Asymptotic cones, bi-Lipschitz ultraflats, and the geometric rank of geodesics

ASYMPTOTIC CONES, BI-LIPSCHITZ UL TRAFLA TS, AND THE GEOMETRIC RANK OF GEODESICS. STEF ANO FRANCA VIGLIA AND JEAN-FRANC ¸ OIS LAF ONT Abstract. Let M be a closed non- po sitively curved Riemannian (NPCR) mani- fold, ˜ M its universal cov er, and X a n ultra limit o f ˜ M . F or γ ⊂ ˜ M a geo desic, let γ ω be a geo desic in X o btained as a n ultra limit of γ . W e show that if γ ω is con tained in a ﬂat in X , then the original g e o desic γ supp orts a non-trivia l, normal, parallel Jacobi ﬁeld. In particular, the rank of a geo des ic can be detected from the ultra- limit of the univ ersa l co ver. W e strengthen this r e sult b y allowing for bi-Lipschitz ﬂats satisfying certain additional h yp otheses. As a pplications we obtain (1) constraints on the b ehavior of quasi- isometries betw een complete, s imply connected, NPCR manifolds, and (2) c o nstraints on the NP CR metr ics supp or ted by certa in ma nifolds, and (3) a corr e s po ndence b e- t ween metric s plittings of complete, simply connected NPCR manifolds , and met- ric splittings of its a symptotic cones. F urthermore, com bining our r e sults with the Ba llma nn-Burns-Spatzier rigidity theorem a nd the cla ssic Mos tow rigidity , we also obtain (4) a new pro o f of Gro mov’s rigidity theorem for higher rank lo ca lly symmetric spaces. 1. Intr oduction. Ultralimits ha v e rev ealed them selv es to b e a par ticularly useful to ol in g eometric group theory . Indeed, a n um b er of sp ectacular results ha v e b een obtained via the use of ultralimits, including: • Gro mo v’s p o lynomial g r o wth theorem [G], [VW] • Kleiner and L eeb’s quasi-isometric rigidit y theorem for lattices in higher ra nk semi-simple Lie g r oups [KlL] • Kap o vic h, Kleiner, and Lee b’s theorem on detecting de R ham decomp ositions for unive rsal cov ers of Hadamard manifolds [KKL] • Kap o vic h and Leeb’s pro of that quasi-isometries preserv e the JSJ decomp o- sition o f Hak en 3-ma nif o lds [KaL] • Drut u and Sapir’s c haracterization of (strongly) relatively h yp erb olic groups in terms of ultralimits [DS] In the prese n t note, w e show that ultralimits of simply connected R iemannian manifolds M of non-p ositive sectional curv ature can b e used to detect the geometric rank o f geo desics in M . More precisely , w e establish the following: 1 Theorem 1.1. L et M b e a simpl y c onne c te d, c omplete, Riemannian manifold of non-p ositive se ctional curvatur e, and let C one ( M ) b e an asymptotic c one of M . F or γ ⊂ M an arbitr ary ge o desic, let γ ω ⊂ C one ( M ) b e the c orr esp onding ge o desic in the asymptotic c one. If ther e exists a ﬂat plane F ⊂ C one ( M ) with γ ω ⊂ F , then ther e ex ists a non-trivial p ar al lel Jac obi ﬁeld J along γ satisfying h J ( t ) , ˙ γ ( t ) i = 0 . I n p articular, the ge o desic γ has higher r ank. Let us brieﬂy explain the lay o ut of the presen t pap er. In Section 2, w e pro vide a quic k review of the requisite notions concerning asymptotic cones, v ar ia tion o f arclength for mulas for geo desic v a r ia tions, and other bac kground material. In Section 3, we pro vide conditions ensuring existence of a no n- trivial, orthogo nal, Jacobi ﬁeld along a g eo desic γ . The conditions in v olv e exis tence of what w e call p oin te d ﬂattening se quenc es of 4-t uples for the geo desic γ . The argumen ts in this section ar e purely diﬀeren tial geometric in nature. In Section 4, w e sho w that if γ ω ⊂ C one ( M ) is con tained in a ﬂat, then p oin ted ﬂattening sequences of 4- tuples can b e constructed along γ (completing the proo f of Theorem 1.1). The argumen ts here rely on some elemen tary argumen ts conce rning asymptotic cones a nd the “la rge-scale geometry” of the manifold M . In Section 5, we establish some impro v emen ts b y allo wing for γ ω ⊂ C one ( M ) to b e con tained in a bi-Lips c hitz ﬂat . The pr ecise result is con tained in: Theorem 1.2. L et M b e a simpl y c onne c te d, c omplete, Riemannian manifold of non-p ositive se ctional curvatur e, and let C one ( M ) b e an asymptotic c one of M . F or γ ⊂ M an a rbitr a ry ge o desic, let γ ω ⊂ C one ( M ) b e the c orr esp onding ge o d e sic in the asymptotic c one. Assume that: • ther e exists g ∈ I som ( M ) w h i ch stabilizes and acts c o c omp actly on γ , and • ther e exists a bi-Lipschitzly emb e dde d ﬂat φ : R 2 ֒ → C one ( M ) ma p ping the x -axis o nto γ ω . Then the original ge o desic γ has highe r r ank. Finally , in Section 6 , w e apply our Theorem 1.2 to obtain v arious geometrical corollaries. These include: • constraints on the p ossible quasi-isometries b etw ee n certain non-p ositiv ely curv ed Riemannian manifolds. • restrictions on the p ossible non-p ositive ly curv ed R iemannian metrics that are supp o rted by certain manifolds. • a pro of that splittings of simply connected non-p ositive ly curv ed Riemannian manifolds corr esp o nd exactly with metric splittings of the asymptotic cones. • a new pro of of Gromov ’s rigidity theorem [BGS]: a closed higher rank lo cally symmetric space supp orts a unique metric o f non-p ositiv e curv ature (up to homothet y). 2 Finally , w e po in t out tha t v a rious authors ha v e studied geometric prop erties en- co ded in the a symptotic cone of non-p ositive ly curv ed manifolds. Pe rhaps the view- p oin t closest to ours is that of Kap ovic h- Kleiner-Leeb pap er [KKL], whic h fo cus o n studying the (lo cal homological) top ology of the asymptotic cone to reco v er geometric information on the original space. W e should also men tion the recen t preprin t of Bestvina-F ujiw ara [BeF u], whic h giv es a b ounded cohomological c haracterization of higher rank symme tric spaces. Although they do not sp eciﬁcally disc uss ultralimits, their discussion of rank 1 isome- tries seems to b ear some philo sophical similarities to o ur w ork. Ac kno wledgemen ts The author’s w ould like to thank V. Guirardel, J. Heinonen, T. Janus zkiewicz, B. Kleiner, R. Spatzier, a nd S. W enger for their helpful commen ts. This w ork w as part ly carried o ut during a visit of the ﬁr st author to the Ohio State Univ ersit y (suppo r ted in part b y the MRI), and a vis it of the second author to the Univ ersit` a di Pisa. The w ork of the ﬁrst autho r w as partly supp orted b y the Europ ean Researc h Council – MEIF-CT-2005- 010975 and MER G-CT-200 7-04655 7 . The w ork of the second author w as par t ly supp orted b y NSF grant DMS-060600 2 . 2. Back gr ound Ma te rial 2.1. In tro duction to asymptotic cones. In t his section, w e pro vide s ome bac k- ground on ultralimits a nd asymptotic cones of metric spaces. Let us start with some basic reminders on ultra ﬁlters. Deﬁnition. A non-princip al ultr aﬁlter on the natur al n umb ers N is a c ol le ction U of subsets of N , sa tisfying the fol low i n g four axiom s: (1) if S ∈ U , and S ′ ⊃ S , then S ′ ∈ U , (2) if S ⊂ N is a ﬁni te subset, then S / ∈ U , (3) if S, S ′ ∈ U , then S ∩ S ′ ∈ U , (4) giv e n any ﬁnite p a rtition N = S 1 ∪ . . . ∪ S k into p airwise disjoint sets, ther e is a unique S i satisfying S i ∈ U . Zorn’s L emma guarantee s the existence of non-principal ultraﬁlters. Now giv en a compact Hausdorﬀ space X and a map f : N → X , there is a unique p oin t f ω ∈ X suc h that ev ery neigh b orho o d U of f ω satisﬁes f − 1 ( U ) ∈ U . This p o in t is called the ω − limit of the sequence { f ( i ) } ; we write ω lim f ( i ) = f ω . In particular, if the target space X is the compact space [0 , ∞ ], w e ha v e that f ω is a we ll-deﬁned real num ber (or ∞ ). Deﬁnition. L et ( X , d, ∗ ) b e a p ointe d metric sp ac e, X N the c ol le ction of X -value d se quenc es, and λ : N → (0 , ∞ ) ⊂ [0 , ∞ ] a se quenc e o f r e al n umb ers satisfying λ ω = ∞ . Given any p air of p oints { x i } , { y i } in X N , we deﬁne the pseudo-distanc e d ω ( { x i } , { y i } ) 3 b etwe en them to b e f ω , wher e f : N → [0 , ∞ ) is the function f ( k ) = d ( x k , y k ) /λ ( k ) . Observe that this p seudo-distanc e takes on values in [0 , ∞ ] . Next, note that X N has a d istinguishe d p oint, c orr e s p onding to the c onstant se- quenc e {∗} . R estricting to the subset of X N c onsisting of se quenc es { x i } satisfying d ω ( { x i } , { ∗} ) < ∞ , an d ide ntifying se quenc es whose d ω distanc e is zer o, one obtains a genuine p ointe d metric sp ac e ( X ω , d ω , ∗ ω ) , which we c al l an asymp totic c o ne of the p ointe d metric sp ac e ( X , d, ∗ ) . W e will usually denote an asymptotic cone by C o ne ( X ). The reader should k eep in mind that the construction of C one ( X ) inv olv es a n um b er of c hoices (basep oin ts, sequence λ i , c hoice of non-principal ultraﬁlters) and that diﬀeren t c hoices could giv e diﬀeren t (non-homeomorphic) asymptotic cone s (see the pap ers [TV], [KSTT], [OS]). W e will require the following facts concerning asymptotic cones of no n-p ositiv ely curv ed spaces: • if ( X , d ) is a CA T(0) space, then C one ( X ) is likew ise a CA T(0) space, • if φ : X → Y is a ( C , K )- quasi-isometric map, then φ induces a bi-Lipsc hitz map φ ω : C one ( X ) → C one ( Y ), • if γ ⊂ X is a geo desic, then γ ω := C one ( γ ) ⊂ C one ( X ) is a geo desic, • if { a i } , { b i } ∈ C one ( X ) a re an a r bit r a ry pair of p oints , t hen the ultra limit of the geo desic segmen ts a i b i giv es a geo desic segmen t { a i }{ b i } joining { a i } to { b i } . Concerning the second p o in t ab o v e, w e r emind the reader that a ( C, K )-quasi-isometric map φ : ( X , d X ) → ( Y , d Y ) b etw ee n metric spaces is a (not necessarily con tin uous) map having the prop ert y tha t : 1 C · d X ( p, q ) − K ≤ d Y ( φ ( p ) , φ ( q )) ≤ C · d X ( p, q ) + K . W e now pro ceed to establish tw o Lemmas whic h will b e used in some of o ur pro ofs. Lemma 2.1 (Cho osing go o d sequences) . L et X b e a CA T(0) sp ac e, C one ( X ) an asymptotic c one of X , γ ⊂ X a ge o desic, and γ ω ⊂ C one ( X ) the c orr esp onding ge o desic in the asymptotic c one. Assume that { A, B , C , D } ⊂ C one ( X ) is a 4 -tuple of p oints having the pr op erty that A, B ∈ γ ω ar e the closest p oints on γ ω to the p oints D , C (r esp e ctively). L e t { C i } , { D i } ⊂ X b e two se quenc e s r epr esenting the p oints C , D ∈ C one ( X ) r esp e ctively. T h en (1) if A i , B i ∈ γ ar e the clos e s t p oints to D i , C i (r esp e ctively), then { A i } , { B i } r epr esent A, B ∈ C one ( X ) r esp e ctively. (2) if { r i } ⊂ R + is a se quenc e of r e al numb e rs satisfying ω lim { r i /λ ( i ) } = d ω ( A, D ) , and D ′ i ∈ − − − → A i D i satisﬁes d ( A i , D ′ i ) = r i , then the se quenc e { D ′ i } r epr esen ts D ∈ C one ( X ) . 4 Lemma 2.1 allo ws us to replace, in certain circumstances , a g iv en se quence of 4- tuples represen ting { A, B , C , D } ⊂ C one ( X ) by a new sequence of 4-tuples that are geometrically b etter b eha v ed (i.e. hav e b etter metric prop erties). Pr o of (Lemma 2.1) . T o establish (1), we assume without loss of generalit y that the constan t sequence {∗} of basepoints used to deﬁne ∗ ∈ C one ( X ) is chose n to lie on γ . Then the triangle inequalit y , comb ined with the fact that A i is the closest p oin t to D i on γ , immediately implies: d ( ∗ , A i ) ≤ d ( A i , D i ) + d ( D i , ∗ ) ≤ 2 d ( D i , ∗ ) This in turns implies that d ω ( { A i } , ∗ ) ≤ 2 d ω ( { D i } , ∗ ) < ∞ , i.e. { A i } do es deﬁne a p oint A ω ∈ C one ( X ). An iden tical a r g umen t show s that { B i } deﬁnes a p o in t B ω ∈ C one ( X ) . F urthermore, since all the p o in ts A i , B i are o n γ , w e hav e that A ω , B ω ∈ γ ω ⊂ C one ( X ). W e now claim that A ω = A and B ω = B . T o see this, we note that the sequence of geo desic segmen ts { D i A i } giv es rise to a geo desic segmen t D A ω joining D ∈ C one ( X ) to the p oint A ω ∈ γ ω ⊂ C one ( X ). Since each D i A i w as a minimal length segmen t joining D i to γ , the segmen t D A ω is lik ew ise a minimal length segmen t joining D to γ ω . But w e know that the closest point on γ ω to D is A ( a nd t his is the unique suc h p oint, as C one ( X ) is CA T(0)). W e conclude that A ω = A , as desired. An iden tical argumen t applies to sho w B ω = B , completing the argumen t for (1). T o establish (2), w e ﬁrst note that the sequence of geo desic rays { − − − → A i D i } deﬁne some geo desic ra y ~ η ⊂ C one ( X ). F urthermore, by construction, we hav e that ~ η originates at A , and passes through D . No w ag ain, an easy a pplication of the tria ngle inequalit y implies that the sequence { D ′ i } represen ts a p oint D ω ∈ C one ( X ), which w e are claiming coincides with the p oint D . Since eac h D ′ i is c hosen to lie on the corresp onding geo desic ray − − − → A i D i , we immediately get D ω ∈ ~ η . Finally , let us calculate the distance b et w een D ω and the p oint A : d ω ( A, D ω ) = ω lim { d ( A i , D ′ i ) /λ ( i ) } = ω lim { r i /λ ( i ) } = d ω ( A, D ) . So w e see that D ω , D are a pair of p oin ts on the geo desic ra y ~ η , ha ving the property that they are b oth at the exact same distance from the basep oin t A of the geo desic ra y . This immediately imp lies that they hav e t o coincide, completing the a rgumen t for (2), and hence the pro of of Lemma 2.1.  Lemma 2.2 (T ranslations on asymptotic cone) . L et X b e a ge o desic sp ac e, γ ⊂ X a ge o desic, and γ ω ⊂ C one ( X ) the c orr esp onding ge o desic in an asymptotic c one C one ( X ) of X . Assume that ther e exists an eleme n t g ∈ I som ( X ) with the pr op erty that g le aves γ invariant, and a c ts c o c omp actly on γ . Then for any p air of p oints p, q ∈ γ ω , ther e is an isome try Φ : C one ( X ) → C o ne ( X ) satisfying Φ( p ) = q . Pr o of (Lemma 2.2) . Let { p i } , { q i } ⊂ γ ⊂ X b e sequences deﬁning the p oints p, q resp ectiv ely . Since g leav es γ in v a rian t, and acts co compactly on γ , there exists a 5 real n um ber R > 0 and a sequence of exponents k i ∈ Z with the prop ert y that for ev ery index i , we ha v e d ( g k i ( p i ) , q i ) ≤ R . No w observ e that the sequence { g k i } of isometries o f X deﬁnes a self-map (deﬁned comp onen t wise) of the space X N of sequences of points in X . Let us denote b y g ω this self-map, whic h we now pro ceed to show induces the desired isometry on C one ( X ). First note that it is immediate that g ω preserv es the pseudo-distance d ω on X N , and has the prop ert y that d ω ( { g k i ( p i ) } , { q i } ) = 0. So to see that g ω descends to a n isometry of C one ( X ), all w e ha v e to establish is tha t for { x i } a sequence satisfying d ω ( { x i } , ∗ ) < ∞ , the image seq uence also satisﬁes d ω ( { g k i ( x i ) } , ∗ ) < ∞ . But w e ha v e the series o f equiv a lences: d ω ( { x i } , ∗ ) < ∞ ⇐ ⇒ d ω ( { x i } , { p i } ) < ∞ ⇐ ⇒ d ω ( { g k i ( x i ) } , { g k i ( p i ) } ) < ∞ ⇐ ⇒ d ω ( { g k i ( x i ) } , { q i } ) < ∞ ⇐ ⇒ d ω ( { g k i ( x i ) } , ∗ ) < ∞ where the ﬁrst and last equiv alences come from applying the triangle inequalit y in the pseudo-metric space ( X N , d ω ), and the second and t hird equiv alence s follow from our earlier comments . W e conc lude that t he induce d isometry g ω on the pseudo-metric space X N of seq uences lea v es inv arian t the subset of sequences at ﬁnite distance from the distinguished constant seq uence, and hence descends t o a n isometry of C one ( X ). Finally , it is immediate from the deﬁnition of the isometry g ω that it will lea v e γ ω in v a rian t, as eac h g k i lea v es γ in v ariant. This concludes the pro of of Lemma 2.2.  Observ e that the elemen t g ∈ I som ( X ) used in Lemma 2.2 giv es rise to a Z -action on X leav ing γ in v arian t. It is w orth pointing out that Lemma 2.2 do es not state that g ∈ I som ( X ) induces an R -action on C one ( X ). The issue is that for each r ∈ R , there is indeed a corresp onding isometry of C o ne ( X ), but these will not in general v ar y contin uously with resp ect to r (as can already b e seen in the case X = H 2 ). 2.2. V ariation of arclength form ulas. The classical v a riation fo rm ulas deal with the energy of curve s within a v ariation. This is primarily due to the fact that the energy functional is “easier” to diﬀeren tiate than the length functional. In the sit- uation w e are in terested in, the asymptotic cones pic k up (asymptotic) distances, and hence w e need to actually work with v ariat io ns for the arclength rather than the energy . W e no w pro ceed to remind the reader of the (p erhaps less fa miliar) v ariation form ulas for arclength. A pro of of the presen t formulas can b e found in Jost’s b o o k [Jo, pgs. 165 -169]. Let us start o ut b y setting up some notatio n. W e consider ge o desi c variations , whic h are maps σ : [0 , 1 ] × ( − ǫ, ǫ ) → M into a Riemannian manif o ld ( s ∈ [0 , 1] will b e the ﬁrst parameter, t ∈ ( − ǫ, ǫ ) the second pa rameter), satisfying the following three prop erties: 6 t S γ 0 Figure 1. Geo desic v aria tion. • the curv es s 7→ γ t ( s ) = σ ( s, t ) is a geo desic for all t , • the curv es γ t are parametrised with constan t sp eed: || ˙ γ t || = L ( t ) where L ( t ) is the length of the geo desic γ t , • the “lateral curv es” t 7→ σ (0 , t ) and t 7→ σ (1 , t ) are geo desics. W e now denote b y S, X the following v ector ﬁelds: S = D σ h ∂ ∂ s i X = D σ h ∂ ∂ t i Finally , w e denote b y ˆ X the vec tor ﬁeld obta ined by taking the pro jection of X orthogonal to S . Figure 1 provide s an illustration of a geo desic v aria t ion. W e hav e included the base geo desic (at the b ottom of the picture) corresp onding to t = 0, and ha v e drawn the p ortio n o f σ corresp onding to t ∈ [0 , ǫ ]. The horizon tal curve s represen t geo desic curv es γ t , while the tw o v ertical curv es are the “lateral curv es”. Along the geo desic γ , we ha v e also illustrated a few v alues of the Jacobi v ector ﬁeld X (p ointing straigh t up). The v aria tion formulas w e will need are: angl First v ariation of arclength: F or t 0 ∈ ( − ǫ, ǫ ), the ﬁrst deriv ative of the length L ( t ) at t 0 is given b y (see [Jo, pg. 167, equation 4.1.4]): dL dt ( t 0 ) = h S, X i (1 ,t 0 ) − h S, X i (0 ,t 0 ) L ( t 0 ) 7 Second v ariation of arclength: F or t 0 ∈ ( − ǫ, ǫ ), t he second deriv ativ e of the length L ( t ) at t 0 is given b y (see [Jo, pg. 167, equation 4.1.7]): d 2 L dt 2 ( t 0 ) = 1 L ( t 0 )  Z 1 0 ||∇ S ˆ X || 2 − K ( S ∧ ˆ X ) L ( t 0 ) 2 || ˆ X || 2 ds  where K ( S ∧ ˆ X ) denotes the sectional curv a ture of the 2-plane spanned b y S and ˆ X . No w observ e that the actual arclength function L i (and hence, its v arious deriv a- tiv es) is in fact indep endent of the parametrization of the “horizon tal geo desics” γ t . P erforming a c hange of v ariable, w e can rew rite the second v a r iation formula in terms of the unit sp eed parametrization: (1) d 2 L dt 2 ( t ) = Z L ( t ) 0 ||∇ ¯ S ˆ X || 2 − K ( ¯ S ∧ ˆ X ) || ˆ X || 2 ds. where now ¯ S denotes the unit v ector in the direction of S , i.e. ¯ S = S/ | | S | | . Notice that b oth X and t he pro jection ˆ X of X orthogonal to S are Jacobi v ector ﬁelds, as they arise from v aria tions by g eo desics (see Section 2.3). W e p oint out an imp ortant conse quence of the second v ariation form ula in the context of non-p ositiv e curv ature. In this setting, equation (1) immediately forces d 2 L dt 2 ( t 0 ) ≥ 0 (since the expression inside the integral is ≥ 0). 2.3. Jacobi ﬁelds, rank of geo desics, and rigidity . F or the conv enience of the reader, w e brieﬂy recall some basic deﬁnitions f r o m Riemannian geometry , referring the reader to [Jo] for more details. Given a geo desic γ in a Riemannian manifold M n of dimension n , a v ector ﬁeld J along γ is said to b e a Jac obi ﬁeld if it satisﬁes the follo wing second order diﬀeren tial equation: J ′′ + R ( J, γ ′ ) γ ′ ≡ 0 where J ′′ refer to the second co v ariant deriv ativ e of J along γ , and R denotes the curv ature op erator . W e will require the follo wing classical results concerning Ja cobi ﬁelds: • Jacobi ﬁelds along γ form a ﬁnite dimensional v ector space (of dimension 2 n ), • a Jacobi ﬁeld is uniquely determined by its v alue (initial conditions) at an y t w o giv en p oints on γ , • give n a geodesic v ariation σ of γ as in the previous section, the “vertical v ector ﬁeld” X is a Jacobi ﬁeld a lo ng γ , • conv ersely , giv en a geo desic segmen t γ in a Riemannian manifold, and a Jacobi ﬁeld J along γ , there exists a geo desic v ariation whose “v ertical v ector ﬁeld” X coincides with J a long γ . Note in particular that the last t w o prop erties ab ov e tell us that Jacobi ﬁelds exactly enc o de the inﬁnitesimal b ehav ior o f geo desic v ariations. A Ja cobi ﬁeld J tha t additionally satisﬁes J ′ ≡ 0 will be called a p ar al lel Jac obi ﬁelds along γ . The r ank of 8 a geodesic γ is deﬁned to b e the dimension of the v ector space o f parallel Jacobi ﬁe lds along γ . Sinc e a conc rete example of a parallel Jacobi ﬁeld is giv en b y t he tangen t v ector ﬁeld V = γ ′ to the geo desic γ , we note tha t r k ( γ ) ≥ 1 for ev ery geo desic γ . A celebrated result in the geometry of non-p ositive ly curv ed Riemannian manifolds is the ra nk rigidity theorem o f Ballman-Burns-Spat zier [Ba2], [BuSp ]: Theorem 2.3 (Ra nk rigidity t heorem) . L et M b e a close d non-p ositively curve d Riemannian manifold, and ˜ M the universal c over of M with the induc e d R iemannian structur e. Assume that ˜ M has higher geometric rank , in the sense that every ge o desi c γ ⊂ ˜ M satisﬁes r k ( γ ) ≥ 2 . Then either: • ˜ M splits isometric al ly as a pr o duct of two simply c onne cte d Riemannian man- ifolds of non-p ositive curvatur e, or • ˜ M is an irr e ducible h i g her r ank symmetric sp ac e of non-c omp act typ e. In Section 6, w e will make extens iv e use of this rigidit y result to obtain the v ario us corollaries men tioned in the in tro duction. 2.4. Distorted subspaces in metric spaces. Let ( X , ρ ), ( Y , d ) b e a pair of metric spaces, and φ : Y → X an injectiv e map. W e deﬁne the distortion of the map φ to b e the suprem um, o v er all tr iples of distinct p oints x, y , z ∈ Y , of the quan tity :    ρ ( φ ( x ) , φ ( y )) ρ ( φ ( y ) , φ ( z )) − d ( x, y ) d ( y , z )    W e denote the distortion of φ b y δ ( φ ). Observ e t hat the distortion δ ( φ ) measures the diﬀerence b et w een relativ e distances in Y , and relative distances in φ ( Y ) ⊂ X . W e say that a metric space ( X , ρ ) con tains an undistorte d c opy of a metric space ( Y , d ) provide d t here exists an injectiv e map φ : ( Y , d ) ֒ → ( X , ρ ) with δ ( φ ) = 0. W e sa y that X contains almost undistorte d c opies if for an y ǫ > 0, one can ﬁnd a map φ ǫ : ( Y , d ) → ( X , ρ ) with δ ( φ ǫ ) < ǫ . Finally , give n a sequence of maps φ i : Y → X , w e sa y that t he sequence is undistorte d in the li m it , prov ided w e hav e lim δ ( φ i ) = 0. Let  denote the 4-p oin t metric space, consisting of the vertex set of the standard unit square in R 2 , with the induced distance, i.e.  consists of four p oints, with the four “side” distances equal to one, and the t w o “ dia g onal” distances equal to √ 2. W e call pairs of points at distance one a side pair of ve rtices. A large part of t his pap er will fo cus on ﬁnding and using ( almost) undistorted copies of  inside sim- ply connected complete Riemannian manifolds of non-p ositive curv ature (and inside their a symptotic cones). Giv en a (cyclicly ordered) 4-tuple of p o in ts { A, B , C , D } inside a space X , we will frequen tly iden tify the 4-tuple with a cop y of  , with the understanding that the ordered 4 -tuple o f p o ints corresp ond to the cyclicly ordered p oin ts in the square. W e no w p oin t out an easy lemma that allows us to o ccasionally “ignore diagonals.” 9 Lemma 2.4. L et { A j , B j , C j , D j } b e a se quen c e of 4 -tuples inside a CA T(0) sp ac e X . Assume that e a c h of the 4 -tuples satisﬁes the c onditions: • the p oint B j is the closest p oin t to C j on the ge o desic se gment A j B j , • the p oint A j is the closest p oin t to D j on the ge o desic se gment A j B j , • we have e quality of the side lengths d ( D j , A j ) = d ( A j , B j ) = d ( B j , C j ) = K j , • d ( C j , D j ) = K j (1 + ǫ j ) , with ǫ j → 0 . Then we have that d ( A j , C j ) /K j → √ 2 a n d d ( B j , D j ) /K j → √ 2 . Pr o of. Let us temp orarily ignore the indices j , and for a 4-tuple { A, B , C , D } of p oin ts as ab o v e, w e let d 1 , d 2 denote the lengths of the tw o diagonals AC , B D . W e no w w an t to con trol the tw o ratios d i /K in terms of ǫ , and in fact, show tha t the ratios tend to √ 2 as ǫ → 0. But this is relativ ely easy to do: consider a comparison triangle ¯ A ¯ B ¯ C ⊂ R 2 for the t r iangle AB C . The fact that t he p oin t B is the closest p oin t to C on the geo desic segmen t AB immediately implies that, in the comparison triangle, w e ha v e ∠ ¯ B ≥ π / 2. This in turn fo rces the inequalit y: d 2 1 = d ( ¯ A, ¯ C ) 2 ≥ d ( ¯ A, ¯ B ) 2 + d ( ¯ B , ¯ C ) 2 = 2 K 2 = ⇒ d 1 ≥ K √ 2 An identical argument establishes d 2 ≥ K √ 2. But on the other hand, we know that CA T(0) spaces satisfy , fo r any 4-tuples of p oin ts { A, B , C , D } the inequalit y: d ( A, C ) 2 + d ( B , D ) 2 ≤ d ( A, B ) 2 + d ( B , C ) 2 + d ( C , D ) 2 + d ( D , A ) 2 . Substituting the kno wn quan tities in to our expression, w e o btain: 2 · ( K √ 2) 2 ≤ d 2 1 + d 2 2 ≤ 3 · K 2 + [ K (1 + ǫ )] 2 Dividing out b y K 2 , w e see that the ratios d 1 /K , d 2 /K are a pair of real num bers ≥ √ 2 whic h satisfy the inequalit y: 4 ≤ ( d 1 /K ) 2 + ( d 2 /K ) 2 ≤ 3 + (1 + ǫ ) 2 . No w taking the indices j back into accoun t, it is no w immediate that as ǫ j → 0, the ratios d 1 /K j → √ 2 and d 2 /K j → √ 2, as desired. This conclude s the pro of of Lemma 2.4.  3. Fro m fla tte ning 4- tuples to p arallel Ja cobi fields. In this se ction, w e fo cus on establishing ho w certain seq uences o f 4-tuples of p o ints can b e used to construct para llel Jacobi ﬁelds along geo desics. More precisely , w e in tro duce the notion of: Deﬁnition (Go o d 4-tuple) . L et ˜ M b e a c omplete, simply c onne cte d, Riemmanian manifold of non-p ositive se ctional curvatur e, and let γ ⊂ ˜ M b e an arbitr ary ge o desic. We say that that a 4-tuple of p oints { A, B , C, D } in the sp ac e ˜ M is go o d (r elative to γ ) pr ovide d that A, B ∈ γ , AD ⊥ γ , B C ⊥ γ , and d ( D , A ) = d ( A, B ) = d ( B , C ) . 10 P B B B C C C D D D 1 1 1 2 3 2 2 3 3 γ Figure 2. Poin ted ﬂattening sequence along a geo desic. In eﬀect, a go o d 4-tuple is a g eo desic quadrilateral in ˜ M , with one side on the geo desic γ , the tw o adjacen t sides p erp endicular t o γ , and with tho se three sides ha ving exactly the same length. Deﬁnition (P ointe d ﬂattening sequence s) . We say that γ has p oin ted ﬂattening 4- tuples if given any p oint P ∈ γ , ther e exists a s e quenc e of { P , B i , C i , D i } of 4-tuples, e ach of which is go o d (r elative to γ ), satisﬁes lim B i = γ ( ∞ ) , and i s undistorte d in the lim i t. Figure 2 ab o v e illustrates t he ﬁrst three 4-tuples of a p oin ted ﬂattening sequence. The sides o f eac h quadrilateral are p erp endicular to the b ot tom geo desic, and the length of the top edge approach es (as a ratio in the limit) the length of the remaining three edges of the quadrilaterals. While the deﬁnition of p ointe d ﬂatt ening sequences of 4 -tuples migh t seem some- what artiﬁcial, the reader will see in Sections 4 and 5 that these are relativ ely easy to detect from the asymptotic cone. The main goal o f this section is to prov e: Theorem 3.1 (P ointed ﬂattening sequence ⇒ higher rank) . L et ˜ M b e a c omplete, simply c onne cte d, Riemmanian manifold of non - p ositive se ctional curvatur e, and let γ ⊂ ˜ M b e an a rb i tr ary ge o desic . If γ has p ointe d ﬂattening 4- tuples, then γ supp o rts a no n-trivial, ortho gonal, p ar al lel Jac o bi ﬁeld. In p articular, r k ( γ ) ≥ 2 . So let us star t with some P ∈ γ , and let { P , B i , C i , D i } b e the sequence of 4-tuples whose existenc e is ensured b y the hypothesis that γ has p oin ted ﬂattening sequences. Observ e that the point P = γ ( r ) divides the geo desic γ in to t w o geodesic rays, and we denote b y ~ γ P the geo desic ra y obtained b y restricting γ to [ r , ∞ ). Our approac h will b e to ﬁrst construct a non-trivial, orthogonal, parallel Jacobi ﬁeld along the geo desic ra y ~ γ P , and then let P tend to γ ( −∞ ). 11 In or der to cons truct the desired Jacobi ﬁeld alo ng the geo desic ra y ~ γ P , w e con- sider geo desic v ariations σ i in the space ˜ M , eac h of whic h is constructed from the corresp onding 4-tuple { P , B i , C i , D i } as fo llows: • α i : [0 , T i ] → ˜ M denotes the unit sp eed geo desic from P to D i , and β i denotes the one from B i to C i . W e set L i ( t ) = d ( α i ( t ) , β i ( t )), so in particular w e ha v e L i (0) = T i . • σ i is para metrised by { ( s, t ) : t ∈ [0 , T i ] , s ∈ [0 , T i ] } . • σ i , when restricted to the in terv a l { 0 } × [0 , T i ], coincides with α i , and when restricted t o the in terv al { 1 } × [0 , T i ], with β i . • for ev ery t ∈ [0 , T i ], the restriction of σ i to the interv al [0 , T i ] × { t } is the constan t sp eed geo desic from α i ( t ) t o β i ( t ). Note t ha t these maps are precis ely v ariations b y geo desics of the t ype discuss ed in section 2.2. Our go al will no w b e to analyze prop erties of the functions L i . W e start with the easy: F act 1: F or an y ﬁxed v alue of i , t he function L i is twic e diﬀeren tiable a nd conv ex . Twice diﬀeren tiabilit y follow s immediately from the formulas for the ﬁrst and sec- ond v ar iation o f ar clength. Con v exit y is immediate from the fact that L ′′ i ( t ) ≥ 0 (see the commen t immediately after equation (1) ) . F act 2: F or an y i a nd an y 0 ≤ x ≤ t ≤ T i , w e hav e L i ( t ) = L i ( x ) + ( t − x ) L ′ i ( x ) + Z t x Z y x L ′′ i ( τ ) d τ dy This is nothing but the F undamen tal Theorem of Calculus applied twice . F act 3: F or an y i a nd an y 0 ≤ t ≤ T i , w e hav e the following expression for L i ( t ): (2) L i ( t ) = L i (0) + Z t 0 Z y 0 L ′′ i ( τ ) d τ dy By F act 2, it is suﬃcien t to argue that eac h of the deriv ativ es L ′ i (0) is equal t o zero. No w r ecall that the ma ps σ i are geo desic v ar ia tions with the prop ert y that each of the “la teral curv es” α i ( t ) = P D i and β i ( t ) = B i C i are geo desics. F urthermore, since the 4-tuple { P , B i , C i , D i } is go o d (relative to γ ) , w e hav e tha t the “lateral curv es” are p erp endicular to the geo desic γ . Now applying the ﬁrst v ariation of arclength form ula (section 2.2), w e immediately see that L ′ i (0) = 0, as desired. Note that a conseq uence of F act 3 is that eac h L i is monotone non- decreasing. Now recall that the v ariations w e are considering come fr om a p o in ted ﬂattening sequen ce of 4-tuples, which in particular means that the corresp onding maps  → ˜ M are 12 undistorted in the limit. In our curren t notatio n, we hav e that d ( C i , D i ) = L i ( T i ) and d ( P , B i ) = T i , hence w e obtain: lim i →∞ L i ( T i ) T i = lim i →∞ d ( C i , D i ) d ( P , B i ) = 1 , F urthermore, recall that L i (0) = d ( P , B i ) = T i , and in particular w e hav e L i (0) /T i = 1 (for all i ). Combin ing this with o ur equation ( 2) in F act 3 (a pplied to t = T i ), w e see tha t : (3) lim i →∞ Z T i 0 Z y 0 L ′′ i ( τ ) T i dτ dy = 0 . The next step is to get rid of the T i factor inside the in tegral. F act 4: W e ha v e that: lim i →∞ Z T i / 2 0 L ′′ i ( τ ) dτ = 0 . T o see this, w e ﬁrst observ e that w e hav e the obvious series of equalities: 1 2 Z T i / 2 0 L ′′ i ( τ ) dτ = T i 2 Z T i / 2 0 L ′′ i ( τ ) T i dτ = Z T i T i / 2 Z T i / 2 0 L ′′ i ( τ ) T i dτ dy No w recall from F act 1 that L ′′ i ( τ ) ≥ 0 (b y conv ex ity ), a nd hence the expression inside eac h of the in tegrands ab o v e is ≥ 0 . But no w, b y p ositivity of each of the functions L ′′ i ( τ ) / T i , containmen t of the do mains of integrations yields the follo wing inequalit y: Z T i T i / 2 Z T i / 2 0 L ′′ i ( τ ) T i dτ dy ≤ Z T i 0 Z y 0 L ′′ i ( τ ) T i dτ d y . Com bining this upp er estimate with equation (3) ab ov e, w e immediately obtain: 0 ≤ lim i →∞ Z T i / 2 0 L ′′ i ( τ ) dτ ≤ 2 · lim i →∞ Z T i 0 Z y 0 L ′′ i ( τ ) T i dτ dy = 0 completing the pro of of F act 4 . Next w e note that a consequence of F act 4 is that the sequenc e of functions L ′′ i ( t ) tends to zero for almo st all t ∈ [0 , T i / 2]. In particular, we can ﬁnd a seq uence { t i } satisfying the fo llo wing t w o conditions: (1) t i ∈ [0 , T i ] a nd lim i →∞ t i = 0 (2) lim i →∞ L ′′ i ( t i ) = 0. Let us denote b y γ i : [0 , 1] → ˜ M the geo desic joining α i ( t i ) to β i ( t i ). Note that these geo desics ar e precisely the curv es σ i ( − , t i ) : [0 , L i ( t i )] → ˜ M , where σ i is our sequence of v a r iations of geo desics. W e next observ e that: 13 F act 5: The geo desic segmen ts γ i tend to the geo desic ra y ~ γ P ⊂ γ . T o see this, w e ﬁrst note since the “lateral curv es” for the v ariat io n σ i are geo desics p erp endicular to γ (and since w e hav e K ≤ 0), w e hav e d ( α i ( t i ) , γ ) = t i = d ( β i ( t i ) , γ ) In particular, w e see that the geo desic segmen ts γ i join a pair of p o in ts whose distance from γ tends to zero. Since geo desic neigh b orho o ds o f γ are con v ex (by the no n- p ositiv e curv at ur e h yp othesis), w e conclude that the distance of any p oint on γ i is at most t i a w a y fro m the g eo desic γ , where t i w as c hosen to tend to 0. F urthermore, w e clearly hav e that lim α i ( t i ) = P , and lim β i ( t i ) = γ ( ∞ ), and hence w e o bta in lim γ i = ~ γ P , as desired. No w alo ng eac h of the geo desic segmen ts γ i , we hav e that the corresp onding geo- desic v ariation σ i giv es r ise to a Jacobi vec tor ﬁeld X i . W e no w fo cus our atten tion to this sequence of Jacobi ﬁelds. F act 6: The Jacobi ﬁeld X i along γ i satisﬁes || X i ( p ) || ≤ 1 for all p ∈ γ i . T o see this, ﬁr st observ e that X i (0) = α ′ i ( t i ) and X i ( L i ( t i )) = β ′ i ( t i ). Since α i and β i are unit sp eed para metrized, this implies that || X i (0) || = || X i ( L i ( t i )) || = 1 . But from the non- p ositiv e curv ature a ssumption and t he Jacobi diﬀeren tial equation, it follo ws that the square-norm of a Jacobi ﬁeld along a geo desic is a con v ex function. Since || X i || = 1 at the endp oin ts of the geo desic γ i , F act 6 follows. F act 7: Up to possibly passing to subsequences , the Jacobi ﬁelds X i along γ i con v erge (uniformly on compact sets), to a Jacobi ﬁeld X along ~ γ P . This follow s from the general fa ct that a Ja cobi ﬁeld is determined by an y tw o of its v alues. T ak e p oints p i , q i in γ i that con v erge to p oin ts p 6 = q of ~ γ P . F rom F act 6, we see that up to p ossibly pa ssing to subsequences , b oth X i ( p i ) and X i ( q i ) hav e a limit. Moreo v er, Jacobi ﬁelds a re solution o f ordinary diﬀerential equations with smo oth co eﬃcien ts (in f a ct with the regularit y of the curv ature tensor) and therefore dep end con tin uously on the initial dat a (the v alues a t p i and q i .) It follows that X i con v erge to a Jacobi ﬁeld X along γ uniformly on compact sets, and in particular p oin t-wise. F act 8: The sequence { t i } can b e c hosen so that the limiting v ector ﬁeld X is p erp endicular to the g eo desic ra y ~ γ P . T o see this, we note that fo r each of the v ar iations σ i , we hav e the tw o asso ciat ed con tin uous v ector ﬁelds S i , X i (see section 2.2) . F urthermore, note that these tw o v ector ﬁelds are o rtho gonal along the base geo desic γ i . Indeed, the v ector ﬁeld S i 14 is just γ ′ i , while the v ector ﬁeld X i is orthogonal to γ i at the t w o endpoints of t he v ar ia tion (recall that α i , β i are ⊥ to γ i ). But from the Jacobi equation, a Jacobi ﬁeld that is orthogo nal to a geo desic at a pair of p oints is orthogonal to the geo desic at every p oin t. Next observ e that the inner pro duct b etw een the v ectors X i and S i v ar ies contin- uously along the domain of σ i . Since we ha v e h X i , S i i ≡ 0 along the g eo desic γ , b y c ho osing t i close enough to zero, one can ensure that lim i →∞ sup x ∈ γ i   h X i , S i i x   = 0 . In particular, for any sequence of p oin ts { p i } ⊂ ˜ M satisfying p i ∈ γ i and lim p i = p ∈ γ , w e hav e t hat: h X ( p ) , γ ′ ( p ) i = h lim i →∞ X i ( p i ) , lim i →∞ γ ′ i ( p i ) i = lim i →∞ h X i ( p i ) , S i ( p i ) i = 0 Applying this to the t w o sequences of p oints w ith distinct limits, we see that the lim- iting vec tor ﬁeld X is o rthogonal to ~ γ P at tw o distinct p o in ts, and hence is orthogonal to ~ γ P at ev ery p oint. In fact, the discussion a b o v e also show s that the Jacobi ﬁeld X deﬁned o n ~ γ P extends to a p erp endicular Jacobi ﬁeld along the entire geo desic γ . F act 9: The Jacobi v ector ﬁeld X along ~ γ P satisﬁes: (4) Z ~ γ P − K ( X ∧ ˙ γ ) | | X || 2 + || ∇ ˙ γ X || 2 ds = 0 . This follo ws imm ediately from F acts 5, 7, condition (2) in our c hoice of the seq uence { t i } (see the discuss ion preceding F act 5), and the second v ar ia tion form ula for L ′′ i ( t i ) (see section 2.2, equation (1 )). Indeed, this is just an application o f the Leb esgue dominated con v ergence theorem (the in tegrand is p ositiv e, b ounded on compact se ts, and we hav e p oint-wise con v ergence.) Observ e that at this p oin t, w e are a lmost done. Since ˜ M has non-p ositiv e sectional curv ature, we see tha t the expression inside the in tegral in equation (4 ) consists of a sum of t w o terms that are ≥ 0 ( p oin tw ise). Since t he o v erall inte gral is zero, and the expression inside t he in tegral v aries con tinuous ly , this tells us that at every p oin t along ~ γ P , w e hav e that: − K ( X ∧ ˙ γ ) || X || 2 = 0 and ||∇ ˙ γ X || 2 = 0 . F urthermore, at the p o in t P w e see that the v ector ﬁeld X is the limit of v ectors of norm =1 (see F act 6), and whose angle with γ ′ tends to π / 2 (see F act 8). In particular this gives : F act 10: The Jacobi ﬁeld X is not the zero v ector ﬁeld, since we ha v e X ( P ) 6 = 0. 15 Finally , let us complete the pro of of t he theorem. Let { P k } b e a seque nce of p oin ts o n γ , with P k = γ ( t k ) for a strictly decreasing sequence of r eal n um b ers t k with lim t k = −∞ (so in part icular, lim P k = γ ( −∞ )) . Let J denote the (2 n − 2)- dimensional vector space of ortho g onal Jacobi ﬁelds along the geo desic γ . Corre- sp onding to eac h P k , w e let J k ⊂ J denote the collec tion of all orthogonal Ja cobi ﬁelds on γ hav ing the prop erty that they are parallel along the geo desic ray ~ γ P k (with no constrain ts on the b eha vior on the rest of γ ). It is obv ious that each J k is actually a v ector subspace of J , and our pro of ensures that each J k con tains a non-zero ve ctor ﬁeld, and in particular, satisﬁes dim J k ≥ 1. F urthermore, whenev er k ≥ k ′ , w e hav e a con tainmen t of geo desic rays ~ γ P k ′ ⊂ ~ γ P k , whic h immediately yields con tainmen ts J k ⊂ J k ′ . Since w e hav e a sequence of nested, non- trivial, v ector sub- spaces of the ﬁnite dimen s i o nal v ector space J , w e conclude that their inte rsection is non-zero. This implies the existence of a globa lly deﬁned, non-trivial, para llel, orthogonal Jacobi ﬁeld along γ , completing the pro o f of the t heorem.  4. From fla ts in the ul tralimit to fla tte ning sequences. In this section, w e fo cus exclusiv ely on ﬁnding conditions o n the ultralimit C one ( ˜ M ) that can b e used to construct p oin ted ﬂa t tening sequences along a geo desic γ . This en tire section will b e devoted to establishing the following: Theorem 4.1 (Undistorted  in ultr alimit ⇒ P oin ted ﬂattening sequence) . L et ˜ M b e a simply c onne cte d, c omplete, R iemmanian manifold of non-p ositive se ctional curvatur e, and let C one ( ˜ M ) b e an asymptotic c one of ˜ M . Given a ge o desic γ ⊂ ˜ M , let γ ω ⊂ C one ( ˜ M ) b e the c orr esp onding ge o de sic in the ultr alimit. Assume that ther e exists a 4-tuple of p oints { A, B , C, D } ⊂ C one ( ˜ M ) , satisfying A, B ∈ γ ω , with ∗ ∈ I nt ( AB ) , an d so that the asso ciate d map  → C one ( ˜ M ) is undistorte d. Then the origi n al ge o d e s ic γ has p ointe d ﬂattening se quenc e s . In the next section, we will establish a strengthening of this result, b y considering the case where γ ω is contained in a bi-Lipsc hitz ﬂat (i.e. a bi- Lipsc hitz ima g e of R 2 equipped with the standard metric). In t his more general con text, and under the presence of some additional constraints w e will see t ha t γ s till has p ointed ﬂa t t ening sequence s. Before starting the pro of of t he t heorem, let us ﬁrst in tro duce an auxiliary notion. Deﬁnition (Flattening sequences) . We say that γ has ﬂattening 4-tuples if ther e exists a se quenc e { A i , B i , C i , D i } of 4-tuples of p o ints e ach of whi ch is go o d (r elative to γ ), has lim A i = γ ( −∞ ) and lim B i = γ (+ ∞ ) , and viewing the 4-tuples as a se quenc e of maps  → ˜ M , we also r e quir e the se quenc e to b e undis torte d in the limit. An illustratio n of a ﬂattening sequence is prov ided in Figure 3. All the quadrilat- erals are p erp endicular along the base geo desic, ha v e three sides of equal length, and 16 Figure 3. Fla t t ening sequence along a geo desic. ha v e t he length of the top edge tending (asymptotically , in the ratio) to the length of the remaining three edges. W e now b egin the pro of o f theorem 4.1, b y establishing: Step 1: The geo desic γ ha s a ﬂattening sequence . Pr o of (Step 1) . In the ultr a limit C one ( ˜ M ), let us pic k out p oin ts { A, B , C, D } to b e the v ertices of an undistorted square, with the pro p ert y that AB ⊂ γ ω , and ∗ ∈ I nt ( AB ). W e now intend to sho w that a suitable appro ximating sequence of 4-tuples in ˜ M will giv e us the desired ﬂattening sequence. Let us start out by pic king a n arbitrary pa ir of approximating sequences { C ′ i } and { D ′ i } for the p oin ts C, D ∈ C one ( ˜ M ). No w obse rv e that corres p o nding to the geo desic γ ⊂ ˜ M , w e hav e a well-deﬁne d pro jection map ρ : ˜ M → γ , where ρ ( x ) is deﬁned to b e the unique p oin t on γ closest to the p oint x . W e now deﬁne the sequence { A i } (resp ectiv ely { B i } ) b y setting A i := ρ ( D ′ i ) (resp ectiv ely B i := ρ ( C ′ i )). Note that eac h of the 4-tuples of p oin ts { A i , B i , C ′ i , D ′ i } clearly satisﬁe s the ﬁrst three prop erties of b eing go o d for the geo desic γ (the p oin ts A i , B i are on γ , and the sides A i D ′ i and B i C ′ i are ⊥ to γ ). T o ensure the last condition, w e pic k p o ints C i ∈ − − → B i C ′ i , D i ∈ − − − → A i D ′ i satisfying d ( D i , A i ) = d ( A i , B i ) = d ( B i , C i ). Note that this construction is exactly the sort considered in Lemma 2.1. In particular, statemen t (1) in Lemma 2.1 tells us that { A i } = A ∈ C one ( ˜ M ) and { B i } = B ∈ C o ne ( ˜ M ), while statemen t (2) in Lemma 2.1 ensures t ha t { C i } = C ∈ C one ( ˜ M ) and { D i } = D ∈ C one ( ˜ M ). Up to this p oin t, we ha v e constructed a sequence { A i , B i , C i , D i } of 4- tuples, eac h of whic h is go o d for the geo desic γ , and additionally having the prop ert y tha t the sequence (ultra) - con v erges to the 4 -tuple { A, B , C, D } ⊂ C one ( ˜ M ). T o conclude, w e simply need to ensure that our sequence also satisﬁes the t w o conditions for a ﬂattening sequence , namely , w e need (1 ) that lim A i = γ ( −∞ ) and lim B i = γ (+ ∞ ), 17 and (2) that lim i →∞ { d ( C i , D i ) /d ( A i , B i ) } = 1 . W e will ensure t hat these additional conditions are satisﬁed b y passing to suitable subseque nces of our original sequenc e. Let us ex plain the argumen t for (2); the argument for (1 ) is analogous (after a p ossible p erm utation o f lab els on the 4-t uples of p oints). Giv en an y k ∈ N , the f act that ω lim { d ( C i , D i ) /λ ( i ) } = d ω ( C , D ) implies t hat the set of indices i for which the follow ing relation holds (5)    d ( C i , D i ) λ ( i ) − d ω ( C , D )    < 1 k forms a subset I k ∈ U . Similarly , t he fact that ω lim { d ( A i , B i ) /λ ( i ) } = d ω ( A, B ) implies that t he set of indices i for whic h the fo llo wing relation holds (6)    d ( A i , B i ) λ ( i ) − d ω ( A, B )    < 1 k lik ewise forms a subset I ′ k ∈ U . Since the ultraﬁlter U is closed under in tersections, w e conclude that I k ∩ I ′ k ∈ U . But ev ery elemen t in U is a n inﬁnite subset of N , and in particular, non-empt y . Hence there is an index i k for whic h b ot h equations (5) and (6) hold. Now conside r the subsequenc e { A i k , B i k , C i k , D i k } of 4-tuples in ˜ M , and o bserv e that f r om (5) and (6), we ha v e that the k th 4-tuple satisﬁes: d ω ( C , D ) − 1 /k d ω ( A, B ) + 1 /k ≤ d ( C i k , D i k ) d ( A i k , B i k ) ≤ d ω ( C , D ) + 1 /k d ω ( A, B ) − 1 /k No w it is immediate that for this subsequence, we obtain: lim k →∞ d ( C i k , D i k ) d ( A i k , B i k ) = d ω ( C , D ) d ω ( A, B ) = 1 where the la st equalit y comes from the fact that the quadrilateral { A, B , C, D } ⊂ C one ( ˜ M ) is an undistorted copy of  . But this is precisely the desired prop ert y (2). So w e no w know t hat the geo desic γ w e w ere in terested in has a ﬂattening sequence. The second step in the pro o f of theorem 4.1 lies in impr oving the choice of our subseque nce, obtaining: Step 2: The geo desic γ ha s p ointe d ﬂattening sequences. Pr o of (Step 2) . In the pro of of Step 1, w e started out b y constructing a sequence of 4-tuples { A i , B i , C i , D i } , eac h of whic h w as go o d for γ . The ﬂattening sequence along γ was then obtained b y pic king a suitable subsequence of this sequence of 4- tuples. W e no w pro ceed to explain ho w, by b eing a bit more careful with our c hoice of subseque nce, w e can construct p oin ted ﬂa ttening sequences. T o this end, let { A i , B i , C i , D i } b e the sequence of go o d 4-tuples for γ obtained in Step 1, a nd let P ∈ γ b e a n arbitrary c hosen p oint. Since ω lim { A i } = A , ω lim { B i } = B , ω lim { P } = ∗ , and w e ha v e a con tainmen t ∗ ∈ I nt ( AB ) ⊂ C one ( ˜ M ), 18 w e immediately see that the set of indices J 1 ⊂ N fo r whic h the corresp onding 4- tuples has the prop erty that P ∈ I nt ( A i B i ) consists of a set in our ultraﬁlter: J 1 ∈ U . No w consider the nearest p oint pro jection ρ : ˜ M → γ , a nd let us ﬁrst consider indices i ∈ J 1 . Since eac h of the 4-tuples in our se quence is go o d, w e clearly ha v e ρ ( D i ) = A i and ρ ( C i ) = B i . Observ e that P disconnects the geo desic γ into t w o comp onen ts (since i ∈ J ), and the p oin ts A i , B i lie in distinct componen ts of γ − { P } . Since ρ ( D i C i ) giv es a path in γ jo ining A i to B i , we conclude that there m ust exist a p oin t E i ∈ D i C i satisfying ρ ( E i ) = P . Observ e that this immediately implies that P E i ⊥ γ ′ . F or t he remaining indice s i / ∈ J 1 , w e set E i = C i . In particular, we now ha v e a sequence of p o in ts { E i } , with E i ∈ D i C i . No w it is easy to v erify that the sequenc e { E i } deﬁnes a p oin t E ∈ C one ( ˜ M ), and since each E i ∈ D i C i , we ha v e that E ∈ D C . F urthermore, the fa ct that ρ ( E i ) = P fo r a collection of indices i ∈ J 1 con tained in our ultraﬁlter U implies that ρ ω ( E ) = ω lim { P } = ∗ ∈ C one ( ˜ M ), where ρ ω : C one ( ˜ M ) → γ ω is the pro jection map f r om C one ( ˜ M ) to γ ω . Observ e that the 4-t uple of p oin ts { A, B , C, D } ⊂ C one ( ˜ M ), corresp o nding to an undistorted  , satisﬁes t he equality : (7) 1 = d ω ( A, C ) 2 + d ω ( B , D ) 2 − d ω ( C , D ) 2 − d ω ( A, B ) 2 2 · d ω ( A, D ) · d ω ( B , C ) But inside a geo desic space, a 4-tuple of (distinct, non-colinear) p oin ts satisﬁes the equalit y in equation (7) if and only if the 4-tuple o f p oin ts a re the v ertices of a ﬂat pa r a llelogram (see Berg-Nik olaev [BeNi, Theorem 15 ]). Applying this to the giv en 4-tuple of p oin ts in C o ne ( ˜ M ), we see that there exists an isometric em b edding P ֒ → C one ( ˜ M ) from a square P ⊂ R 2 , with the prop erty that the v ertices map precisely t o t he p oints { A, B , C , D } . But no w w e immediately see that the p oin t E ∈ C D must be the po int satisfy ing d ω ( E , C ) = d ω ( P , B ), and in particular, that the 4-tuple { P , B , C , E } are the v ertices of a ﬂat rectangle in C one ( ˜ M ), with d ω ( C , E ) = d ω ( P , B ) < d ω ( P , E ) = d ω ( B , C ). So w e can ﬁnd a collection of indices J 2 ∈ U with the prop erty that for all i ∈ J 2 , d ( P , B i ) < d ( P , E i ) a nd d ( P , B i ) < d ( B i , C i ). F or eac h of the indices i ∈ J 2 , w e can no w c ho ose points F i ∈ P E i , G i ∈ B i C i to satisfy d ( P , F i ) = d ( P , B i ) = d ( B i , G i ). F or indices i / ∈ J 2 , we set F i = E i , G i = C i . It is again easy to verit y that the sequence s { F i } , { G i } deﬁne p oints F , G ∈ C o ne ( ˜ M ). F urthermore one can v erify that the 4 - tuple { P , B , G, F } deﬁne an undistorted  in C one ( ˜ M ). Figure 4 con tains an illustration of where the v ario us p oints E i , F i , G i are chose n from the original 4 - tuple { A i , B i , C i , D i } (for indices i ∈ J 1 ∩ J 2 ). W e observ e that for our sequence of 4-tuples we now ha v e: • the sequence of 4-tuples { P , B i , G i , F i } ultra-con v erges to the v ertices o f an undistorted square in C one ( ˜ M ), and 19 A P B F E C i i i i G i i γ D i Figure 4. Constructing p oin ted ﬂattening sequences. • for eac h index i ∈ J 1 ∩ J 2 , the corresp onding 4- t uple { P , B i , G i , F i } is go o d. W e now w an t to pic k a subseque nce of 4-tuples, within the index set J 1 ∩ J 2 , whic h satisﬁes the additional prop ert y that lim k →∞ { d ( F i k , G i k ) /d ( P , B i k ) } = 1. So deﬁne, for k ∈ N , the sets I k , I ′ k ∈ U t o b e the set of indices satisfying the ob vious analogues of equ ations (5 ) and (6) from Step 1. Then w e se e that, from the closure of ultraﬁlters under ﬁnite intersec tions, we hav e for eac h k ∈ N the set I k ∩ I ′ k ∩ J 1 ∩ J 2 lies in our ultraﬁlter U , a nd hence is non-empty . In pa r ticular, we can select indices i k ∈ I k ∩ I ′ k ∩ J 1 ∩ J 2 , and the argumen t giv en at the end of Step 1 extends ve rbatim to see that the subsequenc e { P , B i k , G i k , F i k } s atisﬁes the desired additional prop erty . This completes the pro of of Step 2, and hence of Theorem 4.1.  Finally , w e conclude this section b y p o in ting out that if the geodesic γ ω ⊂ C one ( ˜ M ) is con tained in a ﬂat , then the com bination of Theorem 4.1 and Theorem 3.1 imme- diately tells us that γ must b e of higher rank. In particular, this completes the pro of of Theorem 1.1. 5. Fro m bi-Lipschitz fla ts to fla ttening sequences. In the previous section, w e sa w that w e can use the presence of ﬂats in the asymp- totic cone C one ( ˜ M ) to construct ﬂattening sequences (whic h in turn could b e used to construct p ointe d ﬂattening sequences). F or our a pplications, it will b e imp ortant for us to b e able to use bi-Lipschitz ﬂats instead of gen uine ﬂats. The reason for this is that bi-Lipsc hitz ﬂats in C one ( M ) app ear naturally as ultralimits of quasi-ﬂats in the or ig inal M . T o establish the result, w e will ﬁrst sho w (Lemma 5.1) that w e can use suitable sequence s of maps  → C one ( ˜ M ) that are undistorted in the limit t o construct ﬂattening sequences along γ . W e will then sho w (Theorem 5.2) that in the presence 20 of certain bi-Lipsc hitz ﬂats, one can construct the desired sequence of maps  → C one ( ˜ M ) that are undistorted in the limit. Lemma 5.1 (Almost undistorted metric squares ⇒ ﬂattening sequence) . L et ˜ M b e a c omplete, simply-c onne cte d, Riemannian manifo ld of non-p ositive se ctional curva- tur e, and C one ( ˜ M ) an asymptotic c one of ˜ M . F or γ ⊂ ˜ M a ge o desic, let γ ω ⊂ C one ( ˜ M ) b e the c orr esp onding ge o desic in the asymptotic c one of ˜ M . Assume that for e ach ǫ > 0 , on e c an ﬁnd an ǫ -undistorte d c opy { A, B , C , D } of  in C one ( ˜ M ) satisfying the pr op erties: • A, B ∈ γ ω with ∗ ∈ I nt ( AB ) , • A, B ar e the closest p oints to D , C (r esp e ctively) on the ge o desic γ ω . Then γ has a ﬂattening se quenc e of 4 - tuples. Pr o of. W e w an t to build a sequenc e of g o o d 4-tuples in ˜ M whic h are undistorted in the limit. W e will explain how to ﬁnd, fo r a giv en ǫ > 0, a go o d 4 -tuple in ˜ M with distortion < ǫ . Cho osing such 4-tuples for a sequence of error terms ǫ k → 0 will yield the desired ﬂattening sequence. F rom the h yp otheses in our Lemma, w e can ﬁnd a 4-tuple { A, B , C , D } ⊂ C one ( ˜ M ) satisfying ∗ ∈ I nt ( AB ) ⊂ γ ω , with distortion < ǫ/ 3. Now this 4-tuple in the as- ymptotic cone correspo nds to a sequence { A i , B i , C i , D i } ⊂ ˜ M of 4-tuples satisfying { A i } = A, { B i } = B , { C i } = C , { D i } = D . Applying Lemma 2.1, w e can replace this sequence of 4 - tuples b y ano ther sequence { A ′ i , B ′ i , C ′ i , D ′ i } ⊂ ˜ M satisfying the additional prop ert y that: A ′ i , B ′ i are the closest p o ints on γ to the p oin ts D ′ i , C ′ i resp ectiv ely . Since the distortion of { A, B , C , D } ⊂ C one ( ˜ M ) is < ǫ/ 3, we hav e that for some index i , the distortion of { A ′ i , B ′ i , C ′ i , D ′ i } ⊂ ˜ M is lik ewise < ǫ/ 3 (in fact, the set of suc h indices i has to lie in the ultraﬁlter U ). No w the problem is that there is no guara n tee that this 4-tuple { A ′ i , B ′ i , C ′ i , D ′ i } is go o d for the geo desic γ : while it satisﬁes the orthog onalit y conditions D ′ i A ′ i ⊥ γ a nd C ′ i B ′ i ⊥ γ , the 4- tuple do es not necessarily ha v e the r equisite prop erty that d ( D ′ i , A ′ i ) = d ( A ′ i , B ′ i ) = d ( B ′ i , C ′ i ). So w e mo dify the 4-tuple in the o b vious manner, b y replacing the p oints C ′ i , D ′ i ∈ ˜ M by the p oin ts C ′′ i ∈ − − → B ′ i C ′ i , D ′′ i ∈ − − − → A ′ i D ′ i c hosen so that d ( D ′′ i , A ′ i ) = d ( A ′ i , B ′ i ) = d ( B ′ i , C ′′ i ). This new sequence of 4-tuples now do es hav e the prop erty of b eing go o d for γ . So in order to complete the Lemma, w e just need t o ma ke sure that this new 4-tuple { A ′ i , B ′ i , C ′′ i , D ′′ i } has distortion < ǫ . Note that b y Lemma 2.2 , it is suﬃcien t to sho w that the distance d ( C ′′ i , D ′′ i ) is not to o m uc h larger than d ( A ′ i , B ′ i ). Letting K := d ( A ′ i , B ′ i ), w e ﬁrst o bserv e that f rom the fact that t he (unmo diﬁed) 4-tuple { A ′ i , B ′ i , C ′ i , D ′ i } has distortion < ǫ/ 3 implies that: 1 − ǫ/ 3 < d ( A ′ i , D ′ i ) /K < 1 + ǫ/ 3 21 whic h translates to the estimate: d ( D ′ i , D ′′ i ) = | d ( A ′ i , D ′ i ) − K | < K · ǫ/ 3 An iden tical argumen t give s the estimate d ( C ′ i , C ′′ i ) < K · ǫ/ 3. Now the t r iangle inequalit y give s us the estimate: | d ( C ′′ i , D ′′ i ) − d ( C ′ i , D ′ i ) | ≤ d ( C ′ i , C ′′ i ) + d ( D ′ i , D ′′ i ) < 2 K · ǫ/ 3 Dividing b y K , w e obta in:    d ( C ′′ i , D ′′ i ) K − d ( C ′ i , D ′ i ) K    < 2 ǫ/ 3 But since the original 4-tuple w as ǫ/ 3-undistorted, w e hav e that d ( C ′ i , D ′ i ) /K is within ǫ/ 3 of 1. Hence a pplying the triangle inequalit y o ne last time giv es: 1 − ǫ < d ( C ′′ i , D ′′ i ) K < 1 + ǫ precisely as desired. This concludes the pro of o f Lemma 5.1.  No w assume tha t we ha v e a bi- Lipsc hitz map φ : R 2 ֒ → C one ( ˜ M ). W e will call the image φ ( R 2 ) a bi-Lipschitz ultr aﬂat . The primary application of almost undistorted metric squares lies in establishing the follow ing: Theorem 5.2 (Bi-L ipschitz ultraﬂat ⇒ ﬂattening s equences.) . L e t ˜ M b e a c omplete, simply-c onne c te d, Riemannian m anifold of non-p ositive se ctional curvatur e, and let C one ( ˜ M ) a n asymptotic c one of ˜ M . F o r γ ⊂ ˜ M a ge o des i c , let γ ω ⊂ C one ( ˜ M ) b e the c orr esp onding ge o desic in the asymptotic c one of ˜ M . Assume that: • the se ctional curvatur e of ˜ M is b ounde d b elow, • ther e exists g ∈ I som ( M ) which stabilizes γ , a nd acts c o c omp actly on g , and • ther e is a bi-Lipschitz ultr aﬂat φ : R 2 ֒ → C one ( M ) mapping the x -axis to γ ω . Then γ has a ﬂattening se quenc e of 4 - tuples. Pr o of. Our approac h is to reduce the problem to one whic h can b e dealt with b y meth- o ds similar to those in the previous Lemma 5.1. Let φ : ( R 2 , | | · || ) ֒ → ( C one ( ˜ M ) , d ) b e the giv en bi-Lipsch itz ultraﬂat, and let C b e the bi-Lipsc hitz constant. F or r ∈ R , let us denote b y L r ⊂ R 2 the horizontal line at heigh t r . Note that L 0 coincides with the x -axis in R 2 , and hence, by h yp othesis, m ust map to γ ω under φ . Note that to mak e our v arious express ions more readable, w e are using d to denote distance in C one ( ˜ M ) (as opp o sed to d ω ), and the norm notation t o denote distance inside R 2 . W e no w deﬁne, for eac h r ∈ [0 , ∞ ) a map ψ r : L r → L 0 as follows: giv en p ∈ L r , w e ha v e φ ( p ) ∈ C one ( ˜ M ). Since γ ω ⊂ C one ( ˜ M ) is a geo desic inside the CA T(0) space C one ( ˜ M ), there is a well deﬁned, distance non-increasing, pro jection map π : C one ( ˜ M ) → γ ω , whic h sends any giv en p o in t in C o ne ( ˜ M ) to the (unique) closest 22 X X X X X X 0 1 2 3 k−1 k y 0 y k L r L 0 ψ ψ r r ( ( X 0 X k ) ) Figure 5. Illustration of the pro of of our Assertion . p oin t o n γ ω . Hence give n p ∈ L r , w e hav e the comp osite map π ◦ φ : L r → γ ω . But recall that, b y hypothesis φ maps L 0 homeomorphically to γ ω . W e can now set ψ r : L r → L 0 to b e the comp osite map φ − 1 ◦ π ◦ φ . W e no w ha v e the following: Assertion: F or ev ery r ∈ [0 , ∞ ), there exist a pair of p oints p, q ∈ L r ⊂ R 2 ha ving the prop ert y that || p − q || = 1, and || ψ r ( p ) − ψ r ( q ) | | > 1 / 2. Let us explain ho w to obtain the Assertion . W e ﬁrs t obse rve that for ar bit r a ry x ∈ L r , w e hav e that the distance from x to L 0 is exactly r , and hence fro m the bi-Lipsc hitz estimate, w e ha v e d ( φ ( x ) , γ ω ) = d ( φ ( x ) , φ ( L 0 )) ≤ C r Since π is the ne arest p oint pro jection on to γ ω , this implies that d  φ ( x ) , ( π ◦ φ ) ( x )  ≤ C r . Since ( π ◦ φ )( x ) = φ ( ψ r ( x )), w e can again use the bi- Lipsc hitz estimate to conclude t ha t : C r ≥ d  φ ( x ) , ( π ◦ φ )( x )  = d ( φ ( x ) , φ ( ψ r ( x )) ≥ 1 C · | | x − ψ r ( x ) || Whic h giv es us the estimate: || x − ψ r ( x ) || ≤ C 2 r . Finally , t o establish t he Assertion , let us argue b y con tradiction (w e will ulti- mately con tradict the upp er b ound on || x − ψ r ( x ) || obtained in the previous par a- graph). Consider, for in tegers k ≥ 0, the p oint x k := ( k , r ) ∈ L r , and y k := ( k , 0) ∈ L 0 . Observ e that w e clearly ha v e || x i − x i +1 || = 1, and let us assume, by w ay of con tradiction, that every pair { x i , x i +1 } satisﬁes || ψ r ( x i ) − ψ r ( x i +1 ) || ≤ 1 / 2. W e then observ e that we can easily estimate from ab o v e the distance b et w een ψ r ( x k ) and the origin y 0 : || y 0 − ψ r ( x k ) || ≤ || y 0 − ψ r ( x 0 ) || + k X i =1 || ψ r ( x i − 1 ) − ψ r ( x i ) || Note that since the three p oin ts x 0 , y 0 , a nd ψ r ( x 0 ) form a righ t triangle, tw o of whose sides are con trolled, we can estimate from a b o v e || y 0 − ψ r ( x 0 ) || ≤ r √ C 4 − 1 . 23 P Q B Q P A B A 1 1 1 1 2 2 2 2 γ ω φ (L 1 ) φ (L 2 ) Figure 6. Cho o sing the p oints { A j , B j , Q j , P j } . Com bined with our a ssumption that all the || ψ r ( x k ) − ψ r ( x k +1 ) || ≤ 1 / 2, this yields the estimate: || y 0 − ψ r ( x k ) || ≤ r √ C 4 − 1 + k / 2 Since || y 0 − y k || = k , t he estimate ab ov e immediately giv es us the lo w er b ound: || y k − ψ r ( x k ) || ≥ k/ 2 − r √ C 4 − 1 But now, using the fa ct that the three p oin ts x k , y k , ψ r ( x k ) form a righ t tr iangle, we obtain the low er b ound: || x k − ψ r ( x k ) || ≥ q r 2 + ( k / 2 − r √ C 4 − 1) 2 Note that the low er b ound ab o v e tends to inﬁnity as k → ∞ , and hence for k suﬃ- cien tly large, yields | | x k − ψ r ( x k ) || > C 2 r , whic h con tradicts the previously obta ined upp er b ound || x k − ψ r ( x k ) || ≤ C 2 r . Hence our initial assumption m ust ha v e b een wrong, i.e. there exist a pair { x k , x k +1 } satisfying | | ψ r ( x k ) − ψ r ( x k +1 ) || > 1 / 2, com- pleting the pro of of the Assertion . F or a n illustration of this argument, w e refer the reader to Figure 5. The parallel lines are L 0 at the bottom, L r at the top. The p oin ts x i are represen ted along the line L r , with pairwise distance = 1. The p oints along L 0 represen t the corresp o nding ψ r ( x i ), with a straigh t line segmen t joining eac h x i to t he corresp onding ψ r ( x i ). Our argumen t is merely making fo r mal the fact that if all the success iv e distances along L r are = 1, while all the suc cessiv e distances along L 0 are < 1 / 2, then eve n tually the b old segmen t x k ψ r ( x k ) has arbitr a ry large length (in particular > C 2 r , a contradiction). 24 Let us now use the Assertion to construct almost undistorted metric squares of the ty p e app ear ing in L emma 5.1. F or each index j ∈ N , let us tak e a pair of p oin ts p j , q j ∈ L j whose existenc e is ensured by the Assertion. Consider now the 4-tuple of p o in ts { A j , B j , Q j , P j } in C one ( ˜ M ), deﬁned b y P j = φ ( p j ), Q j = φ ( q j ), A j = ( π ◦ φ )( p j ) , and B j = ( π ◦ φ ) ( q j ). An illus tratio n of a few of these 4-tuples is giv en in Figure 6 ab ov e. The slan ted surface represen ts the bi-Lipsc hitz ﬂat in C one ( ˜ M ), along with the imag e of the horizontal lines L 1 , L 2 ⊂ R 2 under the map φ , and the corresp onding 4- tuples of p oints . No w for eac h intege r j , w e observ e that the corresp onding 4-tuple of p oints in C one ( ˜ M ) satisﬁes the follo wing nice prop erties: (1) d ( P j , Q j ) = d ( φ ( p j ) , φ ( q j )) ≤ C · | | p j − q j || = C , (2) d ( A j , B j ) = d  φ ( ψ j ( p j )) , φ ( ψ j ( q j ))  ≥ 1 C · | | ψ j ( p j ) − ψ j ( q j ) || > 1 / 2 C , (3) A j , B j are t he closest p oints on γ ω to P j , Q j resp ectiv ely , (4) d ( P j , A j ) = d ( P j , γ ω ) = d ( φ ( p j ) , φ ( L 0 )) ≥ j /C , and similarly for d ( Q j , B j ). W e no w pro ceed to explain how we can use this sequence of 4-tuples to construct a seque nce of almost undis torted metric squares along γ ω . This will b e done via a t w o step pro cess, and the mo diﬁcation at each step is illustrated in Figure 7. The ﬁrst step is to replace the original sequence b y a new sequence { A j , B j , P ′ j , Q ′ j } c hosen as follo ws: if d ( P j , A j ) ≤ d ( Q j , B j ), let P ′ j = P j , but pic k Q ′ j to b e the unique p oin t on the geo desic segmen t B j Q j at distance d ( P j , A j ) from the p o in t B j (and p erform t he symmetric pro cedure if d ( P j , A j ) ≥ d ( Q j , B j )). This new seque nce of 4-tuples satisﬁes the same prop erties and estimates (2)-(4) from ab ov e, but o f course, the distance d ( P ′ j , Q ′ j ) no longer satisﬁes estimate (1). W e no w pro ceed to use the triangle inequality to give a new estimate (1 ′ ) for the analogous distance for our new 4-tuple. Assuming that w e are in the case where P ′ j = P j (the o ther case is symmetric), w e a r e truncating the segmen t B j Q j to ha v e the same length a s A j P j ; the amount b eing truncated can b e estimated by the triangle inequalit y: d ( Q ′ j , Q j ) = d ( Q j , B j ) − d ( P j , A j ) ≤ d ( P j , Q j ) + d ( A j , B j ) ≤ 2 C This in turn allows us to estimate from ab ov e the distance: d ( P ′ j , Q ′ j ) = d ( P j , Q ′ j ) ≤ d ( P j , Q j ) + d ( Q j , Q ′ j ) ≤ C + 2 C = 3 C In part icular, our new sequence satisﬁes the follo wing prop ert y: (1 ′ ) for eac h j , w e ha v e the unifo rm estimate d ( P ′ j , Q ′ j ) ≤ 3 C . In addition, our new sequenc e satisﬁes the additio nal pr o p ert y (5) for each j , d ( A j , P ′ j ) = d ( B j , Q ′ j ). Our second step is to furt her mo dify the sequence as follo ws: start ing from the index j ≥ C 2 , consider the new sequence of 4-tuples { A j , B j , Q ′′ j , P ′′ j } c hosen by pic king the p oin ts P ′′ j ∈ A j P ′ j and Q ′′ j ∈ B j Q ′ j to satisfy the following stronger 25 B j A j j A A j B j j P ,, Q j ,, j P j , Q j P , j B Q j Q j , Figure 7. Changing { A j , B j , Q j , P j } to an almost undistorted  . v ersion of (5): d ( A j , P ′′ j ) = d ( A j , B j ) = d ( B j , Q ′′ j ) Note that this new sequence of 4-tuples still satisﬁes prop erties (2) and (3). W e no w mak e the: Claim: The sequence of 4-tuples { A j , B j , Q ′′ j , P ′′ j } , a s a sequence o f maps  → C one ( ˜ M ), is undistorted in the limit. T o establish this, w e need to sho w that the limit (as j → ∞ ) of the ratios of all distances tend to the corresp onding distances in  (i.e. tend to 1 or √ 2 according to whic h ra tio o f distances is considered). W e ﬁrst observ e t hat, for all j ≥ C 2 , w e ha v e b y construction the equalities: d ( A j , P ′′ j ) d ( A j , B j ) = d ( B j , Q ′′ j ) d ( A j , B j ) = 1 whic h accoun ts for the relative distances of three of the four sides. Now let us consider the ratio o f the fourth side to the ﬁrst, i.e. the ratio d ( P ′′ j , Q ′′ j ) /d ( A j , B j ). In order t o estimate this, w e ﬁrst observ e that the p oints P ′′ j , Q ′′ j pro ject to A j , B j under the pro jection map π : C one ( ˜ M ) → γ ω , and hence since this map is distance non-increasing, we o bt a in the estimate d ( A j , B j ) ≤ d ( P ′′ j , Q ′′ j ). T o giv e an upp er b ound, w e make use o f the fact that C one ( ˜ M ) is a CA T(0) space, and hence w e ha v e con v exit y of the distance function. Recall that this t ells us that given a n y t w o geo desic segmen ts α , β : [0 , 1] → C one ( ˜ M ), with pa rametrization prop ortional to 26 arclength, and given any t ∈ [0 , 1], we hav e the estimate: (8) d ( α ( t ) , β ( t )) ≤ (1 − t ) · d ( α (0) , β (0)) + t · d ( α (1) , β (1)) Let us apply this to the t w o geo desic segmen ts α = A j P ′ j and β = B j Q ′ j . In this sit- uation, we see t hat d ( α (0) , β (0)) = d ( A j , B j ). F urthermore, w e ha v e from prop erties (1 ′ ) and (2 ) the estimate: d ( α (1) , β (1)) = d ( P ′ j , Q ′ j ) ≤ 3 C ≤ 6 C 2 · d ( A j , B j ) Substituting these estimates into the conv exit y equation (8), we obtain the follo wing inequalit y: (9) d ( α ( t ) , β ( t )) d ( A j , B j ) ≤ (1 − t ) + 6 C 2 · t Finally , w e recall that d ( A j , P ′′ j ) = d ( A j , B j ) ≤ C , while from prop ert y (4) , w e ha v e that d ( A j , P ′ j ) ≥ j /C . In particular, the parameter t corresp onding t o the p oin t P ′′ j is at most C 2 /j . Now from prop ert y (3), we also kno w that the function d ( α ( t ) , β ( t )) is strictly increasing, giving us the f ollo wing estimate: (10) d ( P ′′ j , Q ′′ j ) d ( A j , B j ) ≤ d ( α ( C 2 /j ) , β ( C 2 /j )) d ( A j , B j ) ≤ (1 − C 2 /j ) + 6 C 2 · C 2 /j ≤ 1 + 6 C 4 /j It is now immediate that this ratio tends to one a s j → ∞ . Applying Lemma 2.4, w e conclude t ha t the sequence of 4 -tuples is undistorted in the limit. T o complete the pro of of Theorem 5.2 , w e would like to apply Lemma 5.1. Lo oking at t he statemen t of the prop osition, w e see that we ha v e one mor e condition w e need to ensure, namely w e require the sequence of ǫ -undistorted squares  ֒ → C one ( ˜ M ) to all satisfy ∗ ∈ I nt ( A j B j ). Note that this is not a priori satisﬁed b y the se quence of 4-tuples we constructed ab ov e. In order to ensure this additional conditio n, w e mak e use of the fact that inside ˜ M , w e assumed that there was a g ∈ I som ( ˜ M ) acting co compactly on the geo desic γ . This allows us to mak e use of Lemma 2 .2 , whic h implies that giv en any pair of p oin ts p, q on γ ω , w e ha v e an isometry of C one ( ˜ M ) lea ving γ ω in v a rian t and taking p to q . T o ﬁnish, w e pic k, for eac h of our previously constructed 4-tuples { A j , B j , Q ′′ j , P ′′ j } , a p oint p j ∈ I nt ( A j B j ) ⊂ γ ω . Then our Lemma 2.2 ensures the existence of a corresp onding isometry Φ j , lea ving γ ω in v a rian t, and mapping p j to the distinguished basep oin t ∗ ∈ C one ( ˜ M ). The seque nce of image 4-tuples Φ j ( { A j , B j , Q ′′ j , P ′′ j } ) no w satisfy all the h yp otheses of Lemma 5.1. Applying the lemma no w completes the pro of of Theorem 5.2.  27 Finally , w e conclude this section by p o in ting out that combining Theorem 5.2, Theorem 4.1 (Step 2), and The orem 3.1 completes the pro of of Theorem 1.2 from the in tro duction. 6. Some applica tions Finally , let us discuss some consequences of o ur main results. Corollary 6.1 ( Constraints on quasi-isometries) . L et ˜ M 1 , ˜ M 2 b e two simply c on - ne cte d, c omplete, Riemannian manifolds of non-p ositive se ctional curvatur e, and as- sume that φ : ˜ M 1 → ˜ M 2 is a quasi-isometry. L et γ ⊂ ˜ M 1 b e a ge o desic, γ ω ⊂ C one ( ˜ M 1 ) the c orr esp onding ge o d esic in the asymptotic c one, and assume that ther e exists a bi-Lipschitz ﬂat F ⊂ C one ( ˜ M 1 ) c ontaining the ge o desic γ ω . T h en the fol l o w- ing dichotomy holds: (1) eve ry ge o desic η at b ounde d distanc e fr om φ ( γ ) satisﬁes η /S tab G ( η ) non- c omp act, w her e G = I som ( ˜ M 2 ) , o r (2) eve ry ge o desic η at b ounde d d istanc e fr om φ ( γ ) ha s r k ( η ) ≥ 2 . Pr o of. This fo llo ws readily f r o m our Theorem 1.2. Assume that the ﬁrst p ossibilit y do es not o ccur, i.e. there exists a geo desic η at b ounded distance from φ ( γ ) with the prop ert y that S tab G ( η ) ⊂ G = I som ( ˜ M 2 ) acts co compactly on η . Then w e w ould lik e to establish that ev ery geo desic η ′ at ﬁnite distance from φ ( γ ) has higher r ank. W e ﬁrst observ e that if there were more than one s uc h g eo desic, then the ﬂat strip theorem w ould imply that an y tw o of them arise as the b oundary of a ﬂat strip, and hence tha t they w ould all ha v e higher rank. So w e only need to deal with the case where there is a unique suc h geo desic, i.e. show that t he geo desic η has r k ( η ) ≥ 2. No w recall that the quasi-isometry φ : ˜ M 1 → ˜ M 2 induces a bi-Lipsc hitz homeomorphism φ ω : C one ( ˜ M 1 ) → C one ( ˜ M 2 ). Since η ⊂ ˜ M 2 w as a geo desic at ﬁnite distance from φ ( γ ), we hav e the containme nt: φ ω ( γ ω ) ⊆ η ω ⊂ C one ( ˜ M 2 ) . Since φ ω ( γ ω ) is a bi-Lipsc hitz copy o f R inside the g eo desic η ω , we conclude that φ maps γ ω homeomorphically on to η ω . But recall that w e a ssumed that γ ω w as con tained inside a bi-L ipschitz ﬂat γ ω ⊂ F ⊂ C one ( ˜ M 1 ), and hence w e see that η ω ⊂ φ ω ( F ) is like wise contained inside a bi-Lipsc hitz ﬂat. F urthermore, since S tab G ( η ) a cts co compactly on η , we see that there exists an elemen t g ∈ G = I som ( ˜ M 2 ) whic h stabilizes and acts cocompactly on η . Hence η satisﬁes the hypotheses of Theorem 1.2, and m ust ha v e r k ( η ) ≥ 2, as desired. This concludes the pro of of Corollary 6.1.  The statemen t of o ur ﬁrst corollary migh t seem somewhat complicated. W e now pro ceed to isola t e the sp ecial case whic h is of most in terest: 28 Corollary 6.2 (Constraints o n p erturbations of metrics) . Assume that ( M , g 0 ) is a close d Riemannian manifold of non-p ositive s e ctional curvatur e, and assume that γ 0 ⊂ M is a close d ge o desic. L et ˜ γ 0 ⊂ ˜ M b e a l i f t of γ 0 , and ass ume that ˜ γ 0 ⊂ F is c ontaine d in a ﬂat F . Then if ( M , g ) is any other R iemannian metric on M with non-p ositive se ctional curvatur e, and γ ⊂ M is a ge o desi c (in the g -metric) fr e ely homotopic to γ 0 , then the lift ˜ γ ⊂ ( ˜ M , ˜ g ) satisﬁes r k ( ˜ γ ) ≥ 2 . W e can think of Corollary 6.2 as a “no n-p erio dic” v ersion o f the Flat T or us theorem. Indeed, in t he case where F is π 1 ( M )-p erio dic, the Flat T orus theorem applied to ( M , g ) implies that ˜ γ is lik ewise con tained in a p erio dic ﬂat ( a nd in particular has rank ≥ 2 ). Pr o of. Since M is compact, the iden tit y map provides a quasi-isometry φ : ( ˜ M , ˜ g 0 ) → ( ˜ M , ˜ g ). The ﬂat F con taining ˜ γ 0 giv es rise to a ﬂat F ω ⊂ C one ( ˜ M , ˜ g 0 ) con taining ( ˜ γ 0 ) ω . In particular, w e can apply the previous Corollary 6.1. Next note that, since γ 0 , γ are freely homoto pic to each other, there is a lift ˜ γ of γ whic h is at ﬁnite distance (in the g -metric) fro m the given ˜ γ 0 ⊂ ( ˜ M , ˜ g ). Indeed, taking the free homotop y H : S 1 × [0 , 1 ] → M b et w een H 0 = γ 0 and H 1 = γ , w e can then ta k e a lift ˜ H : R × [0 , 1] → ˜ M satisfying the initial condition ˜ H 0 = ˜ γ 0 (the giv en lift of γ 0 ). The time one map ˜ H 1 : R → ˜ M will be a lift of H 1 = γ , hence a geo desic in ( ˜ M , ˜ g ). F urthermore, the dis tance (in the g - metric) b etw ee n ˜ γ 0 and ˜ γ will clearly b e b ounded ab ov e b y the suprem um of the g -lengths of the (compact) family of curv es H p : [0 , 1] → ( M , g ), p ∈ S 1 , deﬁned by H p ( t ) = H ( p , t ). No w observ e that by construction, the ˜ γ ⊂ ( ˜ M , ˜ g ) from the previous para graph has S tab G ( ˜ γ ) acting co compactly on ˜ γ , where G = I som ( ˜ M , ˜ g ). Hence the ﬁrst p ossibilit y in the conclusion of Corollary 6.1 cannot o ccur, and w e conclude that ˜ γ has r k ( ˜ γ ) ≥ 2, as desired. This concludes the pro of o f Corollary 6.2.  Next w e recall t ha t the classic de Rham theorem [dR] states that a n y simply connnected, c omplete Riemannian manifold admits a dec omp osition as a metric pro d- uct ˜ M = R k × M 1 × . . . × M k , where R k is a Euclid ean space equipp ed with the standard metric, and each M i is metrically irr educible (and not R or a p oin t). F urthermore, this decompo sition is unique up to permutation of the factors. This resu lt was re- cen tly generalized b y F o ertsc h-Lytc hak to cov er ﬁnite dimensional geo desic metric spaces [FL]. Our next corollary sho ws that, in the presenc e of non-p ositiv e Riemann- ian curv ature, t here is a strong relationship b et w een splittings of ˜ M and splittings of C one ( ˜ M ). Corollary 6.3 (Asymptotic cones detect splittings) . L et M b e a close d Riemann- ian manifold of non-p ositive curvatur e, ˜ M the universal c over of M with ind uc e d 29 Riemannian metric, and X = C one ( ˜ M ) an arbitr ary asymptotic c one of ˜ M . If ˜ M = R k × M 1 × . . . × M n is the de Rham splitting of ˜ M into irr e ducible factors, and X = R l × X 1 × . . . × X m is the F o ertsch-Lytchak splitting of X into irr e ducible factors, then k = l , n = m , and up to a r elab eli n g of the i n dex set, we have that e ac h X i = C one ( M i ) . Pr o of. Let us ﬁrst assume t ha t ˜ M is irreducible (i.e. k=0, n=1), and sho w that C one ( ˜ M ) is also irreducible ( i.e. l=0, m=1) . By w a y o f contradiction, let us assume that X splits as a metric pro duct, and observ e that this clearly implies that ev ery geo desic γ ⊂ X is con tained inside a ﬂat. In par t icular, from our Theorem 1.1, w e see that every geo desic inside ˜ M m ust ha v e higher rank. Applying the Ballmann-Burns- Spatzier ra nk rigidity result, a nd recalling that ˜ M w as irreducible, we conclude that ˜ M is in fact an irreducible higher rank symmetric space. But no w Kleiner-Leeb ha v e sho wn that f o r suc h spaces, t he asymptotic cone is irreducible (see [KlL, Section 6]), giving us the desired con tradiction. Let us no w pro ceed t o the general case: from the metric splitting o f ˜ M , we get a corresponding metric splitting C one ( ˜ M ) = R k × Y 1 × . . . × Y n , where eac h Y i = C one ( M i ). Since eac h M i is irreducible, the previous par a graph tells us that eac h Y i is lik ewise irreducible. So w e no w ha v e tw o pro duct decomp ositions of C one ( ˜ M ) in to irreducible factors. So assuming that eac h Y i is distinct fro m a p oin t and is not isometric to R , w e could app eal to the uniquenes s p ortion of F o ertsc h-Lytc hak [FL, Theorem 1.1] to conclude that, up to relab eling of the index set, each X i = Y i = C one ( M i ), and that the Euclidean factors ha v e t o hav e the same dimension k = l . T o conclude the pr o of of our Corollary , w e establish that if M is a simply connected, complete, Riemannian manifold of non-p ositiv e sec tional cu rv ature, and dim( M ) ≥ 2, then C one ( M ) is distinct from a p oint or R . First, recall that taking an arbitrary geo desic γ ⊂ M (whic h w e ma y assume pass es throug h the basep oin t ∗ ∈ M ), we get a corresp onding geo desic γ ω ⊂ C one ( M ), i.e. an isometric em b edding o f R in to C one ( M ). In particular, w e see that dim( C one ( M )) > 0. T o see that C o ne ( M ) is distinct from R , it is enough to establish the existence of three p o ints p 1 , p 2 , p 3 ∈ C one ( M ) suc h that for eac h index j we hav e: (11) d ω ( p j , p j +2 ) 6 = d ω ( p j , p j +1 ) + d ω ( p j +1 , p j +2 ) But this is easy t o do: tak e p 1 , p 2 to b e the tw o distinct p oin ts on the geo desic γ ω at distance one from the basep oint ∗ ∈ C one ( M ), so t ha t d ω ( p 1 , p 2 ) = 2. Observ e that one can represen t the p oints p 1 , p 2 via the sequen ces of p oin ts { x i } , { y i } along γ ha v- ing the prop ert y that ∗ ∈ x i y i , and d ( x i , ∗ ) = λ i = d ( ∗ , y i ), where λ i is the sequence of scales used in forming the asymptotic cone C one ( M ). No w since dim( M ) ≥ 2, we can ﬁnd another geo desic η thro ugh the basepoint ∗ ∈ M , with the prop erty that η ⊥ γ . T aking t he sequence { z i } to lie on η , and satisfy d ( z i , ∗ ) = λ i , it is easy to see that this sequence deﬁnes a third p oint p 3 ∈ C one ( M ) satisfying d ω ( p 3 , ∗ ) = 1. F rom 30 the triangle inequalit y , w e immediately ha v e that d ω ( p 1 , p 3 ) ≤ 2 and d ω ( p 2 , p 3 ) ≤ 2. On the other hand, since the Riemannian manifold M has non-p ositiv e sectional curv ature, w e can apply T op onogov ’s theorem to eac h of the triangles {∗ , x i , z i } : since w e hav e a right angle at the v ertex ∗ , and w e ha v e d ( ∗ , x i ) = d ( ∗ , z i ) = λ i , T op onogov tells us that d ( x i , z i ) ≥ √ 2 · λ i . Pass ing to the asymptotic cone, this giv es the low er b o und d ( p 1 , p 3 ) ≥ √ 2, a nd an iden tical argumen t giv es t he estimate d ( p 2 , p 3 ) ≥ √ 2. It is now easy to v erify that the three p oin ts p 1 , p 2 , p 3 satisfy (11), and hence C one ( M ) 6 = R , as desired. This concludes the pro of of Corollary 6.3.  Before stating o ur next result, w e recall that the celebrated ra nk rigidit y theorem of Ba llmann-Burns-Spatzier (se e Section 2.3) w as motiv a t ed b y Gro mo v’s w ell-known rigidit y theorem, the pro of of whic h a pp ears in the b o ok [BGS]. Our next corollary sho ws how in fact Gromov’s r igidit y theorem can directly b e deduced from the rank rigidit y theorem. This is our last: Corollary 6.4 (Gromo v’s higher rank rig idity [BGS]) . L et M ∗ b e a c omp ac t lo c al ly symmetric sp a c e of R -r a n k ≥ 2 , with unive rs a l c over ˜ M ∗ irr e ducible, a nd let M b e a c omp act Riemannian m a nifold with se ctional curvatur e K ≤ 0 . If π 1 ( M ) ∼ = π 1 ( M ∗ ) , then M is isome tric to M ∗ , pr ovide d V ol ( M ) = V ol ( M ∗ ) . Pr o of. Since b oth M and M ∗ are compact with isomorphic fundamental groups, the Milnor- ˇ Sv arc theorem gives us quasi-isometries: ˜ M ∗ ≃ π 1 ( M ∗ ) ≃ π 1 ( M ) ≃ ˜ M whic h induce a bi- L ipschitz homeomorphism φ : C one ( ˜ M ∗ ) → C one ( ˜ M ). Now in order to apply the rank rig idity theorem, w e need to establish that ev ery geo desic in ˜ M has rank ≥ 2. W e ﬁrst observ e that the pro of of Corollary 6.2 extends almost v erbatim to the presen t setting. Indeed, in Coro llary 6.2, we used the iden tit y ma p to induce a bi-Lipsc hitz homeomorphism b et w een the asymptotic cones, and then app ealed to Corollary 6.1. The sole diﬀerence in our prese n t contex t is tha t , rather than using the iden tity map, w e use the quasi-isometry b et w een ˜ M and ˜ M ∗ induced b y the isomorphism π 1 ( M ) ∼ = π 1 ( M ∗ ). This in turn induces a bi-Lipsc hitz homeomorphism b et w een asymptotic cones (see Section 2.1). The reader can easily verify that the rest of t he argument in Corollary 6.2 extends to o ur pres en t setting, establishing that ev ery lift t o ˜ M of a p erio dic geo desic in M ha s rank ≥ 2. So w e no w mo v e to the general case, a nd explain wh y every geo desic in ˜ M has higher r a nk. T o see this, assume by w a y of con tradiction that there is a geo desic η ⊂ ˜ M with r k ( η ) = 1. No t e that the geo desic η cannot b ound a half-plane. But once we ha v e the existence of suc h an η , w e can app eal to results of Ballmann [Ba1, Theorem 2.13], whic h imply that η can b e a pproximated (uniformly on compacts) 31 b y lifts of p erio dic geo desics in M ; let { ˜ γ i } → η b e suc h an approxim ating sequence. Since eac h ˜ γ i has r k ( ˜ γ i ) ≥ 2, it s upp ort s a parallel Jacobi ﬁe ld J i , whic h can be tak en to satisfy || J i || ≡ 1 and h J i , ˜ γ ′ i i ≡ 0 . No w we see that: • the limiting v ector ﬁeld J deﬁned a lo ng η exists, due to the control on | | J i || , • the v ector ﬁeld J along η is a parallel Jacobi ﬁeld, since b oth the “parallel” and “Jacobi” condition can b e enco ded by diﬀeren tial equations with smo oth co eﬃcien ts, solutions to whic h will v ary contin uously with respect to initial conditions, and • J will hav e unit length and will b e ortho gonal to η ′ , from the corresp onding condition o n the J i . But this contradicts our a ssumption that r k ( η ) = 1. So w e conclude that ev ery geo desic η ⊂ ˜ M m ust satisy r k ( η ) ≥ 2, as desired. F rom the rank rigidit y theorem, we can now conclude that ˜ M either splits as a pro duct, or is isometric to an ir r educible higher rank symmetric space. Since the asymptotic cone of the irreducible higher rank symmetric space is top olo g ically irreducible (see [KlL, Section 6]) , and C one ( ˜ M ) is homeomorphic to C one ( ˜ M ∗ ), w e hav e that ˜ M cannot split as a pro duct. Finally , we see that π 1 ( M ) ∼ = π 1 ( M ∗ ) acts co compactly , isometrically on tw o ir r educible higher rank symmetric spaces ˜ M and ˜ M ∗ . By Mosto w rigidity [Mo], we hav e that the quotien t spaces are, after suitably rescaling, isometric. This completes our pro of of Gromov ’s higher rank rigidit y theorem.  Finally , let us conclude our pap er with a few commen ts on this last corollary . Remarks: (1) The actual statemen t of Gro mo v’s t heorem in [BGS, pg. (i)] do es no t assume ˜ M ∗ to b e ir r educible, but ra ther M ∗ to b e ir r educible (i.e. there is no ﬁnite co v er o f M ∗ that splits isometrically as a pro duct). This lea v es the p ossibilit y that the univ ers al co v er ˜ M ∗ splits isometrically as a pro duct, but no ﬁnite cov er of M ∗ splits isometric ally as a pro duct. Ho w ev er, in this sp eciﬁc case, the desired result w as already pro v ed b y Eb erlein (see [Eb ]). And in fact, in the original pro of of Gromov’s rigidit y theorem, the v ery ﬁrst step (see [BGS, pg. 154]) consists of app ealing to Eb erlein’s result to reduce to the case where ˜ M ∗ is irreducible. (2) In the course of writing this pap er, the author s learn t of the existenc e of another pro of of Gromov’s rig idit y result, whic h bears some similarity to our reasoning. As the reader has surmised from the pro of of Corollary 6.3, the k ey is t o someho w sho w that M also has to ha v e higher rank. But a sophisticated r esult of Ballma nn- Eb erlein [BaEb] establishes tha t the rank of a non-p ositiv ely curv ed Riemannian manifold M can in f a ct be detected directly from algebraic prop erties of π 1 ( M ), a nd hence the prop ert y of ha ving “higher rank” is in fact algebraic (see a lso the recen t preprin t 32 of Bestvina-F ujiwara [BeF u]). The main adv an tage of our approach is that one can deduce Gromov’s r ig idit y r esult d i r e ctly f r o m ra nk rigidity . (3) W e p oint out that v ario us other mathematicians ha v e obta ined results extending Gromov ’s theorem (a nd whic h do not seem tractable using our metho ds). A v aria- tion cons idered b y Davis-Okun-Zheng ([DOZ], requires ˜ M ∗ to b e r e ducible and M ∗ to b e an irreducible (t he same hypothesis as in Eberlein’s rigidity r esult). How eve r, Da vis-Okun-Zheng allow the metric on M to b e lo cally CA T(0) (rather than Rie- mannian non-p ositive ly curv ed), and are s till able to conclude t ha t M is isometric (after rescaling) to M ∗ . The optimal result in this direction is due to Leeb [L ], giving a c haracterization of certain hig her rank symmetric spaces and Euclidean buildings within the broadest p o ssible class o f metric spaces, the Ha da mard spaces ( complete geo desic spaces for whic h the distance function b et w een pairs of geo desics is alw a ys con v ex). It is w orth mentioning that Leeb’s result relies heav ily on the viewp oint dev elop ed in the Kleiner-Leeb pap er [KlL ]. (4) W e note that our metho d of pro of can also b e used to establish a non-c omp act, ﬁnite volume analogue of the previous corollary . Three of the k ey ingr edients g o- ing into our pro of w ere (i) Ballmann’s result on the densit y of p erio dic geo desics in the tangen t bundle, (ii) Ballmann-Burns-Spatzier’s rank rigidity theorem, and (iii) Mosto w’s stro ng rigidit y t heorem. A ﬁnite volume vers ion of (i) w as obtained by Crok e-Eb erlein-Kleiner (see [CEK, App endix]), under the assumption tha t the fun- damen tal group is ﬁnitely generated. A ﬁnite volume v ersion of (ii) w as obtained b y Eb erlein-Heb er (see [EbH]). The ﬁnite v olume v ersions of Mosto w’s strong rigidit y w ere obtained b y Prasad in the Q -ra nk one case [Pr] and Marg ulis in the Q -ra nk ≥ 2 case [Ma] (se e also [R]). O ne tec hnicalit y in the non-compact case is that iso- morphisms o f fundamen tal g roups no longer induce quasi-isometries of the unive rsal co v er. In particular, it is no longer suﬃcien t to just assume π 1 ( M ) ∼ = π 1 ( M ∗ ), but rather o ne needs a ho mo t op y equiv alence f : M → M ∗ with the prop erty that f lifts to a quasi-isometry ˜ f : ˜ M → ˜ M ∗ . W e leav e the details to the in terested r eader. Reference s [Ba1] W. Ballmann. Axial isometries of manifolds of nonpositive curv ature. Math. A nn. 25 9 (1982 ), 131-1 44. [Ba2] W. Ballma nn. Nonp ositively curved manifolds of higher ra nk. Ann. of Math. (2) 122 (19 85), 597-6 09. [BaEb] W. Ballmann & P . Eb erlein. F undamen tal groups of manifolds of nonp ositive curv ature. J. Diﬀ. Ge om. 25 (1987), 1 -22. [BGS] W. Ballmann, M. Gromov, & V. Sc hro eder. Manifolds of nonp os itive curv ature. Progress in Mathematics 61 , Birkhuser, Boston, 1985. [BeNi] I.D. Berg & I.G. Nikolaev. On a distance b etw een directions in an Alek s androv spac e o f curv ature ≤ K . Michi gan Math. J. 45 (1998), 257-289. [BeF u] M. Bes tvina & K. F ujiwara. 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