Σ and Σ ′ in the 4-dimensional manifold M ; we can assume that the distance a betw een Σ and Σ ′ is strictly smaller than the injectivity radius of the manifold. Note that in this case, there is a unique bijection Σ → Σ ′ , denoted by p 7→ p ′ suc h that the distance | p − p ′ | M = a for any p ∈ Σ . Let ι p : T p → T p ′ b e the parallel translation along the (necessarily unique) minimizing geodesic [ pp ′ ] . Note that there is a pair ( p, p ′ ) suc h that ι p (T p Σ) = T p ′ Σ ′ . Indeed, if this is not the case, then ι p (T p Σ) ∩ T p ′ Σ ′ forms a contin uous line distribution ov er Σ ′ . Since Σ ′ is a tw o-sphere, the latter contradicts the hairy ball theorem. Consider pairs of unit-sp eed geo desics α and α ′ in Σ and Σ ′ that start at p and p ′ resp ectiv ely and go in parallel directions, sa y ν and ν ′ . Set  ν ( t ) = | α ( t ) − α ′ ( t ) | . Use the second v ariation formula together with the low er b ound on Ricci curv ature to show that  ′′ ν (0) has a negative a verage for all tangent directions ν to Σ at p . In particular,  ′′ ν (0) < 0 for a v ector ν as ab ov e. F or the corresp onding pair α and α ′ , it follo ws that there are p oin ts v = α ( ε ) ∈ Σ near p and v ′ = α ′ ( ε ) ∈ Σ ′ near p ′ suc h that | v − v ′ | < | p − p ′ | — a contradiction. Lik ely , any compact p ositively-curv ed 4-dimensional manifold cannot con tain a pair of equidistant spheres. The argumen t ab o ve implies that the distance b et ween suc h a pair has to exceed the injectivit y radius of the manifold. The problem was suggested by Dmitri Burago. Here is a short list of classical problems which use the second v ariation formula in a similar fashion:  Any c omp act even-dimensional orientable manifold with p ositive se c- tional curvatur e is simply-c onne cte d. This is called Synge’s lemma [99].  Any two c omp act minimal hyp ersurfac es in a Riemannian manifold with p ositive Ric ci curvatur e must interse ct.  L et Σ 1 and Σ 2 b e two c omp act ge o desic submanifolds in a manifold with p ositive se ctional curvatur e M and dim Σ 1 + dim Σ 2 ⩾ dim M . 50 CHAPTER 3. COMP ARISON GEOMETR Y Then Σ 1 ∩ Σ 2  = ∅ . These tw o statements hav e b een prov ed by Theo dore F rankel [100].  L et ( M , g ) b e a close d Riemannian manifold with ne gative Ric ci cur- vatur e. Pr ove that ( M , g ) do es not admit an isometric S 1 -action. This is a theorem of Salomon Bo chner [101]. The problem “Geo desic immersion” [page 35] can b e considered as further developmen t of the idea. P ositive curv ature and symmetry . Let M b e a 4-dimensional Rie- mannian manifold with isometric S 1 -action. Consider the quotient space X = M / S 1 . Note that X is a p ositiv ely-curved 3-dimensional Alexandrov space. In particular, the angle ∡ [ x y z ] b etw een any t wo geo desics [ xy ] and [ xz ] is deﬁned. F urther, for any non-degenerate triangle [ xy z ] formed by the minimizing geo desics [ xy ] , [ y z ] , and [ z x ] in X we hav e ( ∗ ) ∡ [ x y z ] + ∡ [ y z x ] + ∡ [ z x y ] > π . Assume that p ∈ X corresp onds to a ﬁxed p oint ¯ p ∈ M of the S 1 - action. Each direction of a geo desic starting at p in X corresp onds to an S 1 -orbit of the induced isometric action S 1 ↷ S 3 on the sphere of unit v ectors at ¯ p . An y such action is conjugate to the action S 1 p,q ↷ S 3 ⊂ C 2 induced by complex matrices  z p 0 0 z q  with | z | = 1 and relatively prime p ositiv e integers p, q . The p ossible quotient spaces Σ p,q = S 3 / S 1 p,q ha ve diameter π 2 and p erimeter of any triangle in Σ p,q is at most π ; this is straigh tforward to chec k but requires some work. Therefore for any three geo desics [ px ] , [ py ] , and [ pz ] in X we hav e ( ∗∗ ) ∡ [ p x y ] + ∡ [ p y z ] + ∡ [ p z x ] ⩽ π . and ( * * * ) ∡ [ p x y ] , ∡ [ p y z ] , ∡ [ p z x ] ⩽ π 2 . Arguing by contradiction, assume that there are 4 ﬁxed p oints q 1 , q 2 , q 3 , and q 4 . Connect eac h pair by a minimizing geo desic [ q i q j ] . Denote by ω the sum of all 12 angles of the type ∡ [ q i q j q k ] . By ( * * * ) , eac h triangle [ q i q j q k ] is non-degenerate. Therefore by ( ∗ ) , we hav e ω > 4 · π . On the other hand, applying ( ∗∗ ) at each v ertex q i , we hav e ω ⩽ 4 · π — a contradiction. 51 The problem is due to W u-Yi Hsiang and Bruce Kleiner [102]. The connection of this pro of to Alexandro v geometry was noticed by Karsten Gro ve [103]. An interesting new twist of the idea is given by Karsten Gro ve and Burkhard Wilking [104]. Energy minimizer. Denote by U the unit tangent bundle ov er R P m and by L the space of pro jectiv e lines in  : R P 1 → R P m . The spaces U and L hav e dimension 2 · m − 1 and 2 · ( m − 1) resp ectiv ely . A ccording to Liouville’s theorem ab out phase volume, the identit y w v ∈U f ( v ) = w ℓ ∈L w t ∈ R P 1 f (  ′ ( t )) holds for any integrable function f : U → R . Let F : R P m → R P m b e a smooth map. Note that up to a multiplica- tiv e constan t, the energy of F can b e expressed the following w ay w v ∈U | dF ( v ) | 2 = w ℓ ∈L w t ∈ R P 1 | [ d ( F ◦  )]( t ) | 2 . Notice that an y noncon tractible curv e in R P m has length at least π . Therefore, by Buny ako vsky inequality , we get w t ∈ R P 1 | [ d ( F ◦  )]( t ) | 2 ⩾ 1 π ·   w t ∈ R P 1 | [ d ( F ◦  )]( t ) |   2 = = 1 π · (length F ◦  ) 2 ⩾ ⩾ π . for any line  : R P 1 → R P m . Hence the result follows. The problem is due to Christopher Crok e [105]. He uses the same idea to sho w that the identit y map on C P m is energy-minimizing in its homotop y class. F or S m , an analogous statement does not hold if m ⩾ 3 . In fact, if a closed Riemannian manifold M has dimension at least 3 and π 1 M = π 2 M = 0 , then the identit y map on M is homotopic to a map with arbitrarily small energy; the latter was shown by Brian White [106]. The same idea is used to prov e the so-called Lo ewner’s inequality [107].  L et g b e a Riemannian metric on R P m that is c onformal ly e quivalent to the c anonic al metric g 0 . Assume that any nonc ontr actible curve in ( R P m , g ) has length at le ast π . Then v ol( R P m , g ) ⩾ v ol( R P m , g 0 ) . 52 CHAPTER 3. COMP ARISON GEOMETR Y A more adv anced application is the sharp isop erimetric inequalit y for 4-dimensional Hadamard manifolds pro v ed by Christopher Crok e [see 108 and also 109]. Curv ature against injectivity radius. W e will sho w that if the in- jectivit y radius of the manifold ( M , g ) is at least π , then the av erage of sectional curv atures on ( M , g ) is at most 1 . This is equiv alen t to the problem. Cho ose a p oint p ∈ M and tw o orthonormal v ectors U, V ∈ T p M . Consider the geo desic γ in M suc h that γ ′ (0) = U . Set U t = γ ′ ( t ) ∈ T γ ( t ) , and let V t ∈ T γ ( t ) b e the parallel translation of V = V 0 along γ . Consider the ﬁeld W t = sin t · V t on γ . Set γ τ ( t ) = exp γ ( t ) ( τ · W t ) ,  ( τ ) = length( γ τ | [0 ,π ] ) , q ( U, V ) =  ′′ (0) . Note that ( ∗ ) q ( U, V ) = π w 0 [(cos t ) 2 − K ( U t , V t ) · (sin t ) 2 ] · dt, where K ( U, V ) is the sectional curv ature in the direction spanned by U and V . Since any geo desics of length π is minimizing, we get q ( U, V ) ⩾ 0 for an y pair of orthonormal v ectors U and V . It follows that the av erage v alue of the right-hand side in ( ∗ ) is non-negative. By Liouville’s theorem ab out phase volume, while taking the av erage of ( ∗ ) , we can switch the order of in tegrals; therefore 0 ⩽ π 2 · (1 − ¯ K ) , where ¯ K denotes the av erage of sectional curv atures on ( M , g ) . Hence the result follows. The problem illustrates the idea of Eb erhard Hopf [110] which was dev elop ed further b y Leon Green [111]. Hopf used it to sho w that a metric on 2-dimensional torus without conjugate p oin ts must b e ﬂat and Green sho wed that the av erage of sectional curv ature on a closed manifold without conjugate points cannot be p ositive. F or more on the sub ject see the pap er of Mikhael Gromov [112]. Appro ximation of a quotient. The pro of will use that for an y Rie- mannian submersion s : M → N the lo wer b ound on sectional curv ature of M can non exceed the low er b ound on sectional curv ature of N . 53 This statement follows from O’Nail’s formula [84, Theorem 3.20] which giv es the following relation b etw een sectional curv atures of M ad N K M ( X, Y ) = K N ( ¯ X , ¯ Y ) + 3 4 | [ ¯ X , ¯ Y ] V | 2 , where X , Y are orthonormal vector ﬁelds on N , ¯ X , ¯ Y their horizon tal lifts to M , [ ∗ , ∗ ] is the Lie brack et and ∗ V is the pro jection to the vertical distribution of the submersion. Indeed, since 3 4 | [ ¯ X , ¯ Y ] V | 2 ⩾ 0 , w e hav e K M ( X, Y ) ⩾ K N ( ¯ X , ¯ Y ) . Note that G admits an em b edding in to a compact connected Lie group H ; in fact, w e can assume that H = SO( n ) , for suﬃciently large n . Supp ose that the curv ature of ( M , g ) is b ounded b elow by κ . The bi-inv ariant metric h on H is non-negativ ely curved. Therefore for any p ositive in teger n the pro duct ( H , 1 n · h ) × ( M , g ) is a Riemannian manifold with curv ature b ounded b elo w b y κ . The diagonal action of G on ( H , 1 n · h ) × ( M , g ) is isometric and free. Therefore the quotient ( H , 1 n · h ) × ( M , g ) /G is a Riemannian manifold, sa y ( N , g n ) . Note that the quotien t map ( H , 1 n · h ) × ( M , g ) → ( N , g n ) is a Riemannian submersion. Therefore ( N , g n ) has sectional curv ature b ounded b elo w by κ . It remains to observe that the spaces ( N , g n ) conv erge to ( M , g ) /G as n → ∞ . The used construction is called Che e ger’s trick . The earliest use of this tric k I found in [113]; it w as used there to sho w that Berger’s spheres ha ve p ositiv e curv ature. This trick is used in the construction of most of the known examples of p ositiv ely and non-negatively curved manifolds [114–118]. The quotient space ( M , g ) /G has a ﬁnite dimension and its curv ature is b ounded b elow in the sense of Alexandrov. It is exp ected that not all ﬁnite-dimensional Alexandrov spaces admit approximation by Riemann- ian manifolds with curv ature b ounded below [some partial results are discussed in 119, 120]. P olar p oints. Choose a unit-sp eed geo desic γ that starts at p ; that is, γ (0) = p . Apply the T oponogov comparison to show that p ∗ = γ ( π ) is a solution. A lternative pr o of. Assume the contrary; that is, for any x ∈ M there is a p oin t x ′ suc h that | x − x ′ | M + | p − x ′ | M > π . Giv en x ∈ M , denote by f ( x ) a p oint that maximizes the follo wing sum: | x − f ( x ) | M + | p − f ( x ) | M . 54 CHAPTER 3. COMP ARISON GEOMETR Y Sho w that f is uniquely deﬁned and contin uous. Cho ose suﬃciently small ε > 0 . Prov e that the set W ε = M \ B ( p, ε ) is homeomorphic to a ball and the map f sends W ε in to itself. By Brouw er’s ﬁxed-p oin t theorem, x = f ( x ) for some x . In this case, | x − f ( x ) | M + | p − f ( x ) | M = | p − x | M ⩽ ⩽ π — a contradiction. The problem is due to Anatoliy Milk a [121]. Isometric section. Arguing b y con tradiction, assume there is an isomet- ric section ι : M → W . It mak es it p ossible to treat M as a submanifold in W . Giv en p ∈ M , denote by N 1 p the unit normal space to M at p . Given v ∈ N 1 p and a real num b er k , set p k · v = s ◦ exp p ( k · v ) . Note that ( ∗ ) p 0 · v = p for any p ∈ M and v ∈ N 1 p . Cho ose suﬃciently small δ > 0 . By Rauch comparison [84, Corollary 1.36], if w ∈ N 1 q is the parallel translation of v ∈ N 1 q along a minimizing geo desic from p to q in M , then ( ∗∗ ) | p k · v − q k · w | M < | p − q | M assuming that | k | ⩽ δ . The same comparison implies that ( * * * ) | p k · v − q k ′ · w | 2 M < | p − q | 2 M + ( k − k ′ ) 2 assuming that | k | , | k ′ | ⩽ δ . Cho ose p and v ∈ N 1 p so that r = | p − p δ · v | takes the maximal p ossible v alue. F rom ( ∗∗ ) it follo ws that r > 0 . Let γ b e the extension of the unit-sp eed minimizing geo desic from p δ · v to p ; denote by v t the parallel translation of v to γ ( t ) along γ . W e can c ho ose the parameter of γ so that p = γ (0) , p δ · v = γ ( − r ) . Set p n = γ ( n · r ) , so p = p 0 and p δ · v = p − 1 . Cho ose a large integer N and set w n = δ · (1 − n N ) · v n · r , q n = p w n n , x n = exp p n ( w n ) , and q n = p w n n = s ( x n ) . By ( * * * ) , there is a constant C indep enden t of N such that | q k − q k +1 | < r + C N 2 · δ 2 . 55 p 0 p 1 p 2 p 3 p 4 q 0 q 1 q 2 q 3 q 4 . . . M x 0 x 1 x 2 x 3 x 4 Therefore | q k +1 − p k +1 | > | q k − p k | − C N 2 · δ 2 . By induction, we get | q N − p N | > r − C N · δ 2 . Since N is large we get | q N − p N | > 0 . Note that w N = 0 ; therefore b y ( ∗ ) , we get q N = p 0 N = p N — a con tra- diction. This is the core of Perelman’s solution of the Soul conjecture [122]. W arp ed pro duct. Given x ∈ Σ , denote b y ν x the normal v ector to Σ at x that agrees with the orien tations of Σ and M . Denote b y κ x the non-negativ e principal curv ature of Σ at x ; since Σ is minimal the other principal curv ature has to b e − κ x . Consider the w arp ed pro duct W = S 1 × f Σ for a p ositiv e smooth function f : Σ → R . Assume that a p oint y ∈ W pro jects to a p oint x ∈ Σ . Straightforw ard computations show that Sc W ( y ) = Sc Σ ( x ) − 2 · ∆ f ( x ) f ( x ) = = Sc M ( x ) − 2 · Ric( ν x ) − 2 · κ 2 x − 2 · ∆ f ( x ) f ( x ) , where Sc and Ric denote the scalar and Ricci curv ature resp ectiv ely . Consider linear operator L on the space of smo oth functions on Σ deﬁned by ( Lf )( x ) = − [Ric( ν x ) + κ 2 x ] · f ( x ) − (∆ f )( x ) . It is suﬃcient to ﬁnd a smo oth function f on Σ suc h that ( ∗ ) f ( x ) > 0 and ( Lf )( x ) ⩾ 0 56 CHAPTER 3. COMP ARISON GEOMETR Y for any x ∈ Σ . Giv en a smo oth function f : Σ → R , extend the ﬁeld f ( x ) · ν x on Σ to a smooth ﬁeld, say v , on whole M . Denote b y ι t the ﬂo w along v for time t and set Σ t = ι t (Σ) . Denote b y H t ( x ) the mean curv ature of Σ t at ι t ( x ) . Note that the v alue ( Lf )( x ) is the deriv ativ e of the function t 7→ H t ( x ) at t = 0 . Therefore the condition ( ∗ ) means that w e can push Σ into one of its sides so that its mean curv ature do es not increase in the ﬁrst order. Since Σ is area-minimizing, suc h push can b e obtained by increasing the pressure on one side of Σ . (Read further if y ou are not convinced.) F ormal end of pr o of. Denote by δ ( f ) the second v ariation of area of Σ t ; that is, consider the area function a ( t ) = area Σ t and set δ ( f ) = a ′′ (0) . Direct calculations show that δ ( f ) = w x ∈ Σ  − [Ric( ν x ) + κ 2 x ] · f 2 ( x ) + |∇ f ( x ) | 2  = = w x ∈ Σ ( Lf )( x ) · f ( x ) . Since Σ is area-minimizing we get ( ∗∗ ) δ ( f ) ⩾ 0 for any f . Cho ose a function f that minimize δ ( f ) for all functions such that r x ∈ Σ f 2 ( x ) = 1 . Note that f is an eigenfunction for the linear operator L ; in particular, f is smooth. Denote b y λ the eigen v alue of f ; by ( ∗∗ ) , λ ⩾ 0 . Sho w that f ( x ) > 0 at any x . Since Lf = λ · f , the inequalities ( ∗ ) follo w. The problem is due to Mikhael Gromov and Blaine La wson [123]. Earlier, in [124], Shing-T ung Y au and Richard Schoen sho wed that the same assumptions imply the existence of a conformal factor on Σ that mak es it p ositiv ely-curved. Both statements are used the same wa y to pro ve that T 3 do es not admit a metric with p ositive scalar curv ature. Both statemen ts admit straightforw ard generalization to higher di- mensions and they can b e used to show the non-existence of a metric with p ositiv e scalar curv ature on T m with m ⩽ 7 . F or m = 8 , the pro of stops w orking since in this dimension area-minimizing h yp ersurfaces migh t ha ve singularities. F or example, any domain in the cone in R 8 deﬁned b y the iden tity x 2 1 + x 2 2 + x 2 3 + x 2 4 = x 2 5 + x 2 6 + x 2 7 + x 2 8 57 is area-minimizing among the hypersurfaces with the same b oundary . No approximation. Cho ose an increasing function ϕ : (0 , r ) → R suc h that ϕ ′′ + ( n − 1) · ( ϕ ′ ) 2 + C = 0 . If Ric g n ⩾ C , then the function x 7→ ϕ ( | q − x | g n ) is subharmonic. Therefore for an arbitrary array of points q i and p ositive reals λ i the function f n : M n → R deﬁned by the formula f ( x ) = X i λ i · ϕ ( | q i − x | M ) is subharmonic. In particular, f n do es not ha v e a lo cal minimum in M n . P assing to the limit as n → ∞ , we get that any function f : R m → R of the form f ( x ) = X i λ i · ϕ ( | q i − x | ℓ p ) do es not ha v e a lo cal minimum in R m . Let e i b e the standard basis in R m . If p < 2 , consider the sum f ( x ) = X ϕ ( | q − x | ℓ p ) , where q = ± ε · e i for all signs and i ’s. Straightforw ard calculations sho w that if ε > 0 is small, then f has a strict lo cal minimum at 0 . If p > 2 , one has to take the same sum for p = P i ± ε · e i for all c hoices of signs. In b oth cases, we arrive at a contradiction. The argument given here is close to the pro of of Abresch–Gromoll inequalit y [125]. The solution admits a straigh tforward generalization whic h implies that if an m -dimensional Finsler manifold F is a Gromov– Hausdorﬀ limit of m -dimensional Riemannian manifolds with uniform lo wer b ound on Ricci curv ature, then F has to b e Riemannian. An alternative solution to this problem can b e built on the almost splitting theorem prov ed by Jeﬀ Cheeger and T obias Colding [126]. Area of spheres. Fix r 0 > 0 . Giv en r > r 0 , choose a p oint q on the distance 2 · r from p . p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p q S Note that an y minimizing geodesic from q to a p oint in B = B ( p, r 0 ) has to cross S = = ∂ B ( p, r ) . The statement follows since v ol B ⩽ C m · r 0 · area S, where C m is a constant dep ending only on the dimension m = dim M . This volume comparison inequality can b e pro ved along the same lines as the Bishop–Gromov inequality . 58 CHAPTER 3. COMP ARISON GEOMETR Y Applying the coarea formula, we see that the volume growth of M is at least linear; in particular, M has inﬁnite volume. The latter was pro ved indep endently b y Eugenio Calabi and Shing-T ung Y au [127, 128]. Flat coordinate planes. Choose ε > 0 such that there is a unique geo desic b et ween any t wo p oin ts at distance < ε from the origin of R 3 . Consider three p oints a , b , and c on the coordinate lines that are ε -close to the origin. The following observ ation is the key to the pro of. ( ∗ ) Ther e is a solid ﬂat ge o desic triangle in ( R 3 , g ) with vertic es at a , b , and c . Since the co ordinate planes are totally geo desic, the parallel transla- tion along a co ordinate line preserv es the directions tangent to a co or- dinate plane. Since the parallel translation preserves the angles b et ween v ectors, the angles b et w ee n co ordinate planes in ( R 3 , g ) are constant. It follows that the angles of the triangle [ abc ] coincide with its mo del angles , that is, the angles in the plane triangle with the same sides. Both curv ature conditions imply that the triangle [ abc ] b ounds a solid ﬂat geo desic triangle in ( R 3 , g ) . Use the family of constructed ﬂat triangles to sho w that at an y x p oin t in the ε 10 -neigh b orho o d of the origin the sectional curv ature v anishes in an op en set of sectional directions. The latter implies that the curv ature is identically zero in this neighborho o d. Mo ve the origin and apply the same argumen t lo cally . This w ay w e get that the curv ature is identically zero everywhere. This problem is based on a lemma discov ered by Sergei Buyalo [Lemma 5.8 in 129; see also 130 and 131]. T w o-conv exity . Morse-style solution. Cho ose ( x, y , z , t ) -co ordinates in R 4 . Consider a generic linear function  : R 4 → R that is close to the sum of co ordinates x + y + z + t . Note that  has non-degenerate critical p oints on ∂ K and all its critical v alues are diﬀerent. F or each s consider the set W s =  w ∈ R 4 \ K    ( w ) < s  . Note that W − 1000 con tains a closed curve, say α , that is conta ctable in R 4 \ K , but not constructible in W − 1000 . Set s 0 to b e the inﬁm um of the v alues s suc h that the α is contactable in W s . Note that s 0 is a critical v alue of  on ∂ K ; denote b y p 0 the corre- sp onding critical p oin t. By 2-conv exity of R 4 \ K , the index of p 0 has to be at most 1 . On the other hand, a disc that contracts α cannot 59 b e mov ed low er s 0 . Therefore the index of p 0 has to b e at least 2 — a con tradiction. A lexandr ov-style pr o of. Assume that the complemen t to K is tw o-con v ex. Note that t wo-con vexit y is preserv ed under linear transformation. Ap- ply a linear transformation of R 4 that mak es the co ordinate planes Π 1 and Π 2 not orthogonal. A ccording to the main result in [132], W = R 4 \ (Int K ) has non- p ositiv e curv ature in the sense of Alexandrov. In particular, the universal metric cov ering ˜ W of W is a CA T(0) space. By rescaling ˜ W and passing to the limit we obtain that the universal Riemannian cov ering Z of R 4 that branches in the planes Π 1 and Π 2 is a CA T(0) space. Note that Z is isometric to the Euclidean cone ov er universal cov ering Σ of S 3 branc hing in t wo great circles Γ i = S 3 ∩ Π i that are not orthogonal. The shortest path in S 3 b et ween Γ 1 and Γ 2 tra veled 4 times back and forth is shorter than 2 · π and it lifts to a closed geo desic in Σ . It follows that Σ is not CA T(1) and therefore Z is not CA T(0) — a contradiction. The Morse-style pro of is based on an idea of Mikhael Gromov [see § ½ in 58], where tw o-conv exity was introduced. Note that the 1 -neigh b orho o d of these t wo planes has t wo-con vex com- plemen t W in the sense of the second deﬁnition; that is, if a closed curve γ lies in the plane Π and is contactable in W , then it is contactable in Π ∩ W . Clearly , the b oundary of this neighborho o d is not smo oth and as it follo ws from the problem, it cannot b e smo othed in the class of t wo-con vex sets. T wo-con vexit y also sho ws up in comparison geometry — the maximal op en ﬂat sets in the manifolds of nonnegative or nonp ositive curv ature are tw o-con vex [131]. Con vex lens. Before going into the pro of, let us describ e a straightfor- w ard idea that do es not work. By the Gauss formula, we get that w D k 1 · k 2 ⩽ w D K, where K denotes the in trinsic curv ature of D . Therefore it w ould be suﬃcien t to show that the right-hand side is small; how ever, the in tegral r D K might b e large for an arbitrarily small angle b etw een the discs; for example, if M = S 3 it might b e arbitrarily close to 2 · π . Denote by ε the maximal angle betw een the discs, we can assume that ε < π 2 . 60 CHAPTER 3. COMP ARISON GEOMETR Y Note that the function h = dist D ′ is con vex in L . Moreo ver, the gradien t ∇ x h p oin ts outside of L for any x ∈ D . Consider the restriction f = h | D . Note that f is a conca v e function that v anishes on ∂ D . Assume that f is smooth. Since the discs are meeting at angle at most ε < π 2 , we hav e that |∇ f | ⩽ sin ε and (hess f )( v , v ) + cos ε · s ( v , v ) ⩽ 0 , where s denotes the second fundamental form of D in M . It follows that k 1 · k 2 = det s ⩽ ⩽ 1 cos 2 ε · det(hess f ) = = 1 2 · cos 2 ε ·  | ∆ f | 2 − | hess f | 2  . Applying the Bo c hner formula for f , we get that w D  | ∆ f | 2 − | hess f | 2 − K · |∇ f | 2  = w ∂ D κ · |∇ f | 2 , where K and κ denotes the curv ature of D and geo desic curv ature of ∂ D in D resp ectively . By the Gauss–Bonnet formula, w e get that w D K + w ∂ D κ = 2 · π . Therefore w D k 1 · k 2 ⩽ sin ε cos 2 ε · π . If f is not smo oth, then one can smooth it using Greene–W u con- struction [88, Theorem 2] and repeat the ab ov e argumen t for the obtained function. This estimate w as used by Nina Lebedev a and the author [133]. F or classical applications of Bo c hner’s form ula including the v anishing theo- rems and estimates for eigenv alues of Laplacian see [134, I I §8]. Small-t wist. Given a p ositiv e integer s , consider the Cliﬀord torus T s Cl = 1 √ s · S 1 × · · · × S 1 | {z } s times ⊂ S 2 · s − 1 ⊂ R 2 · s . Note that the normal curv atures of T s Cl lie in the range from 1 to √ s , and T s Cl comes with a ﬂat metric. Sho w that given a p ositive integer n , one can choose large s = s ( n ) so that T s Cl con tains a geo desic n -dimensional subtorus T n with all normal 61 curv atures iden tically equal to q 3 · n n +2 ; here we consider T n as a sub- manifold in R 2 · s . (F or example, s (2) = 3 ; in this case, T 2 is a subtorus p erpendicular to the main diagonal in T 3 Cl .) No w, choose a closed smo oth manifold M . By the Whitney embed- ding theorem, there is a smo oth embedding M  → R n for n > 2 · dim M . Applying rescaling we can assume that normal curv atures of this embed- ding are arbitrarily small; w e need it to b e smaller than √ 3 − q 3 · n n +2 . Comp osing this embedding with the natural length-preserving co v ering map R n → T n , we get the needed immersion of ι : M → R 2 · s . This construction w as discov ered by Mikhael Gromov [135, 2.A]. In fact, the b ound q 3 · n n +2 is optimal [136]. The immersion ι can b e easily upgraded to em b edding. Applying the Nash embedding theorem instead of the Whitney embedding theorem, one gets that the induced metric on M  → R n is prop ortional to any given Riemannian metric g on M . Here is a closely related problem.  Show that any n -dimensional submanifold in S q with normal curvatur es less than 1 √ 3 is diﬀe omorphic to S n . Note that V eronese embedding R P 2  → S 5 has normal curv atures ex- actly 1 √ 3 in all directions. Therefore the 1 √ 3 b ound is optimal; see also [137]. Chapter 4 Curv ature-free diﬀeren tial geometry The reader should b e familiar with the notions of smo oth manifolds, Rie- mannian metrics, and symplectic forms. Distan t inv olution  Construct a R iemannian metric g on S 3 and an involution ι : S 3 → S 3 such that v ol( S 3 , g ) is arbitr arily smal l and | x − ι ( x ) | g > 1 for any x ∈ S 3 . Semisolution. Given ε > 0 , construct a disk ∆ in the plane with length ∂ ∆ < 10 and area ∆ < ε that admits a contin uous inv olution ι such that | ι ( x ) − x | ⩾ 1 for any x ∈ ∂ ∆ . An example of ∆ can b e guessed from the picture; the in v olution ι makes a length preserving half turn of its b oundary ∂ ∆ . T ake the pro duct ∆ × ∆ ⊂ R 4 ; it is homeomorphic to the 4-ball. Note that v ol 3 [ ∂ (∆ × ∆)] = 2 · area ∆ · length ∂ ∆ < 20 · ε. 62 63 The b oundary ∂ (∆ × ∆) is homeomorphic to S 3 and the restriction of the in volution ( x, y ) 7→ ( ι ( x ) , ι ( y )) has the needed prop erty . All we hav e to do now is to smo oth ∂ (∆ × ∆) a little bit. This example is given by Christopher Crok e [138]. Note that according to Gromov’s systolic inequalit y [107], the inv olution ι ab ov e cannot b e made isometric. The following problem states that a similar construction is not p ossible for S 2 . Another distan t inv olution  Given x ∈ S 2 , denote by x ′ its antip o dal p oint. Supp ose that g is a R iemannian metric on S 2 such that | x − x ′ | g ⩾ 1 for any x ∈ S 2 . Show that the ar e a of ( S 2 , g ) is b ounde d b elow by a ﬁxe d p ositive c onstant. The exp ected solution uses Besico vitch inequality describ ed in the next problem. Besico vitch inequalit y  L et g b e a Riemannian metric on an m -dimensional cub e Q such that any curve c onne cting opp osite fac es has length at le ast 1 . Pr ove that v ol( Q, g ) ⩾ 1 , and the e quality holds if and only if ( Q, g ) is isometric to the unit cub e. Minimal foliation + Minimal surfaces in Riemannian manifolds are deﬁned on page 37.  Consider the pr o duct of spher es S 2 × S 2 e quipp e d with a R iemannian metric g that is C ∞ -close to the pr o duct metric. Pr ove that ther e is a c onformal ly e quivalent metric λ · g and a r e-p ar ametrization of S 2 × S 2 such that for any x, y ∈ S 2 , the spher es { x } × S 2 and S 2 × { y } ar e minimal surfac es in ( S 2 × S 2 , λ · g ) . The exp ected solution requires pseudo-holomorphic curves in tro duced b y Mikhael Gromov [139]. 64 CHAPTER 4. CUR V A TURE-FREE V olume and conv exit y + A function f deﬁned on a Riemannian manifold is called conv ex if, for an y geo desic γ , the comp osition f ◦ γ is a conv ex real-to-real function.  L et M b e a c omplete R iemannian manifold that admits a non-c onstant c onvex function. Pr ove that M has inﬁnite volume. The exp ected solution uses Liouville’s theorem ab out phase v olume. It implies in particular, that the geo desic ﬂow on the unit tangent bundle of a Riemannian manifold preserves the volume. Sasaki metric Let ( M , g ) b e a Riemannian manifold. The Sasaki metric is a natural c hoice of Riemannian metric ˆ g on the total space of the tangent bundle τ : T M → M . It is uniquely deﬁned by the follo wing prop erties: ⋄ The map τ : (T M , ˆ g ) → ( M , g ) is a Riemannian submersion. ⋄ The metric on each tangen t space T p ⊂ T M is the Euclidean metric induced by g . ⋄ Assume that γ ( t ) is a curv e in M and v ( t ) ∈ T γ ( t ) is a parallel v ector ﬁeld along γ . Note that v ( t ) forms a curve in T M . F or the Sasaki metric, w e ha ve v ′ ( t ) ⊥ T γ ( t ) for any t ; that is, the curv e v ( t ) normally crosses the tangent spaces T γ ( t ) ⊂ T M . In other words, we identify the tangent space T u [T M ] for any u ∈ ∈ T p M with the direct sum of v ertical and horizon tal subspaces T p M ⊕ ⊕ T p M . The pro jection of this splitting is deﬁned by the diﬀeren tial dτ : TT M → T M and we assume that the velocity of a curve in T M formed by a parallel ﬁeld along a curve in M is horizontal. Then T u [T M ] is equipp ed with the metric ˆ g deﬁned by ˆ g ( X , Y ) = g ( X V , Y V ) + g ( X H , Y H ) , where X V and X H ∈ T p M denote the vertical and horizon tal components of X ∈ T u [T M ] .  L et g b e a Riemannian metric on the spher e S 2 . Consider the tangent bund le T S 2 e quipp e d with the induc e d Sasaki metric ˆ g . Show that the sp ac e (T S 2 , ˆ g ) lies at a b ounde d distanc e to the r ay R ⩾ 0 = [0 , ∞ ) in the sense of Gr omov–Hausdorﬀ. T w o-systole  Given a lar ge r e al numb er L , c onstruct a Riemannian metric g on the 3-dimensional torus T 3 such that v ol( T 3 , g ) = 1 and area S ⩾ L 65 for any close d surfac e S that do es not b ound in T 3 . A ccording to Gromov’s systolic inequality [107], the volume of ( T 3 , g ) can b e b ounded b elo w in terms of its 1-systole deﬁned to b e the shortest length of a noncontractible closed curve in ( T 3 , g ) . The lo wer b ound on the area of S in the problem is called the 2-systole of ( T 3 , g ) . The problem implies that Gromov’s systolic inequality do es not hav e a direct 2-dimensional analog. Normal exp onen tial map ◦ Let ( M , g ) b e a Riemannian manifold; denote b y T M the tangent bundle o ver M and by T p = T p M the tangen t space at the p oint p . Giv en a v ector v ∈ T p M , denote by γ v the geodesic in ( M , g ) such that γ (0) = p and γ ′ (0) = v . The map exp : T M → M deﬁned by v 7→ γ v (1) is called the exp onential map. The restriction of exp | T p is called the exp onential map at p and is denoted by exp p . Giv en a smo oth immersion L → M , denote by N L the normal bundle o ver L . The restriction exp | N L is called the normal exp onential map of L and is denoted by exp L .  L et M b e a c omplete c onne cte d R iemannian manifold with an immerse d c omplete c onne cte d Riemannian manifold L . Show that the image of the normal exp onential map of L is dense in M . Symplectic squeezing in the torus  L et ω = dx 1 ∧ dy 1 + dx 2 ∧ dy 2 b e the standar d symple ctic form on R 4 , and Z 2 the inte gr al lattic e in the ( x 1 , y 1 ) c o or dinate plane of R 4 . Show that an arbitr ary b ounde d domain Ω ⊂ ( R 4 , ω ) admits a sym- ple ctic emb e dding into the quotient sp ac e ( R 4 , ω ) / Z 2 . Diﬀeomorphism test ◦  L et M and N b e c omplete m -dimensional simply-c onne cte d Riemann- ian manifolds, and f : M → N a smo oth map such that | d f ( v ) | ⩾ | v | for any tangent ve ctor v of M . Show that f is a diﬀe omorphism. 66 CHAPTER 4. CUR V A TURE-FREE V olume of tubular neighborho o ds +  L et M and M ′ b e isometric close d smo oth submanifolds in a Euclide an sp ac e. Show that for al l smal l r > 0 we have v ol B ( M , r ) = vol B ( M ′ , r ) , wher e B ( M , r ) denotes the r -neighb orho o d of M . Disk ∗  Given a lar ge r e al numb er L , c onstruct a Riemannian metric g on the disk D with diam( D , g ) ⩽ 1 and length g ∂ D ⩽ 1 such that the b oundary curve in D is not c ontr actible in the class of close d curves with g -length less than L . Shortening homotop y  L et M b e a c omp act R iemannian manifold with diameter D and p ∈ M . Assume that for some L > D , ther e ar e no ge o desic lo ops b ase d at p in M with length in the interval ( L − D , L + D ] . Show that for any p ath γ 0 in ( M , g ) starting at p , ther e is a homotopy γ t r elative to its endp oints such that a) length γ 1 < L ; b) length γ t ⩽ length γ 0 + 2 · D for any t ∈ [0 , 1] . Examples of manifolds satisfying the ab ov e condition for some L hav e b een found among the Zoll spheres by Floren t Balachev, Christopher Crok e, and Mikhail Katz [140]. Con vex h yp ersurface Recall that a subset K of a Riemannian manifold is called c onvex if ev ery minimizing geo desic connecting tw o p oin ts in K lies completely in K .  L et M b e a total ly ge o desic hyp ersurfac e in a close d R iemannian m - dimensional manifold W . Assume that the inje ctivity r adius of M is at le ast 1 and M forms a c onvex set in W . Show that the maximal distanc e fr om M to the p oints of W c an b e b ounde d b elow by a p ositive c onstant ε m that dep ends only on the dimen- sion m (in fact, ε m = 2 m +3 wil l do). Note that w e did not make any assumption on the injectivit y radius of W . 67 Almost constan t function The unit tangent bundle U M ov er a closed Riemannian manifold M ad- mits a natural choice of v olume. Let us equip U M with the probability measure that is prop ortional to this volume. W e sa y that a unit-sp eed geo desic γ : R → M is r andom if γ ′ (0) takes a random v alue in U M .  Given ε > 0 , show that ther e is a p ositive inte ger m such that for any close d m -dimensional Riemannian manifold M and any smo oth 1 - Lipschitz function f : M → R the fol lowing holds. F or a r andom unit-sp e e d ge o desic γ in M the event | f ◦ γ (0) − f ◦ γ (1) | > ε has pr ob ability at most ε . Semisolutions Another distant inv olution. Let x ∈ S 2 b e a p oint that minimizes the distance | x − x ′ | g . Consider a minimizing geo desic γ from x to x ′ . W e can assume that | x − x ′ | g = length γ = 1 . Let γ ′ b e the antipo dal arc to γ . Note that γ ′ in tersects γ only at the common endpoints x and x ′ . Indeed, if p ′ = q for p, q ∈ γ , then | p − q | ⩾ 1 . Since length γ = 1 , the p oin ts p and q must b e the ends of γ . It follows that γ together with γ ′ forms a closed simple curv e in S 2 that divides the sphere into tw o disks D and D ′ . Let us divide γ in to t wo equal arcs γ 1 and γ 2 ; eac h of length 1 2 . Supp ose that p, q ∈ γ 1 , then | p − q ′ | g ⩾ | q − q ′ | g − | p − q | g ⩾ ⩾ 1 − 1 2 = 1 2 . That is, the minimal distance from γ 1 to γ ′ 1 is at least 1 2 . The same wa y w e get that the minimal distance from γ 2 to γ ′ 2 is at least 1 2 . By Besicovitc h inequalit y , we get that area( D , g ) ⩾ 1 4 and area( D ′ , g ) ⩾ 1 4 . Therefore area( S 2 , g ) ⩾ 1 2 . 68 CHAPTER 4. CUR V A TURE-FREE This inequalit y was pro ved b y Marcel Berger [141]. Christopher Croke conjectured that the optimal bound is 4 π and the round sphere is the only space that achiev es this [see Conjecture 0.3 in 138]. Let us indicate how to improv e the obtained b ound to area( S 2 , g ) ⩾ 1 . Supp ose x , x ′ , γ , and γ ′ are as ab o v e. Consider the function f ( z ) = min t { | γ ′ ( t ) − z | g + t } . Observ e that f is 1-Lipsc hitz. Sho w that tw o p oints γ ′ ( c ) and γ (1 − c ) lie on one connected component of the level set L c =  z ∈ S 2   f ( z ) = c  ; in particular length L c ⩾ 2 · | γ ′ ( c ) − γ (1 − c ) | g . By the triangle inequality , w e ha ve that | γ ′ ( c ) − γ (1 − c ) | g ⩾ 1 − | γ ( c ) − γ (1 − c ) | g = = 1 − | 1 − 2 · c | . It remains to apply the coarea formula area( S 2 , g ) ⩾ 1 w 0 length L c · dc. Besico vitch inequality . Without loss of generality , we may assume that Q = [0 , 1] m . Set A i = { ( x 1 , . . . , x m ) ∈ Q | x i = 0 } . Consider the functions f i : Q → R deﬁned b y f i ( x ) = min { 1 , dist A i ( x ) } . Note that each f i is 1 -Lipschitz; in particular, |∇ f i | ⩽ 1 almost every- where. Consider the map f : x 7→ ( f 1 ( x ) , . . . , f m ( x )) . Note that it maps Q to itse lf and, moreov er, it maps each face of Q to itself. It follows that the restriction f | ∂ Q : ∂ Q → ∂ Q has degree one and therefore f : Q → Q is onto. 69 Let h be the canonical metric on the cub e Q . Denote by J the Jacobian of the map f : ( Q, g ) → ( Q, h ) . Note that | J( x ) | = |∇ x f 1 ∧ · · · ∧ ∇ x f m | ⩽ 1 . By the area formula, we get v ol( Q, g ) ⩾ w x ∈ Q | J( x ) | ⩾ ⩾ vol( Q, h ) = = 1 . In the case of equalit y , we ha v e that ⟨∇ x f i , ∇ x f j ⟩ = 0 for i  = j and |∇ x f i | = 1 for almost all x . It follows then that the map f : ( Q, g ) → ( Q, h ) is an isometry . This inequality w as prov ed b y Abram Besicovitc h [142]. It has a num- b er of applications in Riemannian geometry . F or example, using this inequalit y it is easy to solve the following problem.  Assume a metric g on R m c oincides with the Euclide an metric outside of a b ounde d set K ; assume further that any ge o desic that enters K exits K the same way the Euclide an ge o desic would have done. Show that g is ﬂat. There is a weak er version of the Besico vitch inequality that works for the Hausdorﬀ measure for an y metric on the cub e; nearly the same pro of w orks. Here is one of its applications suggested by Stephan Stadler:  L et X b e a c ontr actible metric sp ac e with zer o ( n + 1) -dimensional Hausdorﬀ me asur e. Assume that ∆ 1 , ∆ 2 ⊂ X ar e two emb e dde d n -disks having the same b oundary. Show that ∆ 1 = ∆ 2 . Minimal foliation. The pro of is based on the observ ation that a self- dual harmonic 2-form on ( S 2 × S 2 , g ) without zeros deﬁnes a symplectic structure. Note that there is a self-dual harmonic 2-form on ( S 2 × S 2 , g ) ; that is, a 2-form ω suc h that dω = 0 and  ω = ω , where  is the Ho dge star op erator. Indeed, take a generic harmonic form ϕ . Note that the form  ϕ is also harmonic. Since  (  ϕ ) = ϕ , the form ω = ϕ +  ϕ does the job. Cho ose p ∈ S 2 × S 2 . W e can use g p to identify the tangent space T p and the cotangent space T ∗ p . There is a g p -orthonormal basis e 1 , e 2 , e 3 , e 4 on T p suc h that ω p = λ p · e 1 ∧ e 2 + λ ′ p · e 3 ∧ e 4 . 70 CHAPTER 4. CUR V A TURE-FREE Note that  ω p = λ ′ p · e 1 ∧ e 2 + λ p · e 3 ∧ e 4 . Since  ω p = ω p , we hav e λ p = λ ′ p . Consider the rotation J p : T p → T p deﬁned by e 1 7→ − e 2 , e 2 7→ e 1 , e 3 7→ − e 4 , e 4 7→ e 3 . Note that J p ◦ J p = − id and ω ( X , Y ) = λ p · g ( X , J p Y ) for any tw o tangent vectors X , Y ∈ T p . Consider the canonical symplectic form ω 0 on S 2 × S 2 whic h is the sum of the pullbacks of the volume form on S 2 b y the tw o co ordinate pro jections S 2 × S 2 → S 2 . Note that for the canonical metric on S 2 × S 2 , the form ω 0 is harmonic and self-dual. Since g is close to the standard metric, we can assume that ω is close to ω 0 . In particular, λ p  = 0 for any p ∈ S 2 × S 2 . It follows that J is a pseudo-complex structure for the symplectic form ω on S 2 × S 2 . The Riemannian metric g ′ = λ · g is conformal to g and ω ( X , Y ) = g ′ ( X, J Y ) for any tw o tangent vectors X , Y at one p oint. In this case, the J -holomorphic curves are minimal with respect to g ′ ; in fact, eac h of them is area-minimizing in its homology class. It remains to reparametrize S 2 × S 2 so that vertical and horizon tal spheres would form pseudo-holomorphic curves in the homology classes of x × S 2 and S 2 × y . F or general metrics, the form ω might v anish at some points. If the metric is generic, then it happ ens on disjoint circles [143]. V olume and conv exit y . W e use the idea from the pro of of the Poincar ´ e recurrence theorem. Let M b e a complete Riemannian manifold that admits a con vex func- tion f . Denote b y τ : U M → M the unit tangent bundle ov er M . Consider the function F : U M → R deﬁned by F ( u ) = f ◦ τ ( u ) . Note that there is a nonempty b ounded op en set Ω ⊂ U M such that d f ( u ) > ε for any u ∈ Ω and some ﬁxed ε > 0 . Denote by ϕ t the geo desic ﬂo w for time t on U M . By Liouville’s theorem ab out phase volume, we hav e ( ∗ ) v ol[ ϕ t (Ω)] = v ol Ω for any t . Giv en u ∈ U M , consider the function h u ( t ) = F ◦ ϕ t ( u ) . Since f is con vex, so is h u . Therefore h ′ u ( t ) > ε for any t ⩾ 0 and u ∈ Ω . 71 Since Ω is a bounded set, the set of v alues F (Ω) is b ounded as w ell. It follows that there is an inﬁnite sequence of time moments 0 = t 0 < t 1 < t 2 < . . . suc h that h v ( t i − 1 ) < h u ( t i ) for any u, v ∈ Ω and i . In particular, w e ha ve ϕ t i (Ω) ∩ ϕ t j (Ω) = ∅ for i  = j . By ( ∗ ) , the latter implies that vol(U M ) = ∞ . Hence v ol M = ∞ . □ The problem is due to Richard Bishop and Barrett O’Neill [144]; it w as generalized by Shing-T ung Y au [145]. Sasaki metric. Cho ose a p oint p ∈ S 2 . Note that any rotation of the tangen t space T p = T p ( S 2 , g ) app ears as a holonomy of a loop at p ; moreo ver, the length of such lo op can b e b ounded by a constant, sa y  . Indeed, ﬁx a smo oth homotop y γ t : [0 , 1] → S 2 , t ∈ [0 , 1] of loops based at p that sweeps out S 2 . By the Gauss–Bonnet formula, the total curv ature of ( S 2 , g ) is 4 · π . Therefore any rotation of T p app ears as the holonom y of γ t for some t and we can take  = max { length γ t | t ∈ [0 , 1] } . Denote by d the diameter of ( S 2 , g ) . F rom the ab ov e, it follows that for any tw o unit tangen t v ectors v ∈ T p and w ∈ T q there is a path γ : [0 , 1] → S 2 from p to q such that length γ ⩽  + d and w is the parallel transp ort of v along γ . In particular, the diameter of the set of all vectors of ﬁxed magnitude in (T S 2 , ˆ g ) has diameter at most  + d . Therefore the map T S 2 → [0 , ∞ ) deﬁned b y v 7→ | v | preserv es the distance up to an error of  + d . Hence the result follows. T w o-systole. Consider the unit cub e with three not intersecting cylin- drical tunnels b et ween the pairs of opp osite faces. In each tunnel, shrink the metric long-wise and expand it cross-wise while keeping the volume the same. 72 CHAPTER 4. CUR V A TURE-FREE More precisely , assume ( x, y, z ) is the co ordi- nate system on the cylindrical tunnel D × [0 , 1] . Then the new metric is deﬁned by g = ϕ · [( dx ) 2 + ( dy ) 2 ] + 1 φ 2 · ( dz ) 2 , where ϕ = ϕ ( x, y ) is a p ositiv e smo oth function on D taking huge v alues around the cen ter and equal to 1 near the b oundary of D . Gluing the opp osite faces of the cub e, we obtain a 3-dimensional torus with a smo oth Riemannian metric. Since the surface S does not b ound in T 3 = S 1 × S 1 × S 1 , one of the three co ordinate pro jections T 3 → T 2 = S 1 × S 1 induces a map of a non-zero degree S → T 2 . It follo ws that area S ⩾ area( D , ϕ · [( dx ) 2 + ( dy ) 2 ]) . F or the righ t choice of the function ϕ , the right-hand side can b e made larger than the given num b er L . Hence the statement follows. I learned this problem from Dmitri Burago. The three-tunnel con- struction was inv ented by Gustav Hedlund in a diﬀerent context [146, 147]. The follo wing problem of Larry Guth [148] is closely related:  Given a smal l ε > 0 , c onstruct a bi-Lipschitz ar e a-nonincr e asing de gr e e-one map [0 , 1] × [0 , 1] × [0 , ε ] → [0 , ε 7 ] × [0 , ε 7 ] × [0 , 1 7 · ε ] . Normal exp onential map. Assume there are p ∈ M and ε > 0 suc h that the image of the normal exponential map to L do es not intersect the ball B ( p, ε ) M ; that is, no geo desic normal to L crosses the ball. Cho ose a positive real n umber R suc h that B ( p, R ) M ∩ L  = ∅ . The sectional curv ature of M in the ball B ( p, R ) is b ounded b elow by a con- stan t, sa y K . Giv en q ∈ L , denote b y v q ∈ T q M the direction of a minimizing geo desic [ q p ] . Note that v q / ∈ N q L . Moreov er, there is δ = δ ( ε, K, R ) > 0 suc h that for any p oint q ∈ B ( p, R ) M ∩ L , and any normal v ector n ∈ N q L , w e ha ve ∡ ( v q , n ) > δ. Otherwise, the geo desic in the direction of n would cross B ( p, ε ) M . It follows that starting at an y p oint q ∈ B ( p, R ) M ∩ L one can construct a unit-sp eed curv e γ in L such that | p − γ ( t ) | ⩽ | p − q | − t · sin δ. F ollowing γ for some time brings us to p ; that is, p ∈ L — a contradiction. 73 The problem w as suggested b y Alexander Lyt- c hak. On the diagram, y ou see an example of an immer- sion suc h that one p oint do es not lie in the image of the corresp onding normal exponential map. It might b e interesting to understand what type of subsets can b e a voided b y suc h images. Symplectic squeezing in the torus. The embedding will b e giv en as a comp osition of a linear symplectomorphism λ with the quotien t map ϕ : R 4 → T 2 × R 2 b y the integral ( x 1 , y 1 ) -lattice. The comp osition ϕ ◦ λ will preserv e the symplectic structure; it remains to ﬁnd λ such that the restriction ϕ ◦ λ | Ω is injective. Without loss of generality , w e can assume that Ω is a ball centered at the origin. Cho ose an oriented 2-dimensional subspace V of R 4 suc h that the integral of ω ov er Ω ∩ V is a p ositive num b er smaller than π 4 . Note that there is a linear symplectomorphism λ that maps planes parallel to V to planes parallel to the ( x 1 , y 1 ) -plane, and that maps the disk V ∩ Ω to a round disk. It follows that the intersection of λ (Ω) with an y plane parallel to the ( x 1 , y 1 ) -plane is a disk of radius at most 1 2 . In particular ϕ ◦ λ | Ω is injective. This construction was given by Larry Guth [149] and attributed to Leonid Poltero vic h. Note that according to Gromo v’s non-squeezing theorem [139], an analogous statemen t with C × D as the target space do es not hold; here D ⊂ C is the open unit disk with the induced symplectic structure. In par- ticular, it shows that the pro jection of λ (Ω) as ab ov e to the ( x 1 , y 1 ) -plane cannot b e made arbitrarily small. Diﬀeomorphism test. Note that the map f is an op en immersion. Let h be the pullback metric on M for f : M → N . Clearly , h ⩾ g . In particular, ( M , h ) is complete and the map f : ( M , h ) → N is a lo cal isometry . Note that any local isometry betw een complete connected Riemannian manifolds of the same dimension is a co vering map. Since N is simply- connected, the result follows. V olume of tubular neighborho o ds. This problem is a direct corollary of the so-called tub e formula given by Hermann W eyl [150]. It expresses the volume of the r -neighborho o d of M as a p olynomial p ( r ) ; the co- eﬃcien ts of p , up to a multiplicativ e constant, are integrals ov er M of some quan tities called the Lipschitz–Kil ling curvatur es — these are cer- tain scalars that can b e expressed in terms of the curv ature tensor at the giv en p oin t. The pro of is done b y straigh tforward calculations. 74 CHAPTER 4. CUR V A TURE-FREE Disk. The follo wing claim is the key step in the pro of. ( ∗ ) Given a p ositive inte ger n , ther e is a binary tr e e T n emb e dde d into the disk D such that any nul l-homotopy of ∂ D p asses a curve that interse cts n diﬀer ent e dges. The pro of of the claim can b e done by induction on n ; the base is trivial. Assuming we constructed T n − 1 , the tree T n can be obtained b y iden tifying three endp oin ts of three copies of T n − 1 . T 1 T 2 T 3 T 4 T ake ε = 1 10 and ﬁx a large integer n . Let us construct a metric on D with the embedded tree T n as in ( ∗ ) such that its diameter and the length of its b oundary are less than 1 and the distance b et ween any tw o edges of T n without a common vertex is at least ε . Cho ose a Riemannian metric g on the cylinder S 1 × [0 , 1] such that ⋄ The ε -neigh b orhoo ds of the b oundary comp onents ha ve product metrics. ⋄ An y v ertical segmen t x × [0 , 1] has length 1 2 . ⋄ One of the b oundary comp onents has length ε . ⋄ The other b oundary component has length 2 · m · ε , where m is the n umber of edges in the tree T n . Equip T n with a length-metric so that each edge has length ε . Glue the cylinder ( S 1 × [0 , 1] , g ) along its long b oundary comp onen t to the tree T n b y a piecewise isometry in suc h a w ay that the resulting space is homeomorphic to a disk and the obtained embedding of T n in D is the same as in the claim ( ∗ ) . By ( ∗ ) , any null-homotop y of the b oundary passes a curve that inter- sects n diﬀerent edges of T n . By construction this curv e is longer than ε 10 · n = 1 100 · n . The obtained metric is not Riemannian, but it is easy to smo oth while k eeping this prop ert y . Since n is large the result follows. This example was constructed by Sidney F rank el and Mikhail Katz [151]. Shortening homotop y . Set p = γ 0 (0) and  0 = length γ 0 . 75 By a compactness argument, there exists δ > 0 such that no geo desic lo op based at p has length in the interv al ( L − D , L + D + δ ] . Assume that  0 ⩾ L + δ . Cho ose t 0 ∈ [0 , 1] suc h that length  γ 0 | [0 ,t 0 ]  = L + δ. Let σ b e a minimizing geodesic from γ ( t 0 ) to p . Note that γ 0 is homotopic to the concatenation γ ′ 0 = γ 0 | [0 ,t 0 ] ∗ σ ∗ ¯ σ ∗ γ | [ t 0 , 1] , where ¯ σ denotes the backw ard parametrization of σ . Applying a curv e shortening pro cess to the lo op λ 0 = γ | [0 ,t 0 ] ∗ σ , we get a homotop y λ t relativ e to its endp oints from the lo op λ 0 to a geo desic lo op λ 1 at p . F rom the ab ov e, length λ 1 ⩽ L − D . p γ ( t 0 ) λ 1 σ The concatenation γ t = λ t ∗ ¯ σ ∗ γ | [ t 0 , 1] is a ho- motop y from γ ′ 0 to another curve γ 1 . F rom the construction, it is clear that length γ t ⩽ length γ 0 + 2 · length σ ⩽ ⩽ length γ 0 + 2 · D for any t ∈ [0 , 1] and length γ 1 = length λ 1 + length σ + length γ | [ t 0 , 1] ⩽ ⩽ L − D + D + length γ − ( L + δ ) = =  0 − δ. Rep eating the pro cedure a suﬃcien t n umber of times, we get curves γ 2 , . . . , γ n connected b y the needed homotopies so that  i +1 ⩽  i − δ and  n < L + δ , where  i = length γ i . If  n ⩽ L , we are done. Othe rwise, rep eat the argument again for δ ′ =  n − L . The problem is due to Alexander Nabutovsky and Regina Rotman [152]. Con vex h yp ersurface. First, let us deﬁne the c one c onstruction of maps into M . Let ∆ ′ b e a simplex with a vertex v . Denote by ∆ the facet in ∆ ′ opp osite to v . Let f : ∆ → M b e a map such that f (∆) ⊂ B ( x, 1) M for some x ∈ M . Given w ∈ ∆ , let γ w : [0 , 1] → M be the minimizing 76 CHAPTER 4. CUR V A TURE-FREE geo desic path from x to f ( w ) . Since the injectivity radius of M is at least 1 , the path γ w is uniquely deﬁned. The map f ′ : ∆ ′ → M deﬁned as f ′ : (1 − t ) · v + t · w 7→ γ w ( t ) is called the c one over f with vertex x . One ma y start with a map f 0 : ∆ 0 → M and iterate the cone construc- tion for the vertices x 1 , . . . x k , to get a sequence of maps f i : ∆ i → M as long as f i − 1 (∆ i − 1 ) ⊂ B ( x i , 1) . A straigh tforward application of the tri- angle inequalit y shows that the latter conditions hold if f 0 (∆ 0 ) ⊂ B ( x i , s ) for each i and s < 2 2+ k . No w w e go back to the solution. Cho ose a ﬁne triangulation of W so that M b ecomes a sub-complex of W . W e can assume that the diameter of eac h simplex in τ is less than an y given ε > 0 . F urthermore, we can assume that all the v ertices of τ can b e colored with m + 2 colors (0 , . . . , m + 1) in suc h a wa y that the v ertices of each simplex hav e diﬀeren t colors; the latter can b e achiev ed b y passing to the barycentric subdivision of τ . Denote by τ i the maximal i -dimensional sub-complex of τ with all the vertices colored b y 0 , . . . , i . Let h b e the maximal distance from p oints in W to M . F or each v ertex v in τ choose a p oint v ′ ∈ M at distance ⩽ h . Note that if v and w are vertices of one simplex, then | v ′ − w ′ | M < 2 · h + ε. Assume that 2 m +3 > h . Cho ose p ositiv e ε < 2 m +3 − h and use it in the construction of the triangulation τ ab o ve. Applying the iterated cone construction for each simplex of τ w e get an extension of the map v 7→ v ′ deﬁned on τ 0 to τ 1 , . . . τ m +1 . According to the ab ov e estimates, the cone constructions are deﬁned at each of the neede d m + 1 iterations. This wa y we get to a retraction W → M . It follows that the fundamen- tal class of M v anishes in the homology ring of M — a contradiction. This problem is a stripp ed version of the b ound on ﬁlling radius given b y Mikhael Gromov [107]. Almost constant function. Giv en a p ositive integer m , denote b y δ m the exp ected v alue of | x 1 | for the random unit v ector x = ( x 1 , . . . , x m ) ∈ ∈ S m − 1 with resp ect to the uniform distribution. Observ e that δ m → 0 as m → ∞ . Indeed, from symmetry and Bun- y ako vsky inequality w e get 1 m = 1 m · E( | x | 2 ) = E( x 2 1 ) ⩾ E( | x 1 | ) 2 = δ 2 m . Since f is 1 -Lipsc hitz, E( | d f ( w ) | ) ⩽ δ m 77 for a random vector w in U M . Note that | f ◦ γ (1) − f ◦ γ (0) | =      1 w 0 d f ( γ ′ ( t )) · dt      ⩽ ⩽ 1 w 0 | d f ( γ ′ ( t )) | · dt. Assume that γ ′ (0) takes a random v alue in U M . By Liouville’s theo- rem ab out phase volume, the same holds for γ ′ ( t ) for any ﬁxed t . There- fore E( | f ◦ γ (1) − f ◦ γ (0) | ) ⩽ E 1 w 0 | d f ( γ ′ ( t )) | · dt ! ⩽ δ m . By Marko v’s inequality , the probability of the even t | f ◦ γ (1) − f ◦ γ (0) | > ε is at most δ m ε . Hence the result follows. I learned this problem from Mikhael Gromov. It gives an example in the Riemannian world of the so-called c onc entr ation of me asur e phe- nomenon [153, 154]. Chapter 5 Metric geometry In this chapter, we consider metric spaces. The relev ant bac kground ma- terial can b e found in [155] or [156]. Let us ﬁx a few standard notations. ⋄ The distance b et ween tw o p oints x and y in a metric space X will b e denoted b y dist x ( y ) , | x − y | or | x − y | X , the latter notation is used to emphasize that x and y b elong to the space X . ⋄ A metric space X is called length-metric sp ac e if, for any tw o p oints x, y ∈ X and any ε > 0 , the p oints x and y can b e connected b y a curv e α with length α < | x − y | X + ε. In this case, we say the metric on X is a length-metric . Em b edding of a compact  Pr ove that any c omp act metric sp ac e is isometric to a subset of a c omp act length-metric sp ac e. Semisolution. Let K b e a compact metric space. Denote by B ( K, R ) the space of real-v alued b ounded functions on K equipp ed with sup-norm; that is, | f | = sup { | f ( x ) | | x ∈ K } . Note that the map K → B ( K, R ) , deﬁned by x 7→ dist x is a distance- preserving embedding. Indeed, by the triangle inequality we ha ve | dist x ( z ) − dist y ( z ) | ⩽ | x − y | K 78 79 for any z ∈ K and the equalit y holds for z = x . In other words, we can and will consider K as a subspace of B ( K , R ) . Denote by W the linear conv ex hull of K in B ( K, R ) ; that is, W is the intersection of all closed conv ex subsets containing K . Clearly , W is a complete subspace of B ( K , R ) . Since K is compact we can c ho ose a ﬁnite ε -net K ε in K . The set K ε lies in a ﬁnite-dimensional subspace; therefore its conv ex hull W ε is compact. Note that W lies in the ε -neighborho o d of W ε . Therefore, W admits a compact ε -net for any ε > 0 . That is, W is totally b ounded and complete and therefore compact. Note that line segments in W are geo desics for the metric induced b y the sup-norm. In particular, W is a compact length-metric space as required. The map x 7→ dist x is called the Kur atowski emb e dding , it w as con- structed in [157]. Essentially the same map w as describ ed b y Maurice F r´ ec het [158, this is the pap er where metric spaces were introduced]. The problem also follows directly from a theorem of John Isb ell, stat- ing that inje ctive envelop es of compact metric spaces are compact; the injectiv e env elop e is an analog of conv ex hull in the category of metric spaces [see 2.11 in 159]. The following related problem is op en even for three-p oint sets. This problem was mentioned by Mikhael Gromov [160, 6.B 1 (f )].  Is it true that any c omp act subset of a c omplete CA T(0) length-sp ac e lies in a c omp act c onvex set? Non-con tracting map ◦ A map f : X → Y b etw een metric spaces is called distanc e non-c ontr acting if | f ( x ) − f ( x ′ ) | Y ⩾ | x − x ′ | X for any tw o p oin ts x, x ′ ∈ X .  L et K b e a c omp act metric sp ac e and f : K → K a distanc e non-c ontr acting map. Pr ove that f is an isometry. Finite-whole extension A map f : X → Y b et ween metric spaces is called non-exp anding if | f ( x ) − f ( x ′ ) | Y ⩽ | x − x ′ | X 80 CHAPTER 5. METRIC GEOMETR Y for any tw o p oin ts x, x ′ ∈ X .  L et X and Y b e metric sp ac es, Y c omp act, A ⊂ X , and f : A → Y a non-exp anding map. Assume that for any ﬁnite set F ⊂ X ther e is a non-exp anding map F → Y that agr e es with f in F ∩ A . Show that ther e is a non-exp anding map X → Y that agr e es with f on A . Horo-compactiﬁcation ◦ Let X be a metric space. Denote by C ( X , R ) the space of contin uous functions X → R equipp ed with the c omp act-op en top olo gy ; that is, for an y compact set K ⊂ X and any open set U ⊂ R the set of all contin uous functions f : X → R such that f ( K ) ⊂ U is declared to b e op en. Cho ose a p oint x 0 ∈ X . Given a p oin t z ∈ X , let f z ∈ C ( X , R ) b e the function deﬁned by f z ( x ) = dist z ( x ) − dist z ( x 0 ) . Let F X : X → C ( X , R ) b e the map sending z to f z . Denote by ¯ X the closure of F X ( X ) in C ( X , R ) ; note that ¯ X is compact. That is, if F X is an embedding, then ¯ X is a compactiﬁcation of X , whic h is called the hor o-c omp actiﬁc ation . In this case, the complement ∂ ∞ X = = ¯ X \ F X ( X ) is called the hor o-absolute of X . The construction ab o v e is due to Mikhael Gromov [161].  Construct a pr op er metric sp ac e X such that F X : X → C ( X , R ) is not an emb e dding. Show that ther e ar e no such examples among pr op er length-metric sp ac es. Appro ximation of the ball b y a sphere  Construct a se quenc e of Riemannian metrics on S 3 c onver ging in the sense of Gr omov–Hausdorﬀ to the unit b al l in R 3 . Macroscopic dimension ◦ Let X b e a lo cally compact metric space and a > 0 . F ollowing Mikhael Gromov [162], we say that the macr osc opic dimen- sion of X at scale a is the smallest integer m such that there is a contin- uous map f from X to an m -dimensional simplicial complex K with diam[ f − 1 { k } ] < a 81 for any p oint k ∈ K . Equiv alen tly , the macroscopic dimension of X on the scale a can b e deﬁned as the smallest in teger m such that X admits an op en cov er with diameter of eac h set less than a and such that each p oin t in X is cov ered b y at most m + 1 sets in the cov er.  L et M b e a simply-c onne cte d Riemannian manifold with the fol lowing pr op erty: every close d curve is nul l-homotopic in its own 1-neighb orho o d. Pr ove that the macr osc opic dimension of M at sc ale 100 is at most 1 . No Lipsc hitz embedding ∗  Construct a length-metric d on R 3 such that the sp ac e ( R 3 , d ) do es not admit a lo c al ly Lipschitz emb e dding into the 3-dimensional Euclide an sp ac e. Sub-Riemannian sphere + Let us explain what is a sub-Riemannian metric. Let ( M , g ) b e a Riemannian manifold. Assume that in the tangent bundle T M a choice of sub-bundle H is given. Let us call the sub-bundle H horizontal distribution . The tangen t v ectors in H will b e called horizontal . A piecewise smo oth curve will b e called horizontal if all its tangent vectors are horizontal. The sub-Riemannian distance b et ween any tw o p oints x and y is de- ﬁned as the inﬁmum of lengths of horizontal curv es connecting x to y . Alternativ ely , the distance can b e deﬁned as the limit of Riemannian distances for the metrics g λ ( X, Y ) = g ( X H , Y H ) + λ · g ( X V , Y V ) as λ → ∞ , where X H denotes the horizon tal part of X ; that is, the orthogonal pro jection of X to H and X V denotes the vertical part of X ; so, X V + X H = X . W e also need an additional condition to ensure the following prop erties ⋄ The sub-Riemannian metric induces the original top ology on the manifold. In particular, if M is connected, then the distance cannot tak e inﬁnite v alues. ⋄ An y curv e in M can be arbitrarily well appro ximated b y a horizon tal curv e with the same endp oints. The most common condition of this type is the so-called c omplete non- inte gr ability ; it means that for an y x ∈ M , one can choose a basis in its tangen t space T x M from the v ectors of the following type A ( x ) , [ A, B ]( x ) , [ A, [ B , C ]]( x ) , [ A, [ B , [ C , D ]]]( x ) , . . . 82 CHAPTER 5. METRIC GEOMETR Y where [ ∗ , ∗ ] denotes the Lie brac ket and the vector ﬁelds A, B , C , D , . . . are horizontal.  Pr ove that any sub-Riemannian metric on S m is isometric to the intrinsic metric of a hyp ersurfac e in R m +1 . It will b e diﬃcult to solve the problem without knowing a pro of of the Nash–Kuip er theorem ab out length preserving C 1 -em b eddings. The original pap ers of John Nash and Nicolaas Kuip er [163, 164] are v ery readable. Length-preserving map + A con tinuous map f : X → Y b et w een metric spaces is called length- pr eserving if it preserves the length of curves; that is, for any curv e α in X we hav e length( f ◦ α ) = length α.  Show that ther e is no length-pr eserving map R 2 → R . The exp ected solution uses Rademacher’s theorem [165] ab out diﬀer- en tiability of Lipschitz functions. Fixed segmen t  L et ρ ( x, y ) = ∥ x − y ∥ b e a metric on R m induc e d by a norm ∥∗∥ . Assume that f : ( R m , ρ ) → ( R m , ρ ) is an isometry that ﬁxes two distinct p oints a and b . Show that f ﬁxes the line se gment b etwe en a and b . Eviden tly , f maps the line segment [ ab ] to a minimizing geo desic con- necting a to b in ( R m , ρ ) . How ever, in general, there migh t b e many minimizing geo desics connecting a to b in ( R m , ρ ) . The problem states that [ ab ] is mapp ed to itself. P ogorelov’s construction ◦  L et µ b e a c entr al ly symmetric R adon me asur e on S 2 which is p ositive on every op en set and vanishes on every gr e at cir cle. Given two p oints x, y ∈ S 2 , set ρ ( x, y ) = µ [ B ( x, π 2 ) \ B ( y , π 2 ) ] . Show that ρ is a length-metric on S 2 , and mor e over, the ge o desics in ( S 2 , ρ ) run along gr e at cir cles of S 2 . 83 Straigh t geo desics Recall that a map f : X → Y betw een metric spaces is called bi-Lipschitz if there is a constant ε > 0 such that ε · | x − y | X ⩽ | f ( x ) − f ( y ) | Y ⩽ 1 ε · | x − y | X . for any x, y ∈ X .  L et ρ b e a length-metric on R m that is bi-Lipschitz e quivalent to the c anonic al metric. Assume that every ge o desic γ in ( R d , ρ ) is aﬃne ; that is, γ ( t ) = v + w · t for c onstant ve ctors v , w ∈ R m . Show that ρ is induc e d by a norm on R m . Hyp erb olic space  Construct a bi-Lipschitz map fr om the hyp erb olic 3 -sp ac e to the pr o duct of two hyp erb olic planes. Quasi-isometry of a Euclidean space + A map f : X → Y b etw een metric spaces is called a quasi-isometry if there is a real constant C > 1 such that 1 C · | x − x ′ | X − C ⩽ | f ( x ) − f ( x ′ ) | Y ⩽ C · | x − x ′ | X + C for any x, x ′ ∈ X and f ( X ) is a C -net in Y ; that is, for an y y ∈ Y there is x ∈ X such that | f ( x ) − y | Y ⩽ C . Note that a quasi-isometry is not assumed to b e contin uous; for ex- ample, any map b etw een compact metric spaces is a quasi-isometry .  L et f : R m → R m b e a quasi-isometry. Show that ther e is a (bi- Lipschitz) home omorphism h : R m → R m at a b ounde d distanc e fr om f ; that is, ther e is a r e al c onstant C such that | f ( x ) − h ( x ) | ⩽ C for any x ∈ R m . The exp ected solution requires the so-called gluing the or em , a corollary of the theorem prov ed by Laurence Sieb enmann [166]. It states that if V 1 , V 2 ⊂ R m are op en and the tw o embedding f 1 : V 1 → R m and f 2 : V 2 → → R m are suﬃcien tly close to each other on the ov erlap U = V 1 ∩ V 2 , then there is an em b edding f deﬁned on an op en set W ′ whic h is slightly smaller than W = V 1 ∪ V 2 and suc h that f is suﬃcien tly close to each f 1 and f 2 at the p oin ts where they are deﬁned. The bi-Lipschitz version requires an analogous statemen t in the cate- gory of bi-Lipschitz embeddings; it was prov ed by Dennis Sulliv an [167]; a detailed pro of is given by Pekk a T ukia and Jussi V¨ ais¨ al¨ a [168, 5.10]. 84 CHAPTER 5. METRIC GEOMETR Y F amily of sets with no section ◦  Construct a family of close d sets C t ⊂ S 1 , t ∈ [0 , 1] that is c ontinuous in the Hausdorﬀ top olo gy but do es not admit a section . That is, ther e is no p ath c : [0 , 1] → S 1 such that c ( t ) ∈ C t for al l t . Spaces with isometric balls  Construct a p air of lo c al ly c omp act length-metric sp ac es X and Y that ar e not isometric, but for some p oints x 0 ∈ X , y 0 ∈ Y and any r adius R the b al l B ( x 0 , R ) X is isometric to the b al l B ( y 0 , R ) Y . A verage distance ◦  Show that for any c omp act length-metric sp ac e X ther e is a numb er  such that for any ﬁnite c ol le ction of p oints ther e is a p oint z that lies of aver age distanc e  fr om the c ol le ction; that is, for any x 1 , . . . , x n ∈ X ther e is z ∈ X such that 1 n · X i | x i − z | X = . Semisolutions Non-con tracting map. Giv en any pair of p oints x 0 , y 0 ∈ K , con- sider t w o sequences x 0 , x 1 , . . . and y 0 , y 1 , . . . suc h that x n +1 = f ( x n ) and y n +1 = f ( y n ) for each n . Since K is compact, we can choose an increasing sequence of integers n k suc h that b oth sequences ( x n i ) ∞ i =1 and ( y n i ) ∞ i =1 con verge. In particular, b oth are Cauc h y sequences; that is, | x n i − x n j | K , | y n i − y n j | K → 0 as min { i, j } → ∞ . Since f is non-con tracting, w e get | x 0 − x | n i − n j | | ⩽ | x n i − x n j | . It follows that there is a sequence m i → ∞ such that ( ∗ ) x m i → x 0 and y m i → y 0 as i → ∞ . Set  n = | x n − y n | K . 85 Since f is non-con tracting, the sequence (  n ) is non-decreasing. By ( ∗ ) ,  m i →  0 as m i → ∞ . It follo ws that (  n ) is a constant sequence. In particular | x 0 − y 0 | K =  0 =  1 = | f ( x 0 ) − f ( y 0 ) | K for any pair of p oints ( x 0 , y 0 ) in K . That is, f is distance-preserving, in particular injective. F rom ( ∗ ) , w e also get that f ( K ) is ev erywhere dense. Since K is compact f : K → K is surjective. Hence the result follows. This is a basic lemma in the in tro duction to Gromo v–Hausdorﬀ dis- tance [see 7.3.30 in 155]. I learned this pro of from T ravis Morrison, a studen t in my MASS class at Penn State, F all 2011. As an easy corollary , one can see that an y surjective non-expanding map from a compact metric space to itself is an isometry . The follow- ing problem due to Aleksander Ca l k a [169] is closely related but more in volv ed.  Show that any lo c al isometry fr om a c onne cte d c omp act metric sp ac e to itself is a home omorphism. Finite-whole extension. Consider the space Y X of all maps X → Y equipp ed with the pro duct top ology . Giv en a ﬁnite set F ∈ X , denote by C F the set of maps h ∈ Y X suc h that the restriction h | F is short and the restriction h | A ∩ F agrees with f : A → Y . By assumption, the sets C F ⊂ Y X are closed and nonempty . Note that for any ﬁnite collection of ﬁnite sets F 1 , . . . , F n ⊂ X , we ha ve C F 1 ∩ · · · ∩ C F n ⊃ C F 1 ∪···∪ F n . In particular, the intersection is nonempty . A ccording to Tikhono v’s theorem [see 170, and the references therein], Y X is compact. By the ﬁnite in tersection property , the intersection T F C F with F ranging along all ﬁnite subsets of X is nonempt y . It re- mains to note that any map h ∈ T F C F solv es the problem. This observ ation was used by Stephan Stadler and me [44]. Horo-compactiﬁcation. F or the ﬁrst part of the problem, take X to b e the set of non-negative integers with the metric ρ deﬁned by ρ ( m, n ) = m + n for m  = n . 86 CHAPTER 5. METRIC GEOMETR Y The second part is prov ed b y contradiction. Assume that X is a proper length space and F X is not an embedding. That is, there is a sequence of p oin ts z 1 , z 2 , . . . and a p oint z ∞ suc h that f z n → f z ∞ in C ( X , R ) as n → ∞ , while | z n − z ∞ | X > ε for ﬁxed ε > 0 and all n . Note that any pair of p oints x, y ∈ X can be connected by a minimizing geo desic [ xy ] . Cho ose ¯ z n on a geo desic [ z ∞ z n ] such that | z ∞ − ¯ z n | = ε . Note that f z n ( z ∞ ) − f z n ( ¯ z n ) = ε and f z ∞ ( z ∞ ) − f z n ( ¯ z n ) = − ε for all n . Since X is prop er, w e can pass to a subsequence of z n so that the sequence ¯ z n con verges; denote its limit by ¯ z ∞ . The ab ov e identities imply that f z n ( ¯ z ∞ ) → f z ∞ ( ¯ z ∞ ) or f z n ( z ∞ ) → f z ∞ ( z ∞ ) — a contradiction. I learned this problem from Linus Kramer and Alexander Lytchak; the example was also men tioned in the lectures of Anders Karlsson and attributed to Uri Bader [see 2.3 in 171]. Appro ximation of the ball by a sphere. Make ﬁne burrows in the standard 3-ball without changing its top ology , but at the same time come suﬃcien tly close to any p oint in the ball. Consider the doubling of the obtained ball along its b oundary . The obtained space is homeomorphic to S 3 . Note that the burro ws can b e made so that the obtained space is suﬃciently close to the original ball in the Gromov–Hausdorﬀ metric. It remains to smo oth the obtained space slightly to get a gen uine Riemannian metric with the needed prop erty . This problem app eared as an exercise in the textb o ok of Dmitri Bu- rago, Y uri Burago, and Segei Iv ano v [155, Ex. 7.5.17]. If M is a compact manifold of dimension at least 3 and X is a rea- sonable compact length space, then the existence of a map M → X that induces a surjective homomorphism on their fundamental groups is a necessary and suﬃcient condition for the existence of Gromo v–Hausdorﬀ appro ximation of X by Riemannian metrics on M . This statemen t was pro ved b y Stev en F erry and Boris Okun [172]. 87 (A doubled cone ov er Haw aiian earring giv es an example of unr e asonable space. It has a nontrivial fundamental group, but ad- mits an approximation by Riemannian met- rics on S 3 .) The tw o-dimensional case is quite diﬀer- en t. There is no sequence of Riemannian metrics on S 2 con verging to the unit disk in the sense of Gromov–Hausdorﬀ. In fact, if X is a limit of ( S 2 , g n ) , then any point x 0 ∈ X either admits a neighborho o d homeomor- phic to R 2 or is a cut p oint; that is, X \ { x 0 } is disconnected [see 3.32 in 79]. Macroscopic dimension. The following claim resembles Besicovitc h inequalit y; it is key to the pro of. ( ∗ ) L et a b e a p ositive r e al numb er. Assume that a close d curve γ in a metric sp ac e X c an b e sub divide d into 4 ar cs α , β , α ′ , and β ′ in such a way that ⋄ | x − x ′ | > a for any x ∈ α and x ′ ∈ α ′ and ⋄ | y − y ′ | > a for any y ∈ β and y ′ ∈ β ′ . Then γ is not c ontr actible in its a 2 -neighb orho o d. T o prov e ( ∗ ) , consider t wo functions deﬁned on X as follows: w 1 ( x ) = min { a, dist α ( x ) } , w 2 ( x ) = min { a, dist β ( x ) } , and the map w : X → [0 , a ] × [0 , a ] , deﬁned by w : x 7→ ( w 1 ( x ) , w 2 ( x )) . Note that w ( α ) = 0 × [0 , a ] , w ( β ) = [0 , a ] × 0 , w ( α ′ ) = a × [0 , a ] , w ( β ′ ) = [0 , a ] × a. Therefore, the comp osition w ◦ γ is a degree 1 map S 1 → ∂ ([0 , a ] × [0 , a ]) . It follows that if h : D → X shrinks γ , then there is a p oin t z ∈ D suc h that w ◦ h ( z ) = ( a 2 , a 2 ) . Therefore h ( z ) lies at distance at least a 2 from α , β , α ′ , β ′ and therefore from γ . Hence the claim ( ∗ ) follows. Cho ose a p oin t p ∈ M . Let us cov er M by the connected comp onen ts of the inv erse images dist − 1 p (( n − 1 , n + 1)) for all integers n . Clearly , an y p oin t in M is cov ered b y at most tw o of these comp onen ts. It remains to sho w that each of these comp onents has diameter less than 100 . 88 CHAPTER 5. METRIC GEOMETR Y τ p x y m Assume the con trary; let x and y b e tw o p oints in one connected comp onent and | x − y | M ⩾ 100 . Connect x to y with a curve τ in this comp onen t. Consider the closed curve σ formed b y τ and tw o geo desics [ px ] , [ py ] . Note that | p − x | > 40 . Therefore there is a p oin t m on [ px ] such that | m − x | = 20 . By the triangle inequality , the sub division of σ in to the arcs [ pm ] , [ mx ] , τ , and [ y p ] satisfy the conditions of the claim ( ∗ ) for a = 10 . Hence the statement follows. The problem is due to Mikhael Gromov [107, App endix 1(E 2 )]. No Lipsc hitz em b edding. Consider a chain of circles c 0 , . . . , c n in R 3 ; that is, c i and c i − 1 are linked for each i . c 0 c 1 . . . c n Assume that R 3 is equipp ed with a length-metric ρ such that the total length of the circles is  and U is an op en b ounded set containing all the circles c i . Note that for an y L -Lipschitz embedding f : ( U, ρ ) → R 3 , the distance from f ( c 0 ) to f ( c n ) is less than L ·  . The ρ -distance from c 0 to c n migh t b e muc h larger than L ·  . Indeed, ﬁx a line segmen t [ ab ] in R 3 . Mo dify the length-metric on R 3 in a small neigh b orho o d of [ ab ] so that there is a chain ( c i ) of circles as ab ov e, that go es from a to b such that (1) the total length, say  , of all the circles c i is arbitrarily small, but (2) the obtained metric ρ is arbitrarily close to the canonical one, say   ρ ( x, y ) − | x − y |   < ε for any t w o p oints x, y ∈ R 3 and ﬁxed in adv anced small ε > 0 . The con- struction of ρ is done b y shrinking the length of each circle and expanding the length in the normal directions to the circles in a small neigh b orhoo d. The latter mak es it imp ossible to use the circles c i as a shortcut; that is, the time needed to go from one circle to another is larger than the time one could sav e by going along the circle. Set a n = (0 , 1 n , 0) and b n = (1 , 1 n , 0) . Note that the line segments [ a n b n ] are disjoint and conv erging to [ a ∞ b ∞ ] , where a ∞ = (0 , 0 , 0) and b ∞ = (1 , 0 , 0) . Apply the abov e construction in non-ov erlapping conv ex neighbor- ho ods of [ a n b n ] for sequences ε n and  n con verging to zero very fast. 89 The obtained length-metric ρ is still close to the canonical metric on R 3 , but it does not admit a locally Lipschitz homeomorphism to R 3 . Indeed, assume that such homeomorphism h exists. Cho ose a b ounded op en set U containing [ a ∞ b ∞ ] ; note that the restriction h | U is L -Lipsc hitz for some L . F rom the ab o ve construction, w e get | h ( a ∞ ) − h ( b ∞ ) | ⩽ | h ( a n ) − h ( b n ) | + + | h ( a ∞ ) − h ( a n ) | + | h ( b n ) − h ( b ∞ ) | ⩽ ⩽ L ·  n + 2 n + 100 · ε n for any p ositive integer n . The right-hand side conv erges to 0 as n → ∞ . Therefore h ( a ∞ ) = h ( b ∞ ) — a contradiction. The problem is due to Dmitri Burago, Sergei Iv anov, and David Sho en- thal [173]. It is exp ected that any metric on R 2 admits lo cally Lipschitz embed- dings in to the Euclidean plane. Also, it seems feasible that an y metric on R 3 admits a lo cally Lipschitz embedding into R 4 . Note that any metric on the cub e in R 3 admits a prop er lo cally Lips- c hitz map to the unit cub e with the canonical metric of degree 1. More- o ver, one can make this map injective on any ﬁnite set of p oin ts. It is instructiv e to visualize this map for the metric of the solution. Sub-Riemannian sphere. If d is a sub-Riemannian metric on S m , then there is a non-decreasing sequence of Riemannian metric tensors g 0 < < g 1 < . . . such that their induced metrics d 1 < d 2 < . . . conv erge to d . The metric g 0 can b e assumed to b e the metric of a round sphere, so it is induced by an embedding h 0 : S m → R m +1 . Applying the construction from the Nash–Kuip er theorem, one can pro duce a sequence of smooth em b eddings h n : S m → R m +1 with the induced metrics g ′ n suc h that | g ′ n − g n | → 0 . In particular, if we denote b y d ′ n the metric corresp onding to g ′ n , then d ′ n → d an n → ∞ . It follo ws from the same construction that if one c ho oses ε n > 0 , dep ending on h n , then we can assume that | h n +1 ( x ) − h n ( x ) | < ε n for any x ∈ S m . Let us in tro duce tw o conditions on the v alues ε n , called we ak and str ong . The weak condition states that ε n < 1 2 · ε n − 1 for any n . This ensures that the sequence of maps h n con verges p oin twise; denote its limit by h ∞ . 90 CHAPTER 5. METRIC GEOMETR Y Denote by ¯ d the length-metric induced by h ∞ . Note that ¯ d ⩽ d . The strong condition on ε n will ensure that actually ¯ d = d . Fix n and assume that h n and therefore ε n − 1 are constructed already . Set Σ = h n ( S m ) , and let Σ r b e the tubular r -neighborho o d of Σ . Equip Σ and Σ r with the induced length-metrics. Since Σ is a smo oth hypersurface, w e can choose r n ∈ (0 , ε n − 1 ] so that the inclusion Σ  → Σ r n preserv es the distance up to error 1 2 n . Then the strong condition states that ε n < 1 2 · r n , whic h is evidently stronger than the weak condition ε n < 1 2 · ε n − 1 ab o ve. Note that if the sequence h n is constructed with the describ ed choice of ε n , then | h ∞ ( x ) − h n ( x ) | < r n for any x ∈ S m . Therefore ¯ d ( x, y ) + 2 · r n + 1 2 n ⩾ d ′ n ( x, y ) for any n and x, y ∈ S m ; hence ¯ d ⩾ d as required. The problem on this list w as ﬁrst disco vered by Enrico Le Donne [174]. A similar construction is described in the lecture notes b y Allan Y ashinski and the author [175] which are aimed for undergraduate students. Y et the results in [176] are closely relev ant. The construction in the Nash–Kuip er em b edding theorem can b e used to prov e strange statemen ts. Here is one example based on the observ ation that the W eyl curv ature tensor v anishes on hypersurfaces in the Euclidean space.  L et M b e a Riemannian manifold diﬀe omorphic to the m -spher e. Show that ther e is a R iemannian manifold M ′ arbitr arily close to M in the Lipschitz metric whose W eyl curvatur e tensor is identic al ly 0. Length-preserving map. Assume the contrary; let f : R 2 → R b e a length-preserving map. Note that f is Lipschitz. Therefore by Rademacher’s theorem [165], the diﬀerential d x f is deﬁned for almost all x . Cho ose a unit v ector u . Giv en x ∈ R 2 , consider the path α x ( t ) = = x + t · u deﬁned for t ∈ [0 , 1] . Note that α ′ x ( t ) = ( d α x ( t ) f )( u ) holds for almost all x and t . It follows that length( f ◦ α x ) = 1 w 0 | ( d α x ( t ) f )( u ) | · dt for almost all x . Therefore | d x f ( v ) | = | v | for almost all x, v ∈ R 2 . In particular, there is x ∈ R 2 suc h that the diﬀerential d x f is deﬁned and | d x f ( e 1 ) | = | e 1 | , | d x f ( e 2 ) | = | e 2 | , | d x f ( e 1 + e 2 ) | = | e 1 + e 2 | 91 for a basis ( e 1 , e 2 ) of R 2 . It follows that d x f has rank 2 — a contradiction. The idea ab o v e can also b e used to solve the following problem.  L et ρ b e a metric on R 2 that is induc e d by a norm. Show that ( R 2 , ρ ) admits a length-pr eserving map to R 3 if and only if ( R 2 , ρ ) is isometric to the Euclide an plane. Fixed segmen t. Note that it is suﬃcien t to show that if f is an isometry suc h that f ( a ) = a and f ( b ) = b for some a, b ∈ R m , then f ( a + b 2 ) = 1 2 · ( f ( a ) + f ( b )) . Without loss of generality , w e can assume that b + a = 0 . Set f 0 = f . Consider the sequence of isometries f 0 , f 1 , . . . recursively deﬁned by f n +1 ( x ) = − f − 1 n ( − f n ( x )) for all n . Note that for all n we hav e f n ( a ) = a , f n ( b ) = b and | f n +1 (0) | = 2 · | f n (0) | . Therefore if f (0)  = 0 , then | f n (0) | → ∞ as n → ∞ . On the other hand, since f n is isometry and f ( a ) = a , we also ha ve | f n (0) | ⩽ 2 · | a | — a contradiction. The idea of the pro of is due to Jussi V¨ ais¨ al¨ a [177]. The problem is the main step in the pro of of the Mazur–Ulam theorem [178], which states that any isometry of ( R m , ρ ) is an aﬃne map. P ogorelov’s construction. The p ositivity and symmetry of ρ are evi- den t. The triangle inequality follows since ( ∗ ) [ B ( x, π 2 ) \ B ( y , π 2 )] ∪ [ B ( y , π 2 ) \ B ( z , π 2 )] ⊇ B ( x, π 2 ) \ B ( z , π 2 ) . Observ e that B ( x, π 2 ) \ B ( y , π 2 ) does not ov erlap B ( y , π 2 ) \ B ( z , π 2 ) and w e get equality in ( ∗ ) if and only if y lies on the great circle arc from x to z . Therefore the second statement follows. This construction was given by Aleksei P ogorelov [179]. It is closely related to the con- struction giv en b y David Hilb ert in [180] which was the motiv ating example for his 4th problem. 92 CHAPTER 5. METRIC GEOMETR Y The following problem was suggested b y Jairo Bo c hi [181]; it lo oks similar, but actually quite diﬀerent.  Consider the set E of al l el lipses with unit ar e a c enter e d at the origin of the plane. Equip E with the metric deﬁne d by ρ ( A, B ) = area( A △ B ) , wher e △ denotes symmetric diﬀer enc e; that is, A △ B = ( A \ B ) ∪ ( B \ A ) . L et ˆ ρ b e the length-metric induc e d by ρ . Show that ( E , ˆ ρ ) is isometric to a L ob achevsky plane. The metrics of the form ρ ( A, B ) = µ ( A △ B ) are v ery sp ecial; evidently , they can b e em b edded into the corresp onding L 1 space. Also, they satisfy the so-called hyp ermetric ine qualities ; that is, X i,j b i · b j · ρ ( A i , A j ) ⩽ 0 for an y sequence of sets A 1 , . . . A n and an y sequence of integers b 1 , . . . , b n suc h that P i b i = 1 . Note that for n = 3 and b 1 = b 2 = − b 3 = 1 we get the usual triangle inequality . F or more on the sub ject, see [182]. Straigh t geo desics. F rom the uniqueness of the straight segment b e- t ween tw o giv en p oints in R m , it follows that any straight line in R m is a geo desic in ( R m , ρ ) . Set ∥ v ∥ x = ρ ( x , ( x + v )) . Note that ∥ λ · v ∥ x = | λ | · ∥ v ∥ x for any x , v ∈ R m , and λ ∈ R . Denote by | x − y | the Euclidean distance b et w een the p oin ts x and y . Since ρ and |∗ − ∗| are bi-Lipschitz equiv alen t, applying the triangle in- equalit y t wice to the p oints x , x + λ · v , x ′ and x ′ + λ · v , we get   ∥ λ · v ∥ x − ∥ λ · v ∥ x ′   ⩽ C · | x − x ′ | for any x , x ′ , v ∈ R m , λ ∈ R and a ﬁxed real constant C . P assing to the limit as λ → ∞ , we obtain that ∥ v ∥ x do es not dep end on x ; hence the result follows. This idea is due to Thomas F o ertsch and Viktor Schroeder [183]. A more general statement w as prov ed b y Petra Hitzelb erger and Alexander Lytc hak [184]. Namely , they sho wed that if any pair of p oin ts in a geo desic metric space X can be separated by an aﬃne function , then X is isometric 93 to a conv ex subset of a normed v ector space. (A function f : X → R is called aﬃne if, for any geo desic γ in X , the comp osition f ◦ γ is aﬃne.) Hyp erb olic space. The h yp erb olic plane H 2 is isometric to ( R 2 , g ) , where g ( x, y ) =  1 0 0 e x  . The same wa y , the hyperb olic space H 3 can b e view ed as ( R 3 , h ) , where h ( x, y , z ) =   1 0 0 0 e x 0 0 0 e x   . In the describ ed co ordinates, consider the pro jections ϕ, ψ : H 3 → H 2 deﬁned by ϕ : ( x, y , z ) 7→ ( x, y ) and ψ : ( x, y , z ) 7→ ( x, z ) . Note that max { | ϕ ( p ) − ϕ ( q ) | H 2 , | ψ ( p ) − ψ ( q ) | H 2 } ⩽ ⩽ | p − q | H 3 ⩽ ⩽ | ϕ ( p ) − ϕ ( q ) | H 2 + | ψ ( p ) − ψ ( q ) | H 2 for any tw o p oints p, q ∈ H 3 . In particular, the map H 3 → H 2 × H 2 deﬁned by p 7→ ( ϕ ( p ) , ψ ( p )) is 2 ∓ 1 -bi-Lipsc hitz. W e used that horo-spheres in the h yp erb olic space are isometric to the Euclidean plane. This observ ation was made by Nikolai Lobachevsky [see 34 in 185]. The same observ ation is used in the following construction disco vered b y K´ aroly B¨ or¨ oczky [see 186 and also 187].  Construct a tessel lation of the hyp erb olic plane with one p olygonal tile of arbitr arily smal l ar e a and/or diameter. Quasi-isometry of a Euclidean space. Cho ose tw o constants M ⩾ 1 and A ⩾ 0 . A map f : X → Y b et ween metric spaces X and Y such that for any x, y ∈ X , we hav e 1 M · | x − y | − A ⩽ | f ( x ) − f ( y ) | ⩽ M · | x − y | + A and any p oin t in Y lies on the distance at most A from a p oint in the image f ( X ) will b e called ( M , A ) -quasi-isometry . Note that ( M , 0) -quasi-isometry is a [ 1 M , M ] -bi-Lipschitz map. More- o ver, if f n : R m → R m is a ( M , 1 n ) -quasi-isometry for each n , then any subsequen tial limit of f n as n → ∞ is a [ 1 M , M ] -bi-Lipschitz map. Therefore given M ⩾ 1 and ε > 0 there is δ > 0 such that for any ( M , δ ) -quasi-isometry f : R m → R m and any p ∈ R m there is an [ 1 M , M ] - bi-Lipsc hitz map h : B ( p, 1) → R m suc h that | f ( x ) − h ( x ) | < ε 94 CHAPTER 5. METRIC GEOMETR Y for any x ∈ B ( p, 1) . Using rescaling, we can get the follo wing equiv alent formulation. Given M ⩾ 1 , A ⩾ 0 , and ε > 0 , there is suﬃciently large R > 0 such that for an y ( M , A ) -quasi-isometry f : R m → R m and an y p ∈ R m there is a [ 1 M , M ] -bi-Lipschitz map h : B ( p, R ) → R m suc h that | f ( x ) − h ( x ) | < ε · R for any x ∈ B ( p, R ) . Co ver R m b y balls B ( p n , R ) and construct a [ 1 M , M ] -bi-Lipschitz map h n : B ( p n , R ) → R m close to the restrictions f | B ( p n ,R ) for each n . The maps h n are 2 · ε · R close to each other on the o v erlaps of their domains of deﬁnition. This makes it p ossible to deform sligh tly each h n so that they agree on the ov erlaps. This can b e done b y Sieb enmann’s theorem [166]. If instead you apply Sulliv an’s theorem [see 167 and also 5.10 in 168], you get a bi-Lipschitz homeomorphism h : R m → R m . The problem was suggested by Dmitri Burago. F amily of sets with no section. Giv en t ∈ (0 , 1] , conside r the real in terv al ˜ C t = [ 1 t + t, 1 t + 1] . Denote by C t the image of ˜ C t under the co vering map π : R → S 1 = R / Z . Set C 0 = S 1 . Note that the Hausdorﬀ distance from C 0 to C t is t 2 . Therefore { C t } t ∈ [0 , 1] is a family of compact subsets in S 1 that is con tinuous in the sense of Hausdorﬀ. Assume there is a contin uous section c ( t ) ∈ C t , for t ∈ [0 , 1] . Since π is a cov ering map, we can lift the path c to a path ˜ c : [0 , 1] → R such that ˜ c ( t ) ∈ ˜ C t for all t . In particular, ˜ c ( t ) → ∞ as t → 0 — a contradiction. The problem was suggested by Stephan Stadler. Here is a simpler, closely related problem.  Show that any Hausdorﬀ c ontinuous family of c omp act sets in R admits a c ontinuous se ction. The existence of sections for a family of sets parametrized by a topological space was considered b y Ernest Michael [188–190]. Spaces with isometric balls. The needed ex- amples can b e constructed b y cutting the upper half-plane along a “dy adic com b” shown on the di- agram; the obtained space should be equipped with the intrinsic metric induced from the  ∞ -norm on the plane. First, let us describ e the comb precisely . Choose an inﬁnite sequence a 0 , a 1 , . . . of zeros and ones. Given an in teger k , cut the upp er half-plane 95 along the line segment b etw een ( k , 0) and ( k , 2 m +1 ) if m is the maximal n umber suc h that k ≡ a 0 + 2 · a 1 + · · · + 2 m − 1 · a m − 1 (mo d 2 m ); If the equality holds for all m , cut the half-plane along the v ertical half-line starting at ( k , 0) . Note that all the obtained spaces, indep endently from the sequence ( a n ) , meet the conditions of the problem for the p oin t x 0 = ( 1 2 , 0) . F urther, note that the resulting spaces for t wo sequences ( a n ) and ( a ′ n ) are isometric only in the following tw o cases ⋄ if a n = a ′ n for all large n , or ⋄ if a n = 1 − a ′ n for all large n . It remains to pro duce tw o sequences that do not hav e these identities for all large n ; tw o random sequences of zeros and ones will do the job with probability one. A v erage distance. If such a num b er do es not exist then the ranges of a verage distance functions hav e empty intersection. Since X is a compact length-metric space, the range of an y contin uous function on X is a closed in terv al. By 1-dimensional Helly’s theorem, there is a pair of such range in terv als that do not intersect. That is, for tw o p oin t-arrays ( x 1 , . . . , x n ) and ( y 1 , . . . , y m ) and their av erage distance functions f ( z ) = 1 n · X i | x i − z | X and h ( z ) = 1 m · X j | y j − z | X , w e ha ve ( ∗ ) min { f ( z ) | z ∈ X } > max { h ( z ) | z ∈ X } . Note that 1 m · X j f ( y j ) = 1 m · n · X i,j | x i − y j | X = 1 n · X i h ( x i ); that is, the av erage v alue of f ( y j ) coincides with the av erage v alue of h ( x i ) , which contradicts ( ∗ ) . This is a result of Oliver Gross [191]. The v alue  is called the r en- dezvous value of X ; in fact, it is uniquely deﬁned. Chapter 6 A ctions and co v erings Bounded orbit Recall that a metric space is called pr op er if all its b ounded closed sets are compact.  L et X b e a pr op er metric sp ac e and ι : X → X an isometry. Assume that for some x ∈ X , the se quenc e x n = ι n ( x ) , n ∈ Z has a c onver gent subse quenc e. Pr ove that the se quenc e x n is b ounde d. Semisolution. Note that we can assume that the orbit { x n } is dense in X ; otherwise, we can pass to the closure of the orbit. In particular, we can choose a ﬁnite n umber of p ositiv e integers n 1 , . . . , n k suc h that the set of p oin ts { x n 1 , . . . , x n k } is a 1 -net for the ball B ( x 0 , 10) ; that is, for an y x ∈ B ( x 0 , 10) there is x n i suc h that | x − x n i | < 1 . Assume that x m ∈ B ( x 0 , 1) for some m . Then B ( x m , 10) = f m ( B ( x 0 , 10)) ⊃ B ( x 0 , 1) . In particular, { x m + n 1 , . . . , x m + n k } is a 1 -net for the ball B ( x 0 , 1) There- fore x m + n i ∈ B ( x 0 , 1) for some i ∈ { 1 , . . . , k } . Set N = max i { n i } . Applying the ab ov e observ ation inductively , we get that at least one p oin t from any string x i +1 , . . . x i + N lies in B ( x 0 , 1) . In particular, the N balls B ( x 1 , 10) , . . . , B ( x N , 10) co ver whole X . Hence the result follo ws. The problem is due to Aleksander Ca l k a [192]. 96 97 Finite action  Show that for any nontrivial c ontinuous action of a ﬁnite gr oup on the unit spher e ther e is an orbit that do es not lie in the interior of a hemispher e. Co vers of the ﬁgure eight Giv en a cov ering f : ˜ X → X of the length-metric space X , one can consider the induced length-metric on ˜ X , deﬁning the length of curv e α in ˜ X as the length of the comp osition f ◦ α ; the obtained metric space ˜ X is called the metric c overing of X . Let us deﬁne the ﬁgur e eight as the length-metric space obtained by gluing together all four ends of tw o unit segments.  Show that any c omp act length-metric sp ac e is a Gr omov–Hausdorﬀ limit of a se quenc e ( e Φ n , ˜ d/n ) wher e ( e Φ n , ˜ d ) → (Φ , d ) , ar e metric c overings of the ﬁgur e eight (Φ , d ) . Diameter of m -fold co vering ∗ The metric cov ering is deﬁned in the previous problem.  L et X b e a length-metric sp ac e, and let ˜ X b e an m -fold metric c overing of X . Show that diam ˜ X ⩽ m · diam X. The ﬁgure below shows a 5-fold cov ering with the diameter of the total space b eing exactly 5 times the diameter of the target. − → Symmetric square ◦ Let X b e a topological space. Note that X × X admits a natural Z 2 -action generated b y the inv olution ( x, y ) 7→ ( y , x ) . The quotient space X × X / Z 2 is called the symmetric squar e of X .  Show that the symmetric squar e of any p ath-c onne cte d top olo gic al sp ac e has a c ommutative fundamental gr oup. 98 CHAPTER 6. A CTIONS AND COVERINGS Sierpi ´ nski gask et ◦ T o construct the Sierpi ´ nski gask et, start with a solid equilateral triangle, sub divide it into four smaller con- gruen t equilateral triangles and remov e the interior of the cen tral one. Rep eat this pro cedure recursively for eac h of the remaining solid triangles.  Find the home omorphism gr oup of the Sierpi ´ nski gasket. Lattices in a Lie group  L et L and M b e two discr ete sub gr oups of a c onne cte d Lie gr oup G , and let h b e a left-invariant metric on G . Equip the gr oups L and M with the metric induc e d fr om G . Assume that L \ G and M \ G ar e c omp act and v ol( L \ ( G, h )) = v ol( M \ ( G, h )) . Pr ove that ther e is a bi-Lipschitz one-to-one mapping f : L → M , not ne c essarily a homomorphism. Piecewise Euclidean quotien t Note that the quotien t of the Euclidean space by a ﬁnite subgroup of SO( m ) is a p olyhe dr al sp ac e as is deﬁned on page 120; on the same page, y ou can ﬁnd the deﬁnition of piecewise linear homeomorphism.  L et Γ b e a ﬁnite sub gr oup of SO( m ) . Denote by P the quotient R m / Γ e quipp e d with the induc e d p olyhe dr al metric. Assume that P admits a pie c ewise line ar home omorphism to R m . Show that Γ is gener ate d by r otations ar ound subsp ac es of c o dimension 2 . The action of the symmetric group S m on C m = R 2 · m b y p erm utation of complex co ordinates pro vides a remark able example. The homeomor- phism C m /S m → C m can b e given by symmetric p olynomials on C m ; that is, ( z 1 , . . . , z m ) 7→ ( a 0 , . . . , a m − 1 ) , where ( z + z 1 ) · · · ( z + z m ) = a 0 + a 1 · z + · · · + a m − 1 · z m − 1 + z m . This homeomorphism is isotopic to a piecewise linear homeomorphism. Subgroups of a free group  Show that every ﬁnitely gener ate d sub gr oup of a fr e e gr oup is an in- terse ction of sub gr oups of ﬁnite index. 99 Short generators ◦  L et M b e a c omp act Riemannian manifold and p ∈ M . Show that the fundamental gr oup π 1 ( M , p ) is gener ate d by the homotopy classes of the lo ops with length at most 2 · diam M . Num b er of generators  L et M b e a c omplete c onne cte d R iemannian manifold with non-ne gative se ctional curvatur e. Show that the minimal numb er of gener ators of the fundamental gr oup π 1 M c an b e b ounde d ab ove in terms of the dimension of M . An equation in a Lie group ◦  L et G b e a c omp act c onne cte d Lie gr oup and n a p ositive inte ger. Show that given a c ol le ction of elements g 1 , . . . , g n ∈ G the e quation x · g 1 · x · g 2 · · · x · g n = e has a solution x ∈ G ; her e e is the identity element in G . Quotien t of the Hilb ert space ∗  Construct an isometric action on the Hilb ert sp ac e with the quotient sp ac e isometric to the spher e S 3 . Semisolutions Finite action. Without loss of generalit y , we may assume that the action is generated by a nontrivial homeomorphism a : S m → S m of prime order p . Assume the con trary; that is, assum e that any a -orbit lies in an op en hemisphere. Then h ( x ) = p X n =1 a n · x  = 0 for any x ∈ S m ; here we consider S m as the unit sphere in R m +1 . Consider the map f : S m → S m deﬁned by f ( x ) = h ( x ) | h ( x ) | . Note that 100 CHAPTER 6. A CTIONS AND COVERINGS (a) if a ( x ) = x , then f ( x ) = x ; (b) f ( x ) = f ◦ a ( x ) for any x ∈ S m . Note further that f is homotopic to the identit y; in particular ( ∗ ) deg f = 1 . The homotopy c an b e deﬁned by ( x, t ) 7→ γ x ( t ) , where γ x is the minimiz- ing geo desic path in S m from x to f ( x ) . By construction, | x − f ( x ) | S m < π 2 ; therefore γ x is uniquely deﬁned. Cho ose x ∈ S m suc h that a ( x )  = x . Note that the group acts without ﬁxed p oints on the inv erse image W = f − 1 ( V ) of a small op en neigh- b orhoo d V ∋ x . Therefore the quotien t map θ : W → W ′ = W / Z p is a p -fold cov ering. By (b), the restriction f | W factors thru θ ; that is, there is f ′ : W ′ → V suc h that f | W = f ′ ◦ θ . Assume that p  = 2 . Note that f ′ and θ hav e well-deﬁned degrees and deg f ≡ deg θ · deg f ′ (mo d p ) . Since θ is a p -fold cov ering, we hav e deg θ ≡ 0 (mo d p ) . Therefore ( ∗∗ ) deg f ≡ 0 (mo d p ) . Finally , observe that ( ∗ ) con tradicts ( ∗∗ ) . In the case p = 2 the same proof works, but the degrees hav e to b e considered mo dulo 2 . Along the same lines, one can get a lo wer b ound for the maximal diameter of the orbits for an y non trivial action of a ﬁnite group on a Riemannian manifold. Applying the problem to the conjugate actions, one gets that if a ﬁxed p oin t set of a ﬁnite group acting on a sphere has nonempty in terior, then the action is trivial. The same holds for any connected manifold. All this w as pro ved b y Max Newman [193]. The following problem from [194] can b e solved using Newman’s the- orem.  L et h b e a home omorphism of a c onne cte d manifold M such that e ach h -orbit is ﬁnite. Show that h has ﬁnite or der. Co vers of the ﬁgure eigh t. First note that any compact length-metric space K can b e approximated by ﬁnite metric graphs. Indeed, ﬁx a ﬁnite ε -net F in K . F or eac h pair x, y ∈ F choose a c hain of p oints x = x 0 , x 1 , . . . , x n = y such that | x i − x i − 1 | K < ε for each i and | x − y | K = | x 0 − x 1 | K + · · · + | x n − 1 − x n | K . 101 Denote by F ′ the union of all these chains with F . Connect a pair of v ertices v , w ∈ F ′ b y an edge of length | v − w | K if | v − w | K < ε . Note that the obtained metric graph is ε close to K in the Gromov–Hausdorﬀ metric. F urther, an y ﬁnite metric graph can be appro xi- mated by a graph made from the fragments shown on the diagram (we hav e to attach eac h pair of free ends of one fragment to a pair of ends in another fragment). It remains to observe that metric graphs obtained from these fragments are ﬁnite cov erings of (Φ , d/n ) . The same idea works if instead of the ﬁgure eight, w e ha ve a compact length-metric space X that admits a map X → Φ inducing an epimorphism of fundamental groups. Such spaces X can be found among compact h yp erb olic manifolds of any dimension ⩾ 2 . All this is due to V edrin ˇ Saho vi´ c [195]. A similar idea was used later to show that any ﬁnitely presen ted group can app ear as a fundamental group of the underlying space of a 3-dimensional hyperb olic orbifold [196]. Diameter of m -fold cov ering. Cho ose points ˜ p, ˜ q ∈ ˜ M . Let ˜ γ : [0 , 1] → → ˜ M b e a minimizing geo desic path from ˜ p to ˜ q . W e need to sho w that length ˜ γ ⩽ m · diam M . Supp ose the con trary . Denote by p , q , and γ the pro jections to M of ˜ p , ˜ q , and ˜ γ resp ectively . Represen t γ as the concatenation of m paths of equal length, γ = γ 1 ∗ . . . ∗ γ m , so length γ i = 1 m · length γ > diam M . Let σ i b e a minimizing geo desic in M connecting the endp oin ts of γ i . Note that length σ i ⩽ diam M < length γ i . Consider m + 1 paths α 0 , . . . , α m deﬁned as the concatenations α i = σ 1 ∗ . . . ∗ σ i ∗ γ i +1 ∗ . . . ∗ γ m . Let ˜ α 0 , . . . , ˜ α m b e their liftings with ˜ q as an endp oin t. The starting p oin t of each curve ˜ α i is one of m inv erse images of p . Therefore tw o curv es, ˜ α i and ˜ α j for i < j , hav e the same starting p oin t in ˜ M . 102 CHAPTER 6. A CTIONS AND COVERINGS Note that the concatenation β = γ 1 ∗ . . . ∗ γ i ∗ σ i +1 ∗ . . . ∗ σ j ∗ γ j +1 ∗ . . . ∗ γ m . admits a lift ˜ β that connects ˜ p to ˜ q in ˜ M . Clearly , length ˜ β < length γ — a contradiction. The question was ask ed b y Alexander Nabutovsky and answered by Sergei Iv ano v [197]. A closely related problem for universal cov erings is discussed by Sergio Zamora in [198]. Symmetric square. Let Γ = π 1 X and ∆ = π 1 (( X × X ) / Z 2 ) . Consider the homomorphism ϕ : Γ × Γ → ∆ induced by the quotien t map X × X → → ( X × X ) / Z 2 . Note that ϕ ( α, 1) = ϕ (1 , α ) for any α ∈ Γ and the restrictions ϕ | Γ ×{ 1 } and ϕ | { 1 }× Γ are onto. It remains to note that ϕ ( α, 1) · ϕ (1 , β ) = ϕ (1 , β ) · ϕ ( α, 1) for any α and β in Γ . The problem was suggested by Rostislav Matvey ev. Sierpi ´ nski gasket. Denote the Sierpi ´ nski gasket by △ . Let us show that any homeomorphism of △ is also an isometry . There- fore its homeomorphism group is the symmetric group S 3 . x y z Let { x, y , z } b e a 3-p oint set in △ such that its com- plemen t has 3 connected comp onents. Show that there is a unique choice for the set { x, y , z } and it is formed b y the midp oin ts of the long sides. It follows that an y homeomorphism of △ p ermutes the set { x, y , z } . Applying a similar argumen t recursively to the smaller triangles, we get that this p ermutation uniquely describ es the homeomorphism. The problem was suggested by Bruce Kleiner. The homeomorphism group of the Sierpi ´ nski carp et is muc h more interesting [199]. Lattices in a Lie group. Denote by V ℓ and W m the V oronoi domains for each  ∈ L and m ∈ M resp ectively; that is, V ℓ = { g ∈ G | | g −  | G ⩽ | g −  ′ | G for any  ′ ∈ L } , W m = { g ∈ G | | g − m | G ⩽ | g − m ′ | G for any m ′ ∈ M } . Note that for any  ∈ L and m ∈ M , we ha ve ( ∗ ) v ol V ℓ = v ol( L \ ( G, h )) = v ol( M \ ( G, h )) = v ol W m . 103 Consider the bipartite graph Γ with the parts L and M such that  ∈ L is adjacent to m ∈ M if and only if V ℓ ∩ W m  = ∅ . By ( ∗ ) the graph Γ satisﬁes the condition of the marriage theorem [200] — any subset S in L has at least | S | neigh b ors in M and the other w ay around; here | S | denotes the num b er of elements in S . Therefore there is a bijection f : L → M such that V ℓ ∩ W f ( ℓ )  = ∅ for any  ∈ L . It remains to observe that f is bi-Lipschitz. The problem is due to Dmitri Burago and Bruce Kleiner [201]. F or a ﬁnitely generated group G , it is not known if G and G × Z 2 can fail to b e bi-Lipschitz. (The groups are assumed to b e equipp ed with the word metric.)  the disc x 0 Piecewise Euclidean quotient. Note that the group Γ is the holonomy group of the quo- tien t space P = R m / Γ . More precisely , one can iden tify R m with the tangen t space to a regular p oin t x 0 of P in such a wa y that for any γ ∈ Γ there is a lo op  based at x 0 that runs in the regular lo cus of P and has the holonomy γ . Cho ose γ and  as ab ov e. Since P is simply- connected, we can shrink  by a disk. By the general p osition argument, we can assume that the disk only passes thru simplices of co dimension 0 , 1 and 2 and intersects the simplices of co di- mension 2 transversely . In other words,  can b e presented as a pro duct of lo ops such that eac h lo op go es around a single simplex of co dimension 2 and comes bac k. The holonomy for each of these lo ops is a rotation around a hyperplane. Hence the result follows. The con verse of the problem also holds; it was prov ed by Christian Lange [202]; his pro of is based on earlier results of Marina Mikhailo v a [203]. Note that the cone o v er the spherical susp ension ov er the Poincar ´ e sphere is homeomorphic to R 5 and it is the quotien t of R 5 b y the binary icosahedral group, which is a subgroup of SO(5) of order 120. Therefore, if one replaces “piecewise linear homeomorphism” with “homeomorphism” in the formulation, then the answer will b e diﬀeren t; a complete classiﬁcation of such actions is given in [202]. Subgroups of a free group. The proof exploits the fact that free groups are fundamental groups of graphs. 104 CHAPTER 6. A CTIONS AND COVERINGS ˜ p ¯ B ( ˜ p, 2 1 2 ) Let F b e a free group and G a ﬁnitely gen- erated subgroup in F . W e need to sho w that G is an in tersection of subgroups of ﬁnite in- dex in F . Without loss of generality , we can assume that F has a ﬁnite num b er of genera- tors, denote it by m . Let W b e the wedge sum of m circles so that π 1 ( W , p ) = F . Equip W with the length- metric such that each circle has unit length. P ass to the metric co vering ˜ W of W such that π 1 ( ˜ W , ˜ p ) = G for a lift ˜ p of p . Cho ose a suﬃciently large integer n and consider the doubling of the closed ball ¯ B ( ˜ p, n + 1 2 ) along its b oundary . Let us denote the obtained doubling by Z n and set G n = π 1 ( Z n , ˜ p ) . Note that Z n is a metric cov ering of W ; this allows us to consider G n as a subgroup of F . By construction, Z n is compact; therefore G n has a ﬁnite index in F . It remains to show that G = \ n>k G n , where k is the maximal word length in the generating set of G . Originally the problem was solved b y Marshall Hall [204–206]. Our pro of is close to the solution of John Stallings [207, 208]. Note that the statemen t do es not hold for inﬁnitely generated subgroups. The same idea can b e used to solve many other problems; here are some examples.  Show that a sub gr oup of a fr e e gr oup is fr e e.  Show that two elements u and v of a fr e e gr oup c ommute if and only if they ar e b oth p owers of an element w . Short generators. Cho ose a length-minimizing lo op γ that represen ts a given element a ∈ π 1 M . p p 1 p i . . . γ i γ i γ i γ i γ i γ i γ i γ i γ i γ i γ i γ i γ i γ i γ i γ i γ i γ i γ i γ i γ i γ i γ i γ i γ i γ i γ i γ i γ i γ i γ i γ i γ i γ i γ i γ i γ i γ i γ i γ i γ i γ i γ i γ i γ i γ i γ i γ i γ i γ i γ i γ i γ i γ i γ i γ i γ i γ i γ i γ i γ i γ i γ i γ i γ i Cho ose ε > 0 . Represent γ as a concatenation of paths γ = γ 1 ∗ . . . ∗ γ n suc h that length γ i < ε for each i . Denote b y p = p 0 , p 1 , . . . , p n = p the endp oin ts of these arcs. Connect p with p i b y a minimizing geo desic σ i . Note that γ is homotopic to a pro duct of lo ops α i = σ i − 1 ∗ γ i ∗ ¯ σ i , 105 where ¯ σ i denotes the path σ i tra veled bac kward. In particular, length α i < 2 · diam M + ε for each i . Note that given  > 0 , there are only ﬁnitely man y elements of the fundamen tal group that can b e realized by lo ops at p with length shorter than  . It follo ws that for the righ t choice of ε > 0 , any lo op α i is homotopic to a lo op of length at most 2 · diam M . Hence the result follows. The statement is due to Mikhael Gromov [Proposition 3.22 in 79]. Num b er of generators. Consider the univ ersal Riemannian cov ering ˜ M of M . Note that ˜ M is non-negatively curv ed and π 1 M acts by isometries on ˜ M . Cho ose p ∈ ˜ M . Given a ∈ π 1 M , set | a | = | p − a · p | ˜ M . Consider the so-called short b asis in π 1 M ; that is, a sequence of ele- men ts a 1 , a 2 , . . . ∈ π 1 M deﬁned in the following wa y: (i) c ho ose a 1 ∈ π 1 M so that | a 1 | takes the minimal v alue, (ii) c ho ose a 2 ∈ π 1 M \ ⟨ a 1 ⟩ so that | a 2 | takes the minimal v alue, (iii) c ho ose a 3 ∈ π 1 M \ ⟨ a 1 , a 2 ⟩ so that | a 3 | takes the minimal v alue, and so on. Note that the sequence terminates at the n -th step if a 1 , . . . , a n gen- erate π 1 M . By construction, w e ha v e | a j · a − 1 i | ⩾ | a j | ⩾ | a i | for any j > i . Set p i = a i · p . Note that | p j − p i | ˜ M = | a j · a − 1 i | ⩾ ⩾ | a j | = = | p j − p | ˜ M ⩾ ⩾ | a i | = = | p i − p | ˜ M . By the T op onogov comparison theorem w e get ∡ [ p p i p j ] ⩾ π 3 . That is, the directions from p to all p i mak e an angle of at least π 3 with eac h other. 106 CHAPTER 6. A CTIONS AND COVERINGS Therefore the num b er of p oints p i can b e b ounded in terms of the dimension of M . Hence the result follows. The short-b asis c onstruction , as w ell as the result ab ov e are due to Mikhael Gromov [19]. An equation in a Lie group. W e will assume that G is equipp ed with a bi-in v arian t metric. In particular, geo desics starting at the iden tity elemen t e ∈ G are given b y homomorphisms R → G . Consider the map ϕ : G → G deﬁned by ϕ ( x ) = x · g 1 · x · g 2 · · · x · g n . W e need to show that ϕ is onto. Note that it is suﬃcient to show that ϕ has a non-zero degree. The map ϕ is homotopic to the map ψ : x 7→ x n . Therefore it is suﬃcien t to show that ( ∗ ) deg ψ  = 0 Note that the claim ( ∗ ) follows from ( ∗∗ ) . ( ∗∗ ) F or any x ∈ G the diﬀer ential d x ψ : T x G → T x n G do es not r evert orientation. Indeed, connect e to a given p oin t y ∈ G b y a geo desic path γ , so γ (0) = e and γ (1) = y . Since γ : R → G is a homomorphism, ψ ( x ) = y for x = γ ( 1 n ) . In particular, the in v erse image ψ − 1 { y } is nonempty for an y y ∈ G . By ( ∗∗ ) , for a regular v alue y , eac h p oin t in the inv erse image ψ − 1 { y } con tributes 1 to the degree of ψ . Hence ( ∗ ) follo ws. It remains to pro ve ( ∗∗ ) . Given an element g ∈ G , denote b y L g and R g the left and righ t shift G → G resp ectiv ely; that is, L g ( x ) = g · x and R g ( x ) = x · g . Iden tify the tangent spaces T x G and T x n G with the Lie algebra g = = T e G using dR x : g → T x G and dR n x : g → T x n G resp ectiv ely . Then for an y V ∈ g , we hav e d x ψ ( V ) = V + Ad x ( V ) + · · · + Ad n − 1 x ( V ) , where Ad x = d e ( L x ◦ R x − 1 ) : g → g . Since the metric on G is bi-in v ariant, Ad x is an isometry of g . It remains to note that the linear transformation V 7→ V + T ( V ) + · · · + T n − 1 ( V ) 107 cannot rev ert orientation for any isometric linear transformation T of the Euclidean space. The last statement is an exercise in linear algebra. The idea of this solution is due to Murray Gerstenhab er and Oscar Rothaus [209]. In fact, the degree of g is n k , where k is the rank of G [210]. Quotien t of Hilb ert space. W e consider S 3 as the set of unit quater- nions; in particular, it has a group structure. Let H b e the set of paths of class W 1 , 2 in S 3 starting at the identit y elemen t e ; that is, the path’s v elo city is square-integrable. The p oint wise m ultiplication of paths deﬁnes a group structure on H . Denote b y Ω the subset of all lo ops in H . It remains to equip H with the structure of a Hilb ert space so that the righ t action of Ω on H is isometric and the quotient is isometric to S 3 . W e will prov e the statement for any connected Lie group G with a bi-in v ariant metric; in particular, for G = S 3 . Denote b y g = T e G the Lie algebra of G . Equip G with a bi-in v ariant metric, and let ⟨∗ , ∗⟩ g b e the corresp onding scalar pro duct in g . Consider the Hilbert space H of all L 2 -functions f : [0 , 1] → g with the scalar pro duct deﬁned by ⟨ f , g ⟩ = 1 w 0 ⟨ f ( t ) , g ( t ) ⟩ g · dt. Construction of the quotient map ϕ : H → G . Giv en v ∈ g , denote by ˜ v the corresp onding righ t-in v ariant tangen t ﬁeld on G . Giv en f : [0 , 1] → g in H , consider the path Γ f : [0 , 1] → G with Γ f (0) = 1 and Γ ′ f ( t ) = ˜ f ( t ) for an y t . The map ϕ : H → G is the ev aluation map ϕ : f 7→ Γ f (1) . Since G is connected, ϕ is onto. Gr oup structur e on H . Note that the functional f 7→ Γ f is an injectiv e map from H to the space of paths in G starting at e . Giv en α ∈ G , we denote b y Ad α : g → g its the adjoint transfor- mation; that is, Ad α = d e Inn α , where Inn α : x 7→ α · x · α − 1 is the inner automorphism of G . Note that Ad α preserv es the scalar pro duct on g . Consider the multiplication  on H deﬁned by ( ∗ ) ( h  f )( t ) = h ( t ) + Ad Γ h ( t ) [ f ( t )] . 108 CHAPTER 6. A CTIONS AND COVERINGS Note that Γ h⋆f ( t ) = Γ h ( t ) · Γ f ( t ) for any t ∈ [0 , 1] . In particular, ( H ,  ) is a group with neutral element 0 . F rom ( ∗ ) , we get ( h  f )( t ) − ( h  g )( t ) = Ad Γ h ( t ) ( f ( t ) − g ( t )) and therefore | ( f  h )( t ) − ( g  h )( t ) | = | f ( t ) − g ( t ) | for an y t . It follows that for any ﬁxed h , the transformation f 7→ h  f is an aﬃne isometry of H . The set Ω = ϕ − 1 { e } is a subgroup of ( H ,  ) ; it can b e viewed as the group of W 1 , 2 -lo ops in G . It remains to note that ϕ : H → G is the quotien t map for the right action of Ω on H . A lternative solution. Again, w e will prov e the statement for any connected Lie group G with a bi-inv arian t metric. Denote by G n the direct pro duct of n copies of G . Consider the map ϕ n : G n → G deﬁned by ϕ n : ( α 1 , . . . , α n ) 7→ α 1 · · · α n . Note that ϕ n is the quotient map for the G n − 1 -action on G n deﬁned by ( β 1 , . . . , β n − 1 ) · ( α 1 , . . . , α n ) = ( α 1 · β − 1 1 , β 1 · α 2 · β − 1 2 , . . . , β n − 1 · α n ) . Denote b y ρ n the pro duct metric on G n rescaled with factor √ n . Note that the quotient ( G n , ρ n ) /G n − 1 is isometric to G = ( G, ρ 1 ) . As n → ∞ the curv ature of ( G n , ρ n ) conv erges to zero and its injec- tivit y radius go es to inﬁnity . Therefore passing to the ultra-limit of G n as n → ∞ w e get a Hilb ert space. It remains to observe that the limit action has the required prop erty . This construction is given by Chuu-Lian T erng and Gudlaugur Thor- b ergsson [see section 4 in 211]; the alternative solution was suggested by Alexander Lytchak. Instead of the group Ω , one could consider the subgroup Ω H of paths γ : [0 , 1] → G such that the pair ( γ (0) , γ (1)) b elongs to a given subgroup H < G × G . In this case, the quotient H / Ω H is isometric to the dou- ble quotient G/ /H ; that is, the quotient of the action on G deﬁned by ( h 1 , h 2 ) · g = h 1 · g · h − 1 2 for ( h 1 , h 2 ) ∈ H < G × G . The following question is op en. 109  Supp ose R is a c omp act simply-c onne cte d Riemannian manifold that is isometric to a quotient of the Hilb ert sp ac e by a gr oup of isometries (or mor e gener al ly R is the tar get of Riemannian submersion fr om a Hilb ert sp ac e). Is it true that R is isometric to a double quotient? That is, is it true that R is a quotient of c omp act Lie gr oup G by a gr oup of isometries? Chapter 7 T op ology In this chapter, we consider geometrical problems with strong top ological ﬂa vor. A t ypical introductory course in top ology , say [212], contains all the necessary material. Isotop y Recall that an isotopy is a contin uous one-parameter family of embed- dings.  L et K 1 and K 2 b e home omorphic close d subsets of the c o or dinate sub- sp ac e R m in R 2 · m . Show that ther e is a home omorphism h : R 2 · m → R 2 · m such that K 2 = h ( K 1 ) . Mor e over, h c an b e chosen to b e isotopic to the identity map. Semisolution. Cho ose a homeomorphism ϕ : K 1 → K 2 . By the Tietze extension theorem, the homeomorphisms ϕ : K 1 → K 2 and ϕ − 1 : K 2 → K 1 can b e extended to contin uous maps; denote these maps by f : R m → R m and g : R m → R m resp ectiv ely . R m R m K 1 K 2 h 1 h 2 h 3 Consider homeomorphisms h 1 , h 2 , and h 3 of R m × R m deﬁned in the following wa y: h 1 ( x, y ) = ( x, y + f ( x )) , h 2 ( x, y ) = ( x − g ( y ) , y ) , h 3 ( x, y ) = ( y , − x ) . 110 111 It remains to observe that each homeomorphism h i is isotopic to the iden tity map and K 2 = h 3 ◦ h 2 ◦ h 1 ( K 1 ) . □ This construction is due to Victor Klee [213] and it is called Kle e’s trick . This trick is used in the ﬁve-line pro of of the Jordan separation theorem b y P atric k Do yle [214]; a pro of of the separation theorem for em b eddings S n  → S n +1 can b e giv en using the same idea [215]. The problem “Monotonic homotopy” on page 133 lo oks similar. Immersed disks T wo immersions f 1 and f 2 of the disk D into the plane will b e called essential ly diﬀer ent if there is no diﬀeomorphism h : D → D such that f 1 = f 2 ◦ h .  Construct two essential ly diﬀer ent smo oth immersions of the disk into the plane that c oincide ne ar the b oundary. P ositive Dehn t wist h − − → Let Σ b e a surface and γ : R / Z → Σ b e a non-contractible closed simple curve. Let U γ b e a neigh b orho o d of γ that admits a parametrization ι : R / Z × (0 , 1) → U γ . A Dehn twist along γ is a homeomorphism h : Σ → Σ that is the identit y outside of U γ and such that ι − 1 ◦ h ◦ ι : ( x, y ) 7→ ( x + y , y ) . If Σ is oriented and ι is orientation preserving, then the Dehn twist describ ed ab o ve is called p ositive .  L et Σ b e a c omp act oriente d surfac e with nonempty b oundary. Pr ove that any c omp osition of p ositive Dehn twists of Σ is not homotopic to the identity r elative to the b oundary. In other wor ds, any pr o duct of p ositive Dehn twists r epr esents a non- trivial class in the mapping class gr oup of Σ . 112 CHAPTER 7. TOPOLOGY Conic neigh b orho o d Let p b e a p oin t in a top ological space X . W e say that an op en neigh b or- ho od U ∋ p is c onic if there is a homeomorphism from a cone to U that sends the vertex to p .  Show that any two c onic neighb orho o ds of one p oint ar e home omorphic to e ach other. Note that tw o cones Cone(Σ 1 ) and Cone(Σ 2 ) might b e homeomorphic while Σ 1 and Σ 2 are not; the existence of suc h examples follo ws from the double susp ension theorem. Unknots ◦  Pr ove that the set of smo oth emb e ddings f : S 1 → R 3 e quipp e d with the C 0 -top olo gy forms a c onne cte d sp ac e. Stabilization  Construct two c omp act subsets K 1 , K 2 ⊂ R 2 such that K 1 is not home omorphic to K 2 , but K 1 × [0 , 1] is home omorphic to K 2 × [0 , 1] . Homeomorphism of a cub e  L et □ b e a cub e in R m and h : □ → □ b e a home omorphism that sends e ach fac e of □ to itself. Extend h to a home omorphism f : R m → R m that c oincides with the identity map outside of a b ounde d set. Finite top ological space ◦  Given a ﬁnite top olo gic al sp ac e F , c onstruct a ﬁnite simplicial c omplex K that admits a we ak homotopy e quivalenc e K → F . Dense homeomorphism ◦  Denote by H b e the set of al l orientation pr eserving home omorphisms S 2 → S 2 e quipp e d with the C 0 -metric. Show that ther e is a home omor- phism h ∈ H such that its c onjugations a ◦ h ◦ a − 1 for al l a ∈ H form a dense set in H . 113 Simple path ◦  L et p and q b e distinct p oints in a Hausdorﬀ top olo gic al sp ac e X . Assume that p and q ar e c onne cte d by a p ath. Show that they c an b e c onne cte d by a simple p ath; that is, ther e is an inje ctive c ontinuous map β : [0 , 1] → X such that β (0) = p and β (1) = q . (This statemen t might b e intuitiv ely obvious, but its pro of is not sim- ple.) P ath on a surface ◦  Show that any p ath with distinct ends in a surfac e is homotopic (r el- ative to the ends) to a simple p ath. Semisolutions Immersed disks. Both circles on the picture b ound essen tially diﬀeren t disks. On the ﬁrst diagram, the dashed lines and the solid lines together b ound three embedded disks; gluing these disks along the dashed lines gives the ﬁrst immersion. The reﬂection of this immersion across the vertical line of symmetry gives another essen tially diﬀeren t immersion. It is a goo d exercise to count the essentially diﬀerent disks in the second example. (The answer is 5.) The existence of examples of that type is gener- ally attributed to John Milnor [216]. An easier problem w ould be to construct t wo essen tially diﬀerent immersions of annuli with the same b oundary curv es; a solution is shown on the picture [for more details and references see 217]. P ositive Dehn twist. Consider the universal cov ering f : ˜ Σ → Σ . The surface ˜ Σ has a b oundary and it comes with the orientation induced from Σ . Cho ose a p oint x 0 on the b oundary ∂ ˜ Σ . Given tw o other p oints y and z in ∂ ˜ Σ , we will write z ≻ y if y lies on the righ t side from a simple curv e from x 0 to z in ˜ Σ . Note that ≻ deﬁnes a linear order on ∂ ˜ Σ \ { x 0 } . W e will write z ⪰ y if z ≻ y or z = y . 114 CHAPTER 7. TOPOLOGY x 0 z y Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Note that any homeomorphism h : Σ → Σ iden- tical on the b oundary lifts to the unique homeomor- phism ˜ h : ˜ Σ → ˜ Σ such that ˜ h ( x 0 ) = x 0 . The following claim is the key step in the pro of: ( ∗ ) If h is a p ositive Dehn twist along a close d curve γ , then y ⪰ ˜ h ( y ) for any y ∈ ∂ ˜ Σ \ { x 0 } and y 0 ≻ ˜ h ( y 0 ) for some y 0 ∈ ∂ ˜ Σ \ { x 0 } . Note that the problem follows from ( ∗ ) . Indeed, the prop erty in ( ∗ ) is a homotopy inv ariant and it survives under comp ositions of maps. If Σ is not an annulus, then by the uniformization theorem we can assume that Σ has a hyperb olic metric with geo desic b oundary; the lifted metric on ˜ Σ has the same prop erties. F urthermore, we can assume that (1) γ is a closed geo desic, (2) the parametrization ι : R / Z × (0 , 1) → U γ from the deﬁnition of Dehn t wist is rotationally symmetric and (3) for an y u ∈ R / Z the arc ι ( u × (0 , 1)) is a geo desic p erpendicular to γ . Consider the p olar co ordinates ( ϕ, ρ ) on ˜ Σ with the origin at x 0 ; since x 0 lies on the b oundary , the angle co ordinate ϕ is deﬁned in [0 , π ] . By construction of the Dehn twist, we get ϕ ( x ) ⩾ ϕ ◦ ˜ h ( x ) for any x  = x 0 , and if the geo desic [ x 0 x ] crosses f − 1 ( U γ ) , then ϕ ( x ) > ϕ ◦ ˜ h ( x ) . In particular, if x lies on the b oundary then ˜ h ( x ) lies on the right of the geo desic [ x 0 x ] ; hence the claim ( ∗ ) follows. ϕ ( x ) x 0 x ˜ h ( x ) f − 1 ( U γ ) f − 1 ( U γ ) f − 1 ( U γ ) f − 1 ( U γ ) f − 1 ( U γ ) f − 1 ( U γ ) f − 1 ( U γ ) f − 1 ( U γ ) f − 1 ( U γ ) f − 1 ( U γ ) f − 1 ( U γ ) f − 1 ( U γ ) f − 1 ( U γ ) f − 1 ( U γ ) f − 1 ( U γ ) f − 1 ( U γ ) f − 1 ( U γ ) f − 1 ( U γ ) f − 1 ( U γ ) f − 1 ( U γ ) f − 1 ( U γ ) f − 1 ( U γ ) f − 1 ( U γ ) f − 1 ( U γ ) f − 1 ( U γ ) f − 1 ( U γ ) f − 1 ( U γ ) f − 1 ( U γ ) f − 1 ( U γ ) f − 1 ( U γ ) f − 1 ( U γ ) f − 1 ( U γ ) f − 1 ( U γ ) f − 1 ( U γ ) f − 1 ( U γ ) f − 1 ( U γ ) f − 1 ( U γ ) f − 1 ( U γ ) f − 1 ( U γ ) f − 1 ( U γ ) f − 1 ( U γ ) f − 1 ( U γ ) f − 1 ( U γ ) f − 1 ( U γ ) f − 1 ( U γ ) f − 1 ( U γ ) f − 1 ( U γ ) f − 1 ( U γ ) f − 1 ( U γ ) f − 1 ( U γ ) f − 1 ( U γ ) f − 1 ( U γ ) f − 1 ( U γ ) f − 1 ( U γ ) f − 1 ( U γ ) f − 1 ( U γ ) f − 1 ( U γ ) f − 1 ( U γ ) f − 1 ( U γ ) f − 1 ( U γ ) f − 1 ( U γ ) f − 1 ( U γ ) f − 1 ( U γ ) f − 1 ( U γ ) f − 1 ( U γ ) f − 1 ( γ ) f − 1 ( γ ) f − 1 ( γ ) f − 1 ( γ ) f − 1 ( γ ) f − 1 ( γ ) f − 1 ( γ ) f − 1 ( γ ) f − 1 ( γ ) f − 1 ( γ ) f − 1 ( γ ) f − 1 ( γ ) f − 1 ( γ ) f − 1 ( γ ) f − 1 ( γ ) f − 1 ( γ ) f − 1 ( γ ) f − 1 ( γ ) f − 1 ( γ ) f − 1 ( γ ) f − 1 ( γ ) f − 1 ( γ ) f − 1 ( γ ) f − 1 ( γ ) f − 1 ( γ ) f − 1 ( γ ) f − 1 ( γ ) f − 1 ( γ ) f − 1 ( γ ) f − 1 ( γ ) f − 1 ( γ ) f − 1 ( γ ) f − 1 ( γ ) f − 1 ( γ ) f − 1 ( γ ) f − 1 ( γ ) f − 1 ( γ ) f − 1 ( γ ) f − 1 ( γ ) f − 1 ( γ ) f − 1 ( γ ) f − 1 ( γ ) f − 1 ( γ ) f − 1 ( γ ) f − 1 ( γ ) f − 1 ( γ ) f − 1 ( γ ) f − 1 ( γ ) f − 1 ( γ ) f − 1 ( γ ) f − 1 ( γ ) f − 1 ( γ ) f − 1 ( γ ) f − 1 ( γ ) f − 1 ( γ ) f − 1 ( γ ) f − 1 ( γ ) f − 1 ( γ ) f − 1 ( γ ) f − 1 ( γ ) f − 1 ( γ ) f − 1 ( γ ) f − 1 ( γ ) f − 1 ( γ ) f − 1 ( γ ) If Σ is an annulus, then the same argument works except we ha ve to c ho ose a ﬂat metric on Σ . In this case, ˜ Σ is a strip b etw een tw o parallel lines in the plane, see the diagram. Note that if the surface has an empty boundary , then the answ er is diﬀeren t: 115  Construct a c omp osition of p ositive Dehn twists on a c omp act oriente d surfac e without b oundary that is homotopic to the identity. I learned the problem from Rostisla v Matvey ev. The describ ed con- struction was giv en b y Hamish Short and Bert Wiest [218] and attributed to William Thurston. It turns out that there is no upp er b ound on the length of the pro duct. Namely , there is a compact orien ted surface Σ with a nonempty boundary and a homeomorphism f : Σ → Σ that do es not mov e b oundary p oints suc h that f is homotopic relative to the b oundary to an arbitrary long pro duct of p ositive Dehn twists. Suc h an example was constructed by Reﬁk ˙ Inan¸ c Ba ykur and Jerem y V an Horn-Morris [219]; see also [220]. Conic neighborho o d. Let V and W b e tw o conic neigh b orho o ds of p . Without loss of generalit y , w e ma y assume that V ⋐ W ; that is, the closure of V lies in W . W e will need to construct a sequence of embeddings f n : V → W such that ⋄ F or any compact set K ⊂ V there is a p ositive integer n = n K suc h that f n ( k ) = f m ( k ) for any k ∈ K and m, n ⩾ n K . ⋄ F or an y p oint w ∈ W there is a p oin t v ∈ V such that f n ( v ) = w for all large n . Note that once such a sequence is constructed, f : V → W deﬁned by f ( v ) = f n ( v ) for all large v alues of n gives the needed homeomorphism. The sequence f n can b e constructed recursively f n +1 = Ψ n ◦ f n ◦ Φ n , where Φ n : V → V and Ψ n : W → W are homeomorphisms of the form Φ n ( x ) = ϕ n ( x ) ∗ x and Φ n ( x ) = ψ n ( x )  x, where ϕ n : V → R ⩾ 0 , ψ n : W → R ⩾ 0 are suitable contin uous functions; “ ∗ ” and “  ” denote the multiplic ation in the cone structures of V and W resp ectiv ely . The problem is due to Kyung Whan Kwun [221]. Unknots. Observ e that it is p ossible to dra w an arbitrary tight knot while keep- ing it smo othly embedded at all times including the last moment. 116 CHAPTER 7. TOPOLOGY This problem was suggested by Greg Kup erb erg. Stabilization. The example can b e guessed from the diagram. K 1 K 2 The tw o sets K 1 and K 2 are sub- spaces of the plane, each one b eing a closed ann ulus with tw o attached line segmen ts. In K 1 one segmen t is at- tac hed from the inside and another from the outside; in K 2 b oth segmen ts are attached from the outside. The pro duct spaces K 1 × [0 , 1] and K 2 × [0 , 1] are solid tori with attac hed rectangles. A homeomorphism K 1 × [0 , 1] → K 2 × [0 , 1] can b e constructed by twisting a part of one solid torus. T o prov e the nonexistence of a homeomorphism K 1 → K 2 consider the sets of cut points V i ⊂ K i and the sets W i ⊂ K i of points that admit a punctured simply-connected neighborho o d. Note that the set V i is the union of the attached line segments and W i is the b oundary of the annulus without points where the segments are attached. Note that V i ∪ W i = ∂ K i ; in particular, a homeomorphism K 1 → K 2 (if it exists) sends ∂ K 1 to ∂ K 2 . Finally , note that each ∂ K i has tw o connected comp onents and V 1 in tersects both components of ∂ K 1 while V 2 lies in one component of ∂ K 2 . Hence K 1 ≇ K 2 . It should b e an old puzzle; I learned it from Maria Goluzina around 1988. Homeomorphism of a cub e. Let us extend the homeomorphism h to R m b y reﬂecting the cub e across its facets. W e get a homeomorphism ˜ h : R m → R m suc h that ˜ h ( x ) = h ( x ) for any x ∈ □ and ˜ h ◦ γ = γ ◦ ˜ h, where γ is any reﬂection with resp ect to the facets of the cub e. Without loss of generalit y , w e ma y assume that the cub e □ is inscribed in the unit sphere cen tered at the origin of R m . In this case, ˜ h has displac ement at most 2 ; that is, | ˜ h ( x ) − x | ⩽ 2 for any x ∈ R m . Cho ose a smo oth increasing function ϕ : R ⩾ 0 → R such that ϕ ( r ) = r for r ⩽ 1 and ϕ ( r ) → 2 as r → ∞ . Equip R m with polar coordinates ( u, r ) , where u ∈ S m − 1 , r ⩾ 0 . Consider a homeomorphism Φ from R m to an op en ball of radius 2 deﬁned b y Φ( u, r ) = ( u, ϕ ( r )) . 117 □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ Set f ( x ) = " x if | x | ⩾ 2 , Φ ◦ ˜ h ◦ Φ − 1 ( x ) if | x | < 2 . It remains to observe that f : R m → R m is a solution. This problem is stripp ed from the pro of of Robion Kirby [222]. The condition that eac h face is mapp ed to itself can be remo v ed and instead of homeomorphism one could tak e an y em b edding close to the identit y . An in teresting twist to this idea was given by Dennis Sulliv an [see 167 and also 5.10 in 168]. Instead of the discrete group of motions of the Eu- clidean space, he uses a discrete group of motions of the hyperb olic space in the conformal disk mo del. T o see the idea, note that the construction of ˜ h can b e done for a Coxeter p olytop e in the h yp erb olic space instead of a cub e. 1 Then the constructed map ˜ h coincides with the identit y on the absolute and therefore the last “shrinking” step in the pro of ab o ve is not needed. Moreo ver, if the original homeomorphism is bi-Lipschitz, then the Sulliv an construction pro duces a bi-Lipschitz homeomorphism — this is its main adv antage. Finite top ological space. Given a point p ∈ F , denote by O p the minimal op en set in F containing p . Note that we can assume that F is a connected T 0 -space; in particular, O p = O q if and only if p = q . Let us write p ≼ q if O p ⊂ O q . Evidently , ≼ is a partial order on F . Let us construct a simplicial complex K b y taking F as the set of v ertices and declaring a collection of v ertices to b e a simplex if it can b e linearly ordered with resp ect to ≼ . Giv en k ∈ K , consider the minimal simplex ( f 0 , . . . , f m ) ∋ k ; w e can assume that f 0 ≼ · · · ≼ f m . Set h : k 7→ f 0 ; it deﬁnes a map K → F . It remains to chec k that h is con tinuous and induces isomorphisms for all the homotopy groups. In a similar fashion, one can construct a ﬁnite top ological space F for any given simplicial complex K suc h that there is a weak homotopy equiv alence K → F . Both constructions are due to Pa v el Alexandrov [225, 226]. Dense homeomorphism. Note that there is a countable set of homeo- morphisms h 1 , h 2 , . . . that is dense in H such that each h n ﬁxes all the p oin ts outside an op en round disk, say D n . 1 By Vinberg’s theorem [223, 224] h yp erb olic space of large dimension has no Coxeter polytop es, but the idea works after some modiﬁcations. 118 CHAPTER 7. TOPOLOGY Cho ose a countable disjoin t collection of round disks D ′ n . Consider the homeomorphism h : S 2 → S 2 that ﬁxes all the p oints outside of S n D ′ n and for each n , the restriction h | D ′ n is conjugate to h n | D n . Note that for large n , the homeomorphism h is conjugate to a home- omorphism close to h n . Therefore h is a solution. The problem was mentioned b y F rederic Le Ro x [227] on a problem section at a conference in Ob erw olfach, where he also conjectured that this is not true for the area-preserving homeomorphisms. An aﬃrmative answ er to this conjecture was giv en b y Daniel Dore, Andrew Hanlon, and Sobhan Seyfaddini [228, 229]. In particular, it implies the following seemingly evident but nontrivial statement.  Given ε > 0 , ther e is δ > 0 such that Ω ∩ h (Ω)  = ∅ for any top olo gic al disk Ω ⊂ S 2 with ar e a at le ast ε and any ar e a-pr eserving home omorphism h : S 2 → S 2 with displac ement at most δ ; that is, such that | h ( x ) − x | S 2 < δ for any x ∈ S 2 . Simple path. W e will give tw o solutions, the ﬁrst one is elementary and the second one is inv olved. First solution. Let α b e a path connecting p to q . P assing to a subin terv al if necessary , we can assume that α ( t )  = p, q for t  = 0 , 1 . An op en set Ω in (0 , 1) will b e called suitable if, for any connected comp onen t ( a, b ) of Ω , w e hav e α ( a ) = α ( b ) . Since the union of nested suitable sets is suitable, we can ﬁnd a maximal suitable set ˆ Ω . Deﬁne β ( t ) = α ( a ) for any t in a connected comp onen t ( a, b ) ⊂ Ω . Note that β is contin uous and monotonic; that is, for any x ∈ [0 , 1] the set β − 1 { β ( x ) } is connected. It remains to reparametrize β to make it injective. In other words, we need to construct a non-decreasing surjective function τ : [0 , 1] → [0 , 1] suc h that τ ( t 1 ) = τ ( t 2 ) if and only if there is a connected comp onen t ( a, b ) suc h that t 1 , t 2 ∈ [ a, b ] . The construction is similar to the construction of the devil’s staircase. Se c ond solution. Note that one can assume that X coincides with the image of α . In particular, X is a connected lo cally connected compact Hausdorﬀ space. An y such space admits a length-metric. This statement is not at all trivial; it was conjectured by Karl Menger [230] and pro v ed independently b y R. H. Bing [231, 232] and Edwin Moise [233]. 119 It remains to consider a geo desic path from p to q . The problem w as inspired b y a lemma pro ved by Alexander Lytchak and Stefan W enger [see 7.13 in 234]. P ath on a surface. Denote the surface b y Σ ; assume that the path runs from p to q . The follo wing picture suggests an idea for an induction on the num b er of self-crossings. T o do the pro of formally , let us presen t the path as a concatenation α ∗ β of tw o paths such that α is simple and β does not pass thru p . W e can assume that β : [0 , 1] → Σ is smo oth. Cho ose a smo oth time-dep enden t vector ﬁeld V t on Σ such that V t ( β ( t )) = β ′ ( t ) and V t ( p ) = 0 for any t ∈ [0 , 1] . Consider the ﬂow Φ t : Σ → Σ along V t ; that is, Φ 0 ( x ) = x and d dt (Φ t ( x )) = V t (Φ t ( x )) for any t ∈ [0 , 1] and x ∈ Σ . The map Φ 1 : Σ → Σ is a diﬀeomorphism; in particular, Φ 1 sends the simple path α to a simple path α 1 = Φ 1 ◦ α . By construction α 1 (1) = q . Since V t ( p ) = 0 for an y t , we hav e α 1 (0) = p . That is, the path α 1 runs from p to q . It remains to show that α 1 is homotopic to α ∗ β relative to the ends. Set α τ = Φ τ ◦ α and denote by β τ the path running along β from β ( τ ) to q ; that is, β τ ( t ) = β ( τ + 1 1 − τ · t ) . The concatenation α τ ∗ β τ pro vides a homotop y from α ∗ β to α 1 ∗ β 1 . Since β 1 is a constan t path, α ∗ β is homotopic to α 1 . Hence the statemen t follo ws. This is a stripp ed version of the problem suggested by Jaros l a w Ke ˛dra [2]; it was used by Michael Khanevsky [Lemma 3 in 235]. Chapter 8 Piecewise linear geometry A p olyhe dr al sp ac e is a complete length-metric space that admits a lo cally ﬁnite triangulation suc h that eac h simplex is isometric to a simplex in a Euclidean space. By a triangulation of a polyhedral space, we alwa ys mean a triangulation of that type. A p oint in a p olyhedral space is called r e gular if it has a neighborho o d isometric to an op en set in a Euclidean space; otherwise, it is called singular . If we replace the Euclidean spaces by the unit spheres or the h yp erb olic spaces, we arrive at the deﬁnition of spheric al and hyp erb olic p olyhe dr al sp ac es resp ectively . The term pie c ewise typically means that there is a triangulation with some prop erty on each triangle. F or example, if P and Q are p olyhedral spaces, then ⋄ a map f : P → Q is called pie c ewise distanc e-pr eserving if there is a triangulation T of P such that for any simplex ∆ ∈ T the restriction f | ∆ is distance-preserving; ⋄ a map h : P → Q is called pie c ewise line ar if b oth spaces P and Q admit triangulations suc h that each simplex of P is mapp ed to a simplex of Q by an aﬃne map. In particular, a pie c ewise line ar home omorphism is a piecewise linear map which is a homeomor- phism. Spherical arm lemma Recall that a p olygon without self-intersections is called simple .  L et A = [ a 1 . . . a n ] and B = [ b 1 . . . b n ] b e two simple spheric al p olygons with e qual c orr esp onding sides. Assume that A lies in a hemispher e and ∡ a i ⩾ ∡ b i for e ach i . Show that A is c ongruent to B . 120 121 Semisolution. Cut out A from the sphere and glue B in its place. Denote b y Σ the obtained spherical p olyhedral space. Note that ⋄ Σ is homeomorphic to S 2 . ⋄ Σ has curv ature ⩾ 1 in the sense of Alexandrov; that is, the total angle around each singular p oint is less than 2 · π . ⋄ All the singular p oints of Σ lie outside of an isometric copy of a hemisphere S 2 + ⊂ Σ . Denote by n the num b er of singular p oints in Σ . It is suﬃcient to sho w that n = 0 . Assume the contrary; that is, n ⩾ 1 . W e can assume that n takes the minimal p ossible v alue. Clearly , n > 1 ; that is, Σ cannot hav e a single singular p oint. There- fore w e can choose tw o singular p oints p, q ∈ Σ . Cut Σ along a geo desic [ pq ] . The obtained hole can b e patched so that we obtain a new p olyhe- dral space Σ ′ of the same type but with n − 1 singular p oin ts. Since n is minimal, we arrive at a contradiction. Namely , if the total angles around p and q are 2 · π − α and 2 · π − β resp ectiv ely , consider the spherical triangle △ with the base | p − q | Σ and the adjacent angles α 2 , β 2 . The needed patch is obtained b y doubling △ along its lateral sides. A lternative end of the pr o of. By the Alexandro v em b edding theorem, Σ is isometric to the surface of a conv ex p olyhedron P in the unit sphere S 3 . The cen ter of the hemisphere has to lie in a facet of P , say F . It remains to note that F contains the equator and therefore P has to b e a hemisphere in S 3 or an intersection of tw o hemispheres. In b oth cases, its surface is isometric to S 2 . The problem is due to Victor Zalgaller [236]; the result of Victor T o- p onogo v in [237] gives a smo oth analog of this statement. The patch construction ab ov e was introduced b y Aleksandr Alexandro v in his pro of of conv ex embeddability of p olyhedra [see VI, §7 in 238]. The alternative end of the pro of is taken from [131]. T riangulation of 3-sphere  Construct a triangulation of S 3 with 100 vertic es such that any two vertic es ar e c onne cte d by an e dge. F olding problem  L et P b e a c omp act 2 -dimensional p olyhe dr al sp ac e. Construct a pie c e- wise distanc e-pr eserving map f : P → R 2 . 122 CHAPTER 8. PIECEWISE LINEAR GEOMETR Y Piecewise distance-preserving extension  Pr ove that any 1-Lipschitz map fr om a ﬁnite subset F ⊂ R 2 to R 2 c an b e extende d to a pie c ewise distanc e-pr eserving map R 2 → R 2 . Closed p olyhedral surface  Construct a close d p olyhe dr al surfac e Σ in R 3 with nonp ositive curva- tur e; that is, the total angle ar ound e ach vertex of Σ is at le ast 2 · π . Minimal p olyhedral disk By a p olyhedral disk in R 3 w e mean a triangulation of a plane polygon P with a map P → R 3 that is aﬃne on each triangle. The area of the p olyhedral disk is deﬁned as the sum of areas of the images of the triangles in the triangulation.  Consider the class of p olyhe dr al disks glue d fr om n triangles in R 3 with a ﬁxe d br oken line as the b oundary. L et Σ n b e a disk of minimal ar e a in this class. Show that Σ n is a saddle surface ; that is, a plane c annot cut al l the e dges c oming fr om one of the interior vertic es of Σ n . Coheren t triangulation ◦ A triangulation of a con vex p olygon is called coherent if there is a conv ex function that is linear on each triangle and changes its gradient on every edge of the triangulation.  Find a non-c oher ent triangulation of a triangle. Sphere with one edge ∗  Construct a p olyhe dr al sp ac e that is home omorphic to S 3 and such that its singular lo cus is forme d by a cir cle. T riangulation of a torus  Show that the torus do es not admit a triangulation such that one vertex has 5 e dges, one has 7 e dges and al l other vertic es have 6 e dges. No simple geo desics ◦  Construct a c onvex p olyhe dr on P whose surfac e do es not have a close d simple ge o desic. 123 Semisolutions T riangulation of 3-sphere. Cho ose 100 distinct p oin ts p 1 . . . , p 100 on the moment curve γ : t 7→ ( t, t 2 , t 3 , t 4 ) in R 4 . Denote b y P the conv ex hull of { p 1 , . . . , p 100 } . The surface of P is homeomorphic to S 2 . Therefore it is suﬃcient to sho w that any tw o vertices of P are connected by an edge. The latter follo ws from the following claim. ( ∗ ) Given two p oints p and q on γ , ther e is a hyp erplane H in R 4 that interse cts γ only at p and q and le aves γ on one side. T o prov e the claim, assume that p = γ ( t 1 ) and q = γ ( t 2 ) . Consider the p olynomial f ( t ) = a + b · t + c · t 2 + d · t 3 + t 4 = ( t − t 1 ) 2 · ( t − t 2 ) 2 . Clearly , f ( t ) ⩾ 0 , and the equality holds only at t 1 and t 2 . It follows that the aﬃne function  : R 4 → R deﬁned by  : ( w , x, y , z ) 7→ a + b · w + c · x + d · y + z is nonnegative at the p oin ts of γ and v anishes only at p and q . Therefore the zero-set of  is the required hyperplane H in ( ∗ ) . The p olyhedron P abov e is an example of the so-called cyclic p olytop es . F olding problem. Giv en a triangulation of P , consider the V oronoi domain V v for eac h v ertex v ; that is, V v is the set of all points in P closer to v than to any other vertex. Note that the triangulation can b e sub divided if necessary so that the V oronoi domain of each vertex is isometric to a conv ex subset in the cone with the vertex at its tip. The b oundaries of all the V oronoi domains form a graph with straigh t edges. Let us triangulate P so that each triangle has one of those edges as the base and the opp osite vertex is the center of an adjacent V oronoi domain; such a vertex will b e called the main vertex of the triangle. θ a b x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w V v V v V v V v V v V v V v V v V v V v V v V v V v V v V v V v V v V v V v V v V v V v V v V v V v V v V v V v V v V v V v V v V v V v V v V v V v V v V v V v V v V v V v V v V v V v V v V v V v V v V v V v V v V v V v V v V v V v V v V v V v V v V v V v V v V w V w V w V w V w V w V w V w V w V w V w V w V w V w V w V w V w V w V w V w V w V w V w V w V w V w V w V w V w V w V w V w V w V w V w V w V w V w V w V w V w V w V w V w V w V w V w V w V w V w V w V w V w V w V w V w V w V w V w V w V w V w V w V w V w Cho ose a solid triangle △ = [ v ab ] in the constructed triangulation; let v b e its main v ertex. Given a p oint x ∈ △ , set ρ ( x ) = | x − v | and θ ( x ) = min { ∡ [ v a x ] , ∡ [ v b x ] } . 124 CHAPTER 8. PIECEWISE LINEAR GEOMETR Y Let us map x to the p oint with p olar co ordinates ( ρ ( x ) , θ ( x )) in the plane. Note that for each triangle △ , the constructed map △ → R 2 is piece- wise distance-preserving. It remains to c hec k that these maps agree on the common sides of the triangles. This construction was given by Victor Zalgaller [239]. Sv etlana Krat generalized the statement to higher dimensions [240]. Piecewise distance-preserving extension. Let a 1 , . . . , a n and b 1 , . . . . . . , b n b e t wo collections of p oin ts in R 2 suc h that | a i − a j | ⩾ | b i − b j | for all pairs i , j . W e need to construct a piecewise distance-preserving map f : R 2 → R 2 suc h that f ( a i ) = b i for each i . Assume that the problem is already solv ed for n < m ; let us do the case n = m . By assumption, there is a piecewise linear length-preserving map f : R 2 → R 2 suc h that f ( a i ) = b i for each i > 1 . Consider the set Ω =  x ∈ R 2   | f ( x ) − b 1 | > | x − a 1 |  . Since | a i − a 1 | ⩾ | b i − b 1 | , we get a i / ∈ Ω for i > 1 . Note that we can assume that the map f and therefore the set Ω are b ounded. Indeed, let □ b e a square containing all the p oints b i . There is a piecewise isometric map h : R 2 → □ obtained by folding plane along the lines of the grid deﬁned by □ . Then the comp osition h ◦ f is b ounded and it satisﬁes all the prop erties of f describ ed ab ov e. If Ω = ∅ , then f ( a 1 ) = b 1 ; that is, f is a solution. It remains to consider the case Ω  = ∅ . Note that Ω is star-shap ed with resp ect to a 1 . Indeed, if x ∈ Ω , then | a 1 − x | < | b 1 − f ( x ) | . If y ∈ [ a 1 x ] then | a 1 − y | + | y − x | = | a 1 − x | and since f is length-preserving we get | f ( x ) − f ( y ) | ⩽ | x − y | . By the triangle inequalit y , | a 1 − y | < | b 1 − f ( y ) | ; that is, y ∈ Ω . a 1 Ω T i E i E i E i E i E i E i E i E i E i E i E i E i E i E i E i E i E i E i E i E i E i E i E i E i E i E i E i E i E i E i E i E i E i E i E i E i E i E i E i E i E i E i E i E i E i E i E i E i E i E i E i E i E i E i E i E i E i E i E i E i E i E i E i E i E i The boundary ∂ Ω can b e sub divided in to a ﬁnite collection of line segments { E i } so that f maps rigidly eac h E i . Note that | f ( x ) − b 1 | = | x − a 1 | for any x ∈ E i . Denote by T i the trian- gle with the base E i and the v ertex a 1 . F rom the ab ov e, there is a rigid motion m i of T i suc h that m i ( x ) = f ( x ) for an y 125 x ∈ E i and m i ( a 1 ) = b 1 . Let us redeﬁne the map f in Ω by sending x to m i ( x ) for any x ∈ T i . The maps m i agree on the common sides of triangles T i . Therefore we ha ve pro duced a new piecewise isometric map f ′ : R 2 → R 2 satisfying all the requirements. The same pro of works in all dimensions. The statement was prov ed b y Ulrich Brehm and rediscov ered by Ar- seniy Akop y an and Alexey T arasov [see 241, 242, and Section 2 in 175]. Closed p olyhedral surface. An example can b e constructed by drilling a polyhedral cav e in your fav orite con vex polyhedron. On the diagram, y ou see the result of this construction for the o ctahedron. Cho ose a conv ex p olyhedron K . W e can assume that the in terior of K contains the origin 0 ∈ R 3 . Remov e from K the in terior of K ′ = 4 5 · K . Note that one can drill from each ver- tex of K a p olyhedral tunnel to the corre- sp onding v ertex of K ′ so that the surface Σ of the obtained non-con vex p olytop e is a solution. The problem suggested by Jaros l a w Ke ˛dra. The given construction ab ov e pro duces a surface of gen us at least 3. Another ex- ample shown on the diagram is isometric to a ﬂat torus. It is a b ent version of the so-called Schwarz b o ot [243]. It is made by joining a few iden tical cylinders, each made from six triangles. The existence of suc h a torus also fol- lo ws from a general result of Y uri Burago and Victor Zalgaller [244]. They show in particular that any 1-Lipschitz smo oth em b edding of the ﬂat torus in R 3 can b e appro ximated by a piecewise distance-preserving embedding. The following related problem was prop osed by Brian Bo wditch.  Construct a p olyhe dr al metric on the 3-spher e such that the total angle ar ound any e dge of its triangulation is at le ast 2 · π . A solution can b e built using the construction of Jo el Hass [245]. An- other solution w as given by Karim Adiprasito [246]; he prov ed that an example can b e found among spaces that admit a cubulation into unit 126 CHAPTER 8. PIECEWISE LINEAR GEOMETR Y cub es. One can also show that it is imp ossible to ﬁnd suc h an example by starting with a doubled cub e (as well as other simple p olyhedral metrics on the sphere) and passing to cov erings branching along unknots а ﬁnite n umber of times. Minimal p olyhedral disk. Arguing by contradiction, assume that a p olyhedral disk Σ minimizing the area is not saddle; that is, there is an in terior vertex v of Σ suc h that all the edges from v can b e cut with a plane. Note that we can mov e v in suc h a wa y that the lengths of all its edges decrease. q p v w q p v w Since the area is minimal, this defor- mation do es not decrease the area. T ak- ing the deriv ative of the total area along this deformation implies that Σ contains t wo adjacent non-coplanar triangles [ pv w ] and [ q v w ] such that ∡ [ p v w ] + ∡ [ q v w ] > π . In this case, replacing the triangles [ pv w ] and [ q v w ] by the triangles [ v pq ] and [ w pq ] leads to a p olyhedral surface with a smaller area. That is, Σ is not area-minimizing — a contradiction. F or a general p olyhedral surface, a de- formation decreasing the lengths of all edges ma y not decrease the area. More- o ver, the surface that minimizes the area among all surfaces with a ﬁ xed triangula- tion might not be saddle; the symmetric ten t sho wn on the diagram provides an ex- ample [see 247 for more details]. Coheren t triangulation. An example is sho wn on the diagram. The triangulation of the triangle [ x ′ y ′ z ′ ] has a homothetic triangle [ xy z ] and the edges [ xx ′ ] , [ y y ′ ] , [ z z ′ ] , [ y x ′ ] , [ z y ′ ] , [ xz ′ ] . z z ′ x x ′ y y ′ Assume this triangulation is coheren t; let f b e the corresp onding piecewise linear con v ex func- tion. Without loss of generality , we can assume that f v anishes on the b oundary of the big trian- gle. F rom the con vexit y of f at the edges [ x ′ y ] , [ y ′ z ] , and [ z ′ x ] , we get f ( x ) > f ( y ) > f ( z ) > f ( x ) 127 — a contradiction. The problem is discussed in the bo ok of Israel Gelfand, Mikhail Kapra- no v, and Andrei Zelevinsky [see 7C in 248]. The giv en example is closely related to the so-called Sch¨ onhar dt p olyhe dr on , an example of a non- con vex p olyhedron that do es not admit a triangulation [249]. Sphere with one edge. An example can be found among ﬂat orb- ifolds; in other words, the required p olyhedral space can b e chosen to b e a quotient of R 3 b y a crystallographic action. Consider the action on R 3 generated b y order-3 rota- tions around tw o diagonals of the cub es sho wn on the di- agram. Note that this action is crystallographic; in fact, it preserves the cubical grid. Therefore the quotien t of R 3 b y this action, say P , is a compact p olyhedral space. F or the describ ed action, the isotrop y group of any p oin t is either trivial or has order 3. Therefore any p oint in P admits a neigh b orho o d that is isometric to an op en set either in R 3 or R 3 / Z 3 . Both of these spaces are top ological 3-manifolds; therefore, P is a 3-dimensional manifold as well. Note that P is simply-connected; this follows since the action is gener- ated by rotations. By the Poincar ´ e conjecture, P is a top ological sphere. The singular lo cus of P is the image of the axes of all the order-3 rotations. Note that the action is transitiv e on the set of all these axes. It follo ws that the singular lo cus P s of P is connected; that is, P s is a circle. This is the so-called P 2 1 3 action. Among 219 crystallographic actions, this is the only one with the quotient space that has the required property; see [250]. Note that a 3-fold cov ering of P that is branching in P s is a ﬂat manifold. In particular, this cov ering is not simply-connected. Therefore P s is not a trivial knot; it is, in fact, the ﬁgure-eight knot. There are a few crystallographic actions with singu- lar lo cus formed b y links. The simplest is the Borromean rings; it is the singular lo cus of a ﬂat orbifold obtained b y gluing eac h face of a cub e to itself along the reﬂections with resp ect to the middle lines shown on the picture; the corresp onding action is called I 2 1 2 1 2 1 . Other examples (nonorbifold, spherical, h yp erb olic) are discussed b y Mic hel Boileau and Joan Porti [251, Chapter 9]; more examples are given b y Alexander Mednykh [252]. I get in trigued by this problem b ecause of the following connection: spaces of directions in 4-dimensional polyhedral K¨ ahler manifolds are 3- spheres with a spherical polyhedral metric with a singular lo cus formed 128 CHAPTER 8. PIECEWISE LINEAR GEOMETR Y b y a knot or link. In addition, these spaces hav e an isometric R -action induced by the complex structure. P olyhedral K¨ ahler manifolds admit a natural stratiﬁcation into ﬂat manifolds of even co dimensions. So the existence problem of p olyhedral K¨ ahler metrics on a giv en manifold or with a given stratiﬁcation might b e considered as a higher-dimensional analog of the original problem. A p olyhedral metric on a manifold is called K¨ ahler if it comes with a complex structure on each simplex of maximal dimension that agrees on each simplex of co dimension 1 (plus a minor condition). This class of spaces w as deﬁned and studied by Dmitri Pano v [253]. In particular, he constructed many examples of p olyhedral K¨ ahler metrics on C P 2 — try to construct one, it is not that easy . The following problem is completely op en:  Is it p ossible to appr oximate the c anonic al metric on C P 2 by p olyhe dr al metrics with nonne gative curvatur e in the sense of Alexandr ov? T riangulation of a torus. Assume the con trary; let τ b e a triangulation of the torus with the v ertex z 5 meeting 5 triangles, vertex z 7 meeting 7 triangles, and every other vertex meeting 6 triangles. Let us equip the torus with the ﬂat metric suc h that each triangle is equilateral. The metric will hav e t wo singular cone p oints z 5 and z 7 . The total angle around z 5 is 5 3 · π and the total angle around z 7 is 7 3 · π . Note that ( ∗ ) the holonomy gr oup of the obtaine d p olyhe dr al metric on the torus is gener ate d by the r otation by π 3 . Indeed, since parallel translation along any lo op preserves the direc- tions of the sides of any triangle; it can only p ermute it cyclically , which corresp onds to rotations by m ultiples of π 3 . On the other hand, the holon- om y of the lo op that surrounds z 5 is a rotation by π 3 . Consider a closed geo desic γ 1 minimizing the length among all not n ull-homotopic circles. Let γ 2 b e another closed geo desic that minimizes the length and is not homotopic to any p o wer of γ 1 . Note that γ 1 and γ 2 in tersect at a single p oint; otherwise, one could shorten one of them keeping the deﬁning prop erty . Note that γ i do es not con tain z 5 . In fact, no geo desic can pass thru an y singular p oin t with a total angle smaller than 2 · π . Assume that γ i passes thru z 7 . Then by ( ∗ ) , one of the tw o angles cut b y γ i at z 7 is π . It follows that one can push γ i aside so that it do es not longer pass thru z 7 , but remains to b e a closed geo desic with the same length. 129 z 5 z 7 Cut the torus along γ 1 and γ 2 . In the obtained quadrilateral, connect z 5 to z 7 b y a minimizing geo desic and cut along it. This wa y we obtain an ann ulus Ω with a ﬂat metric. Note that a neighborho o d of the ﬁrst b ound- ary comp onen t is a parallelogram — it has equal opp osite sides and its angles add up to 2 · π . In particular, Ω admits an isometric immersion into the plane. The second b oundary comp onen t has to b e mapp ed to a diangle with straigh t sides and angles π 3 . Such diangle do es not exist in the plane — a con tradiction. The problem w as originally disco v ered and solv ed b y Stanisla v Jendroľ and Ernest Jucovi ˇ c [254], their pro of is combinatorial. The solution de- scrib ed ab o ve w as giv en b y Rostisla v Matv eyev [255]. A complex-analytic pro of w as found by Iv an Izmestiev, Rob ert Kusner, G ¨ unter Rote, Boris Springb orn, and John Sulliv an [256]. There are ﬂat metrics on the torus with only tw o sin- gular p oin ts of total angles 5 3 · π and 7 3 · π . Such an example can b e obtained by iden tifying the hexagon on the picture according to the arro ws. How ever, the holonomy group of the obtained torus is generated b y the rotation b y π 6 . In particular, the observ ation ( ∗ ) is essential in the pro of. The same argumen t shows that the holonomy group of a ﬂat torus with exactly t wo singular p oin ts of total angle 2 · (1 ± 1 n ) · π has more than n elements. In the solution, w e did the case n = 6 . If one denotes by v m the num b er of v ertices in a triangulation of the torus with m incoming edges, then by Euler’s form ula, w e get ( ∗∗ ) X m ( m − 6) · v m = 0 . Note that this equation says nothing ab out v 6 . It turns out that for almost any sequence v 3 , v 4 , . . . satisfying ( ∗∗ ) one can adjust v 6 so that it corresp onds to a triangulation of the torus — the sequence 0 , 0 , 1 , v 6 , 1 , 0 , 0 , . . . discussed in the problem is the only exception. The follo wing problem is harder. Recall that the curv ature of a p oint s in a p olyhedral surface is deﬁned as 2 · π − θ , where θ denotes the total angle around s . Note that all regular p oints in a p olyhedral surface hav e zero curv ature. 130 CHAPTER 8. PIECEWISE LINEAR GEOMETR Y  L et Σ b e a spheric al p olyhe dr al sp ac e home omorphic to the 2-spher e and ω 1 , . . . , ω n b e the curvatur es of its singular p oints. Set δ i = min  | ω i 2 − 2 · k · π |   k ∈ Z  . Show that ther e is a close d p olygonal line in the unit spher e with sides δ 1 , . . . , δ n . This problem was stated and solved by Gabriele Mondello and Dmitri P anov [257]. The solution requires another holonomy group — it assigns an elemen t of the double cov ering of SO(3) (which is SU(2) = S 3 ) to an y lo op in Σ that av oids singularities. No simple geo desics. The curv ature of a vertex on the s urface of a con vex p olyhedron is deﬁned as 2 · π − θ , where θ is the total angle around the vertex. By the Gauss–Bonnet formula, a simple closed geo desic cuts the sur- face into tw o disks e ac h with total curv ature 2 · π . Therefore it is suﬃ- cien t to construct a conv ex p olyhedron with curv atures of the v ertices ω 1 , . . . , ω n suc h that 2 · π cannot b e obtained as a sum of some of the ω i . An example of that type can b e found among the tetrahedrons. The problem is due to Gregory Galp erin [258]; it was rediscov ered b y Dmitry F uchs and Serge T abac hniko v [see 20.8 in 8]. The following problem is closely related.  Assume that the surfac e of c onvex p olyhe dr on P c ontains arbitr ary long close d simple ge o desics. Show that P is an isosceles tetrahedron ; that is, the opp osite e dges of the tetr ahe dr on ar e e qual. The latter statemen t was pro ved b y Vladimir Protasov [see 259 and also 260, 261]. Chapter 9 Discrete geometry In this chapter, we consider geometrical problems with a strong com bina- toric ﬂav or. No sp ecial prerequisite is needed. Round circles in 3-sphere  Supp ose that C is a ﬁnite c ol le ction of p airwise linke d r ound cir cles in the unit 3-spher e. Pr ove that ther e is an isotopy of C that moves al l of them into gr e at cir cles. Semisolution. F or each circle in C , consider the plane containing it. Note that the circles are link ed if and only if the corresp onding planes intersect at a single p oint inside the unit sphere S 3 ⊂ R 4 . Consider the collection of circles formed by the intersections of the planes with the sphere of radius R ⩾ 1 . Rescale the sphere and pass to the limit as R → ∞ . This wa y we get the needed isotopy . This problem w as discussed b y Geneviev e W alsh [262]. The same idea was used b y Michael F reedman and Ric hard Skora to show that an y link made from pairwise not linked round circles is trivial; in particular, Borromean rings cannot b e realized by round circles [see Lemma 3.2 in 263]. Bo x in a b o x  Supp ose a r e ctangular p ar al lelepip e d with sides a, b, c lies inside another r e ctangular p ar al lelepip e d with sides a ′ , b ′ , c ′ . Show that a ′ + b ′ + c ′ ⩾ a + b + c. 131 132 CHAPTER 9. DISCRETE GEOMETR Y Piercing the cub e  L et Π b e a k -plane that p asses thru the c enter of a unit n -cub e Q . Show that k -ar e a of the interse ction Π ∩ Q is at le ast 1. Harnac k’s circles  Pr ove that a smo oth algebr aic curve of de gr e e d in R P 2 c onsists of at most n = 1 2 · ( d 2 − 3 · d + 4) c onne cte d c omp onents. T w o p oin ts on eac h line  Construct a set in the Euclide an plane that interse cts e ach line at exactly 2 p oints. Balls without gaps  L et B 1 , . . . , B n b e b al ls of r adii r 1 , . . . , r n in a Euclide an sp ac e. As- sume that no hyp erplane divides the b al ls into two non-empty sets without interse cting at le ast one of the b al ls. Show that the b al ls B 1 , . . . , B n c an b e c over e d by a b al l of r adius r = r 1 + · · · + r n . Co vering lemma  L et { B i } i ∈ F b e any ﬁnite c ol le ction of b al ls in R m . Show that ther e is a sub c ol le ction of p airwise disjoint b al ls { B i } i ∈ G , G ⊂ F such that v ol [ i ∈ F B i ! ⩽ 3 m · v ol [ i ∈ G B i ! . W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 1 W 2 W 3 W 4 W 5 W 6 Kissing n umber ◦ Let W 0 b e a con vex b o dy in R m . W e say that k is the kissing numb er of W 0 (brieﬂy k = kiss W 0 ) if k is the maximal integer such that there are k b o dies W 1 , . . . , W k suc h that (1) eac h W i is congruen t to W 0 , (2) W i ∩ W 0  = ∅ for eac h i and (3) no pair W i , W j has common interior p oints. As you may ha ve guessed from the diagram, the kissing num b er of the round disk in a plane is 6 . 133  Show that for any c onvex b o dy W 0 in R m we have that kiss W 0 ⩾ kiss B , wher e B denotes the unit b al l in R m . Monotonic homotop y  L et F b e a ﬁnite set and h 0 , h 1 : F → R m b e two maps. Consider R m as a subsp ac e of R 2 · m . Show that ther e is a homotopy h t : F → R 2 · m fr om h 0 to h 1 such that the function t 7→ | h t ( x ) − h t ( y ) | is monotonic for any p air x, y ∈ F . F acet cov er  Show that any fac et of a c onvex p olyhe dr on c an b e c over e d by the r emaining fac ets. Cub e  Half of the vertic es of an m -dimensional cub e ar e c olor e d in white and the other half in black. Show that the cub e has at le ast 2 m − 1 e dges c onne cting vertic es of diﬀer ent c olors. Geo desic lo op  Show that the surfac e of a cub e in R 3 do es not admit a ge o desic lo op with a vertex as the b ase p oint. Righ t and acute triangles  L et x 1 , . . . , x n ∈ R m b e a c ol le ction of p oints such that any triangle [ x i x j x k ] is right or acute. Show that n ⩽ 2 m . Upp er appro ximant  L et µ b e a Bor el pr ob ability me asur e on the plane. Show that given ε > 0 , ther e is a ﬁnite set of p oints S that interse cts every c onvex ﬁgur e of me asur e at le ast ε . Mor e over, we c an assume that | S | — the numb er of p oints in S , dep ends only on ε (in fact, one c an take | S | = ⌈ 1 ε 5 ⌉ ). 134 CHAPTER 9. DISCRETE GEOMETR Y Righ t-angled p olyhedron + A p olyhedron is called right-angle d if all its dihedral angles are right.  Show that in al l suﬃciently lar ge dimensions, ther e is no c omp act c onvex hyp erb olic right-angle d p olyhe dr on. Here is a summary of the Dehn–Sommerville equations that can help. Let P be a simple Euclidean m -dimensional p olyhedron; that is, ex- actly m facets meet at each vertex of P . Denote by f k the n um b er of k -dimensional faces of P ; the array of integers ( f 0 , . . . f m ) is called the f -v ector of P . Cho ose a linear function  that takes diﬀerent v alues on the v ertices of P . The index of a v ertex v is deﬁned as the num b er of edges [ v w ] of P suc h that  ( v ) >  ( w ) . The num b er of v ertices of index k will b e denoted b y h k . The arra y of integers ( h 0 , . . . h m ) is called the h -vector of P . Eac h k -face of P con tains a unique vertex that maximizes  . If the v ertex has index i , then i ⩾ k , and then it is the maximal v ertex of exactly  i k  faces of dimension k . This observ ation can b e pac ked in the following p olynomial iden tity: X k h k · ( t + 1) k = X k f k · t k . This identit y implies that the h -vector do es not dep end on the choice of  . Changing the sign of  , we get that, the h -vector is the same for the rev ersed order; that is, ( ∗ ) h k = h m − k for all k . The iden tities ( ∗ ) for all k are called the Dehn–Sommerville equations. They give a complete list of linear equations for h -vectors (and therefore f -v ectors) of simple p olyhedrons. F or more on the sub ject, see [264, Chapter 9]. Real ro ots of random p olynomial ◦ Consider the moment curve γ n : t 7→ (1 , t, . . . , t n ) in R n +1 . Let  n = length γ n | γ n | .  Show that ℓ n π is the exp e cte d numb er of r e al r o ots of a p olynomial of de gr e e n with indep endent normal ly distribute d r e al c o eﬃcients. 135 Space coloring ◦  L et A b e a set of c olor e d p oints in R d . W e ar e al lowe d to c olor any line c ontaining at le ast k alr e ady c olor e d p oints. Supp ose that in a ﬁnite numb er of steps, we c an c olor any p oint in R d . What is the minimal numb er of p oints in A as a function of d and k ? Semisolutions Bo x in a b ox. Let Π b e a parallelepiped with dimensions a , b , and c . Denote by v ( r ) the volume of the r -neighborho o d of Π , Note that for all p ositive r we hav e ( ∗ ) v Π ( r ) = w 3 (Π) + w 2 (Π) · r + w 1 (Π) · r 2 + w 0 (Π) · r 3 , where ⋄ w 0 (Π) = 4 3 · π is the v olume of the unit ball, ⋄ w 1 (Π) = π · ( a + b + c ) , ⋄ w 2 (Π) = 2 · ( a · b + b · c + c · a ) is the surface area of Π , ⋄ w 3 (Π) = a · b · c is the volume of Π , Let Π ′ b e another parallelepiped with dimensions a ′ , b ′ and c ′ . If Π ⊂ Π ′ , then v Π ( r ) ⩽ v Π ′ ( r ) for any r . F or r → ∞ , these inequalities imply a + b + c ⩽ a ′ + b ′ + c ′ . □ A lternative pr o of. Note that the av erage length of the pro jection of Π to a line is Const · ( a + b + c ) for some Const > 0 . (In fact, Const = 1 2 , but w e will not need it.) Since Π ⊂ Π ′ , the a v erage length of the pro jection of Π cannot exceed the av erage length of the pro jection of Π ′ . Hence the statemen t follows. The problem was discussed by Alexander Shen [265]. A formula analogous to ( ∗ ) holds for an arbitrary con vex b o dy B of arbitrary dimension m . It was discov ered by Jakob Steiner [266]. The co eﬃcien t w i ( B ) in the p olynomial (with diﬀerent normalization con- stan ts) app ears under diﬀerent names, most commonly intrinsic volumes and quermassinte gr als . Up to a normalization constant, they can also b e deﬁned as the a v erage area of the pro jections of B to the i -dimensional planes. In particular, if B ′ and B are con vex b o dies such that B ′ ⊂ B , then w i ( B ′ ) ⩽ w i ( B ) for any i . This generalizes our problem quite a bit. F urther generalizations lead to the theory of mixe d volumes [267]. 136 CHAPTER 9. DISCRETE GEOMETR Y The equality w 1 (Π) = π · ( a + b + c ) still holds for all parallelepip eds, not only rectangular ones. In particular, if one parallelepiped lies inside another then the sum of all edges of the ﬁrst one cannot exceed the sum for the second. Piercing the cub e. Observe that there is an o dd increasing 1-Lipsc hitz function ϕ : R → ( − 1 2 , 1 2 ) that pushes the measure with the densit y e − π x 2 to the Leb esgue measure on ( − 1 2 , 1 2 ) . W e can assume that Q =  ( x 1 , . . . , x n ) ∈ R n   | x i | ⩽ 1 2  . Apply- ing ϕ to each co ordinate, we get a 1-Lipschitz diﬀeomorphism from R n to the in terior of Q that pushes the measure with the Gauss densit y ρ ( x 1 , . . . , x n ) = e − π ( x 2 1 + ··· + x 2 n ) to the Leb esgue measure on Q . Note that the inv erse image Π ′ = ϕ − 1 (Π) is a symmetric k -surface in R n . Since ϕ is 1 -Lipschitz, we get that the area of Π ∩ Q cannot b e smaller than the ρ -weigh ted k -area of Π ′ . Moreov er, we hav e equality for co ordinate k -planes. Denote by S r the sphere of radius r in R n cen tered at the origin. Note that Π ′ ∩ S r in tersects eac h great ( n − k − 1) -sphere in S r . By Crofton’s formula, we get that ( k − 1) -area of Π ′ ∩ S r cannot b e smaller than ( k − 1) -area of the great ( k − 1) -sphere in S r . By the coarea formula, the ρ -weigh ted k -area of Π ′ cannot b e smaller than ρ -weigh ted k -area of a k -plane that passes thru the origin — hence the result. The problem was p osed by An ton Go o d and solved by Jeﬀrey V aaler [268]. The presented solution was found by Arseniy Akop y an, Alfredo Hubard, and Roman Karasev [269]; note that it prov es the following more general statement: If f : Q → R n − k is an o dd c ontinuous map, such that Z = f − 1 { 0 } is smo oth k -surfac e, then the k -ar e a of Z is at le ast 1. The map R n → Q was ﬁrst used by Gilles Pisier [270, p. 182] Harnac k’s circles. Let σ ⊂ R P 2 b e an algebraic curv e of degree d . Consider the complexiﬁcation Σ ⊂ C P 2 of σ . Without loss of generalit y , w e ma y assume that Σ is regular. Note that all regular complex algebraic curves of degree d in C P 2 are isotopic to eac h other in the class of regular algebraic curves of degree d . Indeed, the set of equations of degree d that corresp ond to singular curves has real codimension 2. It follo ws that the set of equations of degree d that corresp ond to regular curves is connected. Therefore, one can construct an isotopy from one regular curve to any other b y changing contin uously the parameters of the equations. In particular, it follows that all regular complex algebraic curves of degree d in C P 2 ha ve the same gen us, denote it by g . Perturbing a singular curve formed by d lines in C P 2 , we can see that g = 1 2 · ( d − 1) · ( d − 2) . 137 The real curv e σ forms the ﬁxed p oint set in Σ by the complex con- jugation. In particular Σ \ σ has at most tw o connected comp onents. Therefore, the num b er of comp onen ts of σ cannot exceed g + 1 . This problem is a background for Hilbert’s 16th problem. The in- equalit y w as originally prov ed by Axel Harnack using a diﬀeren t metho d [271]. The idea to use complexiﬁcation is due to F elix Klein [272]. In fact an y num b er of connected comp onents up to g + 1 is realizable, with one exception: if d is o dd, then σ has at least one connected comp onent. T w o p oints on each line. T ak e any complete ordering of the set of all lines so that each b eginning interv al has cardinality less than con tinuum. Assume we ha ve a set of p oints X of cardinality less than contin uum suc h that each line intersects X in at most 2 p oints. Cho ose the least line  in the ordering that intersects X in 0 or 1 point. Note that the set of all lines in tersecting X at tw o p oin ts has cardinality less than contin uum. Therefore we can choose a p oin t on  and add it to X so that the remaining lines are not ov erloaded. It remains to apply the well-ordering principle. This problem has an endless list of v ariations. The following problem lo oks similar but far more inv olv ed; a solution follows from the pro of of P aul Monsky that a square cannot b e cut into triangles with equal areas [273].  Sub divide the plane into thr e e everywher e dense sets A , B , and C such that e ach line me ets exactly two of these sets. Balls without gaps. Assume the mass of each ball is proportional to its radius. Denote b y z the center of mass of the balls. It is suﬃcien t to sho w the following. ( ∗ ) The b al l B ( z , r ) c ontains al l B 1 , . . . , B n . Assume this is not the case. Then there is a line  thru z , suc h that the orthogonal pro jection of a ball B i to  do es not lie completely inside the pro jection of B . (This observ ation reduces the problem to the one- dimensional case.) Note that the pro jection of all balls B 1 , . . . , B n has to be connected and it contains a line segment longer than r on one side from z . In this case, the center of mass of the balls pro jects inside of this segment — a con tradiction. The statemen t was conjectured by P aul Erd˝ os. The solution w as given b y A dolph and Ruth Go o dmans [see 274 and also 275]. Co vering lemma. The required collection { B i } i ∈ G is constructed using the gr e e dy algorithm . W e c ho ose the balls one b y one; on eac h step we 138 CHAPTER 9. DISCRETE GEOMETR Y tak e the largest ball that does not intersect those which w e hav e chosen already . Note that eac h ball in the original collection { B i } i ∈ F in tersects a ball in { B i } i ∈ G with a larger radius. Therefore ( ∗ ) [ i ∈ F B i ⊂ [ i ∈ G 3 · B i , where 3 · B i denotes the ball concen tric to B i and three times larger radius. Hence the statement follows. The constant 3 m can b e impro v e d sligh tly [276]. F or m = 1 the optimal constan t is 2 . Possibly , for any m , the optimal constant is 2 m ; it can not b e smaller, an example can b e found among collections of unit balls that con tain a ﬁxed p oin t. The inclusion ( ∗ ) is called the Vitali c overing lemma . The following statemen t is called the Besic ovitch c overing lemma ; it has a similar pro of.  F or any p ositive inte ger m , ther e is a p ositive inte ger M such that any ﬁnite c ol le ction of b al ls { B i } i ∈ F in R m c ontains a sub c ol le ction { B i } i ∈ G such that (1) c enter of any b al l in { B i } i ∈ F lies inside one of a b al l fr om { B i } i ∈ G and (2) the c ol le ction { B i } i ∈ G c an b e sub divide d into M sub c ol- le ctions of p airwise disjoint b al ls. Both lemmas were used to prov e the so-called c overing the or ems in measure theory , whic h state that “undesirable sets” hav e v anishing mea- sures. Their applications o verlap but aren’t iden tical, the Vitali c overing the or em w orks for nice measures in arbitrary metric spaces while the Besi- c ovitch c overing the or em works in nice metric spaces with arbitrary Borel measures. More precisely , Vitali works in arbitrary metric spaces with a doubling me asur e µ ; the latter means that µ [2 · B ] ⩽ C · µB for a ﬁxed constant C and any ball B in the metric space. On the other hand, Besicovitc h works for all Borel measures in the so-called dir e ction- al ly limite d metric spaces [see 2.8.9 in 277]; these include Alexandrov spaces with curv ature b ounded b elo w. Kissing n umber. Set n = kiss B . Let B 1 , . . . , B n b e copies of the ball B that touch B and don’t hav e common interior p oin ts. F or each B i consider the vector v i from the center of B to the center of B i . Note that ∡ ( v i , v j ) ⩾ π 3 if i  = j . F or each i , consider the supp orting hyperplane Π i of W with the outer normal vector v i . Denote b y W i the reﬂection of W with resp ect to Π i . 139 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 0 W i W j Π i Π j Note that W i and W j ha ve no common in terior p oin ts if i  = j ; the latter gives the needed inequality . The pro of is given by Charles Halb erg, Eugene Levin, and Ernst Straus [278]. It is not known if the same inequality holds for the orientation-preserving v ersion of the kissing num b er. Monotonic homotopy . Note that we can assume that h 0 ( F ) and h 1 ( F ) b oth lie in the co ordinate m -spaces of R 2 · m = R m × R m ; that is, h 0 ( F ) ⊂ R m × { 0 } and h 1 ( F ) ⊂ { 0 } × R m . Direct calculations show that the following homotopy is monotonic h t ( x ) =  h 0 ( x ) · cos π · t 2 , h 1 ( x ) · sin π · t 2  . □ This homotop y was discov ered by Ralph Alexander [279]. It has a n umber of applications, one of the most b eautiful is giv en by K´ aroly Bezdek and Rob ert Connelly [280]; they prov ed the Kneser–Poulsen and Klee–W agon conjectures in the t w o-dimensional case. The dimension 2 · m is optimal; that is, for an y p ositive integer m , there are tw o maps h 0 , h 1 : F → R m that cannot b e connected by a monotonic homotop y h t : F → R 2 · m − 1 . The latter w as sho wn by Maria Belk and Rob ert Connelly [281] F acet cov er. Cho ose a con vex polyhedron P and its facet F . Denote by Π the hyperplane of F . F or each p oint p ∈ F consider the maximal ball B p ⊂ P that con- tains p . Note that B p touc hes another facet of P at some p oint q . Sho w that the restriction of the partially deﬁned map q 7→ p to an y other facet F ′ can be extended to a distance-preserving map F ′ → Π . By construction union of all such images cov er F . This problem was considered by Igor Pak and Rom Pinchasi [282]; the presen ted pro of was given by Arseniy Akop yan [283]. A slightly diﬀerent v ersion of this problem made it to the All-Russian olympiad of 2012 [284, № 116774]. In the three-dimensional case, a more inv olved, but straigh tforw ard solution can b e built on the fact that orthogonal pro jection of a conv ex ﬁgure can b e cov ered by the ﬁgure itself. The latter statemen t is not as simple as one might think [see 285, and the references therein]. Cub e. Consider the cub e [ − 1 , 1] m ⊂ R m . Any vertex of this cub e has the form q = ( q 1 , . . . , q m ) , where q i = ± 1 . 140 CHAPTER 9. DISCRETE GEOMETR Y F or eac h v ertex q , consider the intersection of the corresp onding hy- p eroctant with the unit sphere; that is, consider the set V q =  ( x 1 , . . . , x m ) ∈ S m − 1   q i · x i ⩾ 0 for each i  . Let A ⊂ S m − 1 b e the union of all the sets V q for black q . Note that v ol m − 1 A = 1 2 · v ol m − 1 S m − 1 . By the spherical isop erimetric inequality , v ol m − 2 ∂ A ⩾ vol m − 2 S m − 2 . It remains to observe that v ol m − 2 ∂ A = k 2 m − 1 · v ol m − 2 S m − 2 , where k is the num b er of edges of the cub e with one black end and the other in white. The problem was suggested by Greg Kup erb erg. Geo desic lo op. Let γ b e a geo desic from vetrex to vertex; denote by v its midp oin t. Sho w that there is symmetry of the cub e that ﬁxes v , reverts γ , and mo ves all the vertices of the cub e. Conclude that the endp oints of γ are diﬀeren t. I learned this problem from Jaros l a w Ke ˛dra; it was rediscov ered inde- p enden tly by Diana Davis, Victor Do ds, Cynthia T raub, and Jed Y ang [286]. The presented solution is taken from the paper of Serge T roubet- zk oy [287]. This idea can b e used to solve the following harder problems.  Show the same for the surfac e of the n -dimensional cub e, n ⩾ 4 .  Show the same for the surfac e of the tetr ahe dr on, o ctahe dr on, and ic osahe dr on. F or the do decahedron suc h lo ops exist; a developmen t of one example is on the diagram. V ertices of an inscrib ed tetrahedron are circled. A 141 classiﬁcation of all suc h lo ops is found b y Ja y adev S. Athrey a, Da vid Aulicino, and W. Patric k Ho op er [288]. These and related problems are discussed by Dmitry F uchs [289]. Righ t and acute triangles. Denote by K the conv ex hull of { x 1 , . . . . . . , x n } . Without loss of generality , we can assume that dim K = m . x i x j z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z i z i z i z i z i z i z i z i z i z i z i z i z i z i z i z i z i z i z i z i z i z i z i z i z i z i z i z i z i z i z i z i z i z i z i z i z i z i z i z i z i z i z i z i z i z i z i z i z i z i z i z i z i z i z i z i z i z i z i z i z i z i z i z i z i z j z j z j z j z j z j z j z j z j z j z j z j z j z j z j z j z j z j z j z j z j z j z j z j z j z j z j z j z j z j z j z j z j z j z j z j z j z j z j z j z j z j z j z j z j z j z j z j z j z j z j z j z j z j z j z j z j z j z j z j z j z j z j z j z j K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K Note that for any distinct p oints x i , x j , and an y in terior p oin t z in K , we ha ve ( ∗ ) ∡ [ x i x j z ] < π 2 . Indeed, if ( ∗ ) do es not hold, then ⟨ x j − x i , z − x i ⟩ < 0 . Since z ∈ K we ha ve ⟨ x j − x i , x k − x i ⟩ < < 0 for some v ertex x k . That is, ∡ [ x i x j x k ] > π 2 — a contradiction. Denote by h i the homothety with center x i and co eﬃcient 1 2 . Set K i = h i ( K ) . Let us show that K i and K j ha ve no common in terior p oints. Assume the contrary; that is, z = h i ( z i ) = h j ( z j ); for some interior p oints z i and z j in K . Note that ∡ [ x i x j z j ] + ∡ [ x j x i z i ] = π , whic h con tradicts ( ∗ ) . Note that K i ⊂ K for any i ; it follows that n 2 m · v ol K = n X i =1 v ol K i ⩽ vol K. Hence the result follows. The problem was p osted by P aul Erd˝ os [290] and solv ed b y Ludwig Danzer and Branko Gr ¨ un baum [291]. Grigori P erelman noticed that the same pro of w orks for a similar prob- lem in Alexandro v spaces [292]. The latter led to interesting connections with the crystallographic groups [293]; in particular, it gives an approach to the following op en problem.  L et Γ ↷ R n b e an eﬀe ctive pr op erly disc ontinuous isometric action. Denote by m the numb er of maximal ﬁnite sub gr oups Γ up to c onjugation. Is it true that m ⩽ 2 n ? Surprisingly , the maximal num b er of p oin ts that make only acute tri- angles gro ws exp onentially with m as well. The latter w as shown by Paul Erd˝ os and Zolt´ an F ¨ uredi [294] using the pr ob abilistic metho d . Later, an 142 CHAPTER 9. DISCRETE GEOMETR Y elemen tary constructiv e argumen t w as found and improv ed by Dmitriy Zakharo v, grizzly (an anon ymous mathematician), Bal´ azs Gerencs ´ er, and Viktor Harangi [295–297]; the curren t low er b ound is 2 m − 1 + 1 , whic h is exp onen tially optimal. Upp er approximan t. W e as sume that the measure is given by a distri- bution. The general case is done by a straightforw ard mo diﬁcation. Cut the plane b y tw o lines into 4 angles of equal measure. Let p b e the in tersection p oint of the tw o lines. Note that every con vex set av oiding p is fully contained in three angles out of the four. In particular, its measure cannot exceed 3 4 . Apply this construction recursively for the restriction of the measure to eac h triple of angles. After n steps we get a (1 + . . . + 4 n − 1 ) -p oin t set that intersects each conv ex ﬁgure F of measure ( 3 4 ) n . This is a stripp ed version of a theorem prov ed by Boris Bukh and Gabriel Niv asc h [298]. Righ t-angled p olyhedron. Let P b e a right-angled h yp erb olic p oly- hedron of dimension m . Note that P is simple; that is, exactly m facets meet at each vertex of P . F rom the pro jective mo del of the hyperb olic plane, one can see that for any simple compact hyperb olic p olyhedron, there is a simple Eu- clidean p olyhedron with the same com binatorics. In particular, the Dehn– Sommerville equations hold for P . Denote by ( f 0 , . . . f m ) and ( h 0 , . . . h m ) the f - and h -v ectors of P . Re- call that h i = h m − i , f 1 = h 1 + 2 · h 2 + · · · + m · h m , f 2 = h 2 + 3 · h 3 + · · · +  m 2  · h m . Observ e that h i ⩾ 1 ; together with the identities ab o ve it implies that ( ∗ ) f 2 > m − 2 4 · f 1 . Since P is hyperb olic, each 2-dimensional face of P has at least 5 sides. It follows that f 2 ⩽ m − 1 5 · f 1 . The latter contradicts ( ∗ ) for m ⩾ 6 . This is the core of the pro of of nonexistence of compact hyperb olic Co xeter’s p olyhedrons of large dimensions giv en by Ernest Vinberg [223, 224]. Pla ying a bit more with the same inequalities, one gets the nonexis- tence of righ t-angled hyperb olic p olyhedrons, in all dimensions starting 143 from 5 . In the 4 -dimensional case, there is a regular right-angled hyper- b olic p olyhedron with 120-c el ls — a 4 -dimensional uncle of the do decahe- dron. The following related question is op en:  L et m b e a lar ge inte ger. Is ther e a c o c omp act pr op erly disc ontinuous isometric action on the m -dimensional L ob achevsky sp ac e that is gener- ate d by ﬁnite or der elements (for example, involutions)? Real roots of random p olynomial. Cho ose a p olynomial p ( t ) = a 0 + · · · + a n · t n . Consider the hyperplane Π in R n +1 deﬁned by the equation a 0 · x 0 + · · · + a n · x n = 0 . Note that the n umber of real ro ots of p equals the n um b er of intersections of Π with the moment curve. It remains to apply the spherical Crofton formula. The observ ation is due to Alan Edelman and Eric K ostlan [299]. Space coloring. The answer is  d + k − 1 d  . Cho ose d + k − 1 hyperplanes in R d in general p osition. Let A b e the set of all  d + k − 1 d  in tersection p oints of ev ery d -tuple of the chosen h yp erplanes. Show that starting with A one can color every p oint in R d . T o prov e the low er b ound, note that  d + k − 1 d  is the dimension of the space of p olynomials p : R d → R of total degree at most k − 1 . Therefore, if A has less than  d + k − 1 d  p oin ts, then there is a non-zero p olynomial p of degree at most k − 1 such that A lies in its zero-set Z . Observe that if a line do es not lie in Z , then it has at most k − 1 common p oin ts with Z . Therefore, it is imp ossible to color a p oint outside of Z . This is a problem of Iv an Mitrofano v and F edor Petro v [300, № 10]; a three-dimensional version of this problem appeared at the 28 th An- n ual V o jtˇ ec h Jarn ´ ık International Mathematical Comp etition, Category I I [301]. The problem illustrates the so-called p olynomial metho d ; for more on the sub ject c heck the b ook of Lary Guth [302]. Index 120-cells, 143 Antoine’s nec klace, 16 asymptotic line, 24 Baire category theorem, 9 Busemann function, 38 Cheeger’s trick, 53 concentration of measure, 77 conic neighborhoo d, 112 conjugate points, 39 conv ex set, 21, 66 Crofton’s formula, 13 curv ature, 23 curv ature op erator, 38 curv ature vector, 23 curve, 5 cyclic polytop e, 123 Dehn twist, 111 displacement, 116 doubling measure, 138 equidistant sets, 39 exponential map, 65 ﬁgure eight, 97 focal p oint, 48 G-delta set, 9 geodesic, 23 gutter, 24 Hausdorﬀ distance, 9 Hausdorﬀ measure, 8 Heisenberg group, 48 horizontal distribution, 81 horo-compactiﬁcation, 80 horo-sphere, 39 hypermetric inequality , 92 index of vertex, 134 inv erse mean curv ature ﬂow, 45 kissing num b er, 132 Kuratowski em b edding, 79 lakes of W ada, 16 length-metric, 78 length-preserving, 82 Liberman’s lemma, 28 macroscopic dimension, 80 mean curv ature, 24 metric cov ering, 97 minimal surface, 24, 37 mixed volumes, 135 moment curv, 134 monotonicity form ula, 31 net, 83 non-contracting map, 79 non-expanding map, 79 piecewise, 120 polyhedral K¨ ahler manifold, 128 polyhedral space, 120 polynomial volume growth, 39 probabilistic method, 141 proper metric space, 96 quasi-isometry , 83 regular point, 120 saddle surface, 23, 122 Sch warz bo ot, 125 Sch¨ onhardt p olyhedron, 127 second fundamental form, 38 selection theorem, 9 short basis, 105 simple polygon, 120 simple polyhedron, 134 symmetric square, 97 total curv ature, 22 totally geodesic, 35 tw o-conv ex set, 42 vertex of curv e, 10 warped product, 41 144 Bibliograph y [1] P . 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