Sub-Riemannian geometry of parallelizable spheres

SUB-RIEMANNIAN GEOMETR Y OF P ARALLELIZABLE SPHERES MAURICIO GODOY MOLINA IRINA MAR K INA Abstra ct. The first aim of the prese nt pap er is t o compare va rious sub-R iemannian structures ov er the three dimen sional sphere S 3 originating from d ifferent constructions. Namely , we describe th e sub-Riemannian geometry of S 3 arising through its right Lie group action o v er itself, the one inherited from the natural complex structure of the open unit ball in C 2 and th e geometry that app ears when considering the Hopf map as a principal bundle. The main result of this comparison is that in fact those three structures coincide. In t h e second place, we present tw o brack et generating distributions for the sev en dimensional sphere S 7 of step 2 with ranks 6 and 4. These yield to sub -Riemannian structures for S 7 that are n ot present in th e literature until no w. One of t he distributions can be obtained b y considering the CR geometry of S 7 inherited from the natural complex structure of the open un it ball in C 4 . The other one originates from t h e qu aternionic analogous of th e Hopf map. 1. Introduction One of the main ob jectiv es of classical sub-Riemannian geometry is to study manifolds whic h are path-connected by curve s whic h are admissible in a certain s ense. In ord er to define what do es admissib ilit y mean in this cont ext, w e b egin by setting notatio ns that will b e u s ed thr oughout this pap er. Let M b e a smo oth connected manifold of dimens ion n , together with a smo oth d istribution H ⊂ T M of rank 2 ≤ k < n . Suc h v ector bun dles are often called horizontal in the literature. An absolutely con tin uous cu r v e γ : [0 , 1] → M is called admissible or horizontal if ˙ γ ( t ) ∈ H a.e. Distributions satisfying the condition that their Lie-hull equals the whole tangen t sp ace of the manifold at eac h p oin t pla y a cen tral r ole in the s earch for horizon tall y path- connected manifolds. Suc h vecto r bundles are s aid to satisfy the br acket gene r ating c on- dition . T o b e more p recise, define the follo wing real vec tor b undles H 1 = H , H r +1 = [ H r , H ] + H r for r ≥ 1 , whic h naturally giv e rise to the flag H = H 1 ⊆ H 2 ⊆ H 3 ⊆ . . . . 2000 Mathematics Subje ct Classific ation. 53C17, 32V15. Key wor ds and phr ases. sub-Riemannian geometry , CR geometry , Hopf bun dle, Ehresmann connection, parallelizable spheres, qu aternions, o ctonions. The authors are partially su pp orted by the gran t of th e Norw egian Research Council # 177355/V30 and by t he grant of the European Science F oundation Netw orking Programme HCAA. 1 2 MAURICIO GODOY M.,IRINA MARKINA Then w e s ay that a distribu tion is br ac ket generating if for all x ∈ M there is an r ( x ) ∈ Z + suc h that (1) H r ( x ) x = T x M . If the dimensions dim H r x do not dep end on x for any r ≥ 1, we say that H is a regular distribution. Th e least r suc h that (1) is satisfied is called th e step of H . W e will fo cus on regular distributions of step 2. In [16] the reader can fi n d d etailed defin itions an d broad discussion ab out terminology . The follo wing classical result sho ws the precise rela tion b et ween the notion o f path- connectedness b y means of horizon tal curv es and the assumption that H is a b rac k et generating distribution. Theorem 1 ([9, 20]) . If a distribution H ⊂ T M is br acket gener ating, then the set o f p oints that c an b e c onne cte d to p ∈ M by a horizontal p ath is the c onne c te d c omp onent of M c ontaining p . Th us, the searc h for horizonta lly p ath-connected manifolds can b e reduced to the searc h of appropriate d istributions defined on them. W e define the class of manifolds whic h will b e of our concern. Definition 1. A sub-Riemannian structur e over a manifold M is a p air ( H , h· , ·i ) , wher e H is a br acket ge ne r ating distribution and h· , ·i a fib er inner pr o duct define d on H . In this setting, the length of an absolutely c ontinuous horizontal curve γ : [0 , 1] → M is ℓ ( γ ) := Z 1 0 k ˙ γ ( t ) k dt, wher e k ˙ γ ( t ) k 2 = h ˙ γ ( t ) , ˙ γ ( t ) i whenever ˙ γ ( t ) exists. The triple ( M , H , h· , ·i ) is c al le d sub- R iemannian manifold. Thereby , restricting ourselv es to connected sub-Riemannian manifolds endo w ed with brac k et g enerating distribu tions, it is p ossible to define the n otion of sub-Riemannian distance b et ween t w o p oints. Definition 2. The sub-R iemannian distanc e d ( p, q ) ∈ R ≥ 0 b etwe en two p oints p, q ∈ M is given by d ( p, q ) := in f ℓ ( γ ) , wher e the infimum is taken over al l abso lutely c ontinuous horizonta l curves joining p to q . An absolutely c ontinuous horizontal curve that r e alizes the distanc e b etwe en two p oints is c al le d a horizontal length minimizer. Remark: The connecte dness of M and the fact that H is brac k et generating, assure that d ( p, q ) is a finite nonnegativ e n u m b er. Nev ertheless, the br ac ket generating h yp othesis, required for the previous d efinition, is p ossible to b e w eak ened . In fact, in [22] the au- thor finds a necessa ry and s u fficien t requirement to horizon tal path-connectedness f or a manifold in terms of the corresp onding d istribution. Clearly , th is theorem con tains, as a particular case, the brac k et generating condition. Historically , the first examples of sub-Riemannian m an if olds that h av e b een considered w ere Lie groups, see e.g . [2, 5, 7 , 12, 15], b ecause due to its algebraic structure, it is SUB-RIEMANNIAN GEOME T R Y OF P ARALLELIZABLE SPHERES 3 p ossible to red u ce the problem of finding globally defi n ed brac k et generating distributions to find ing suc h d istributions at th e id en tit y by considering right (or left) in v arian t vect or fields. An imp ortant role h as b een pla y ed b y considerin g domains in Euclidean spaces with sp ecial algebraic structures (such as the Heisen b erg group s, H − t yp e groups as their natural generalizations to C lifford algebras, En gel group s, Carnot groups, etc.). P artic ular atten tion ha v e had the three dimen s ional un imo dular Lie group s wh ic h w ere s tudied, for example, in [2], [5] and [12]. T h e main purp ose of this comm unicatio n is to presen t recen t results concerning differen t su b-Riemannian structures of th e second simplest family of examples of manifolds, namely , spher es. W e also b e inspired by the article [23], w here the close relation b et w een the Hopf map and physical applications is pr esen ted. The follo wing celebrated theorem in top ology , see [1], gives a v ery stron g restriction on the problem of fin ding globally defin ed sub-Riemannian stru ctur es o v er s pheres. Theorem 2 ([1]) . L et S n − 1 = { x ∈ R n : k x k 2 = 1 } b e the unit spher e in R n , with r esp e ct to the usual Euclide an norm k · k . Then S n − 1 has pr e cisely  ( n ) − 1 glob al ly define d and non vanishing ve ctor fields, wher e  ( n ) is define d in the fol lowing way: if n = (2 a + 1)2 b and b = c + 4 d wher e 0 ≤ c ≤ 3 , then  ( n ) = 2 c + 8 d . In p art icular, two classic al c onse que nc es fol low: S 1 , S 3 and S 7 ar e the only spher es with maximal nu mb er of line arly indep endent glob al ly define d non v anishing ve ctor fields, and the even dimensional spher es have no glob al ly define d and non vanishing ve ctor fields. The condition that a manifold M has maximal num b er of linearly indep enden t globally defined non v anishing v ector fields is usually rephrased as sa ying that M is p ar al lelizable . The fact that S 1 , S 3 and S 7 are the only paralleli zable spheres was pro v en in [6] and that the eve n dimensional sp heres hav e no globally defin ed and n on v anishing vec tor fields is a consequence of the Hopf index theorem, see [24]. This theorem p ermits to conclude at least t w o ma jor p oin ts of discussion: there is no p ossible global basis of a su b -Riemannian stru cture for s pheres with even dimension and it is imp ossible to fin d a glo bally defin ed basis for brack et generating distributions, except for S 3 and S 7 . The fact that S 3 and S 7 can b e seen as the set of quaternions and o ctonions of unit length will play a core role in man y argumen ts throu gh ou t th is pap er. The m ain results th at w e present h ere are: a comparison b et ween three su b-Riemannian structures of S 3 , namely , the one arisin g through its right Lie group action o v er itself as the set of unit quaternions, the one in h erited from the natural complex structure of the op en unit ball in C 2 and the geometry that app ears wh en considering the Hopf map as a pr incipal bun dle. Notic e that this structure admits a tange n t co ne isomorphic to the one dimensional Heisen b erg group in a sense of Gromov- Margulis-Mitc hell-Mosto w con- struction of the tangen t cone [13, 18, 19]. Con s idering C R-structure of S 7 , inherited fr om the natural complex structur e of the op en unit ball in C 4 , w e obtain a 2-step brac ket generating d istribution of the rank 6. T h is construction in timately rela ted to the Hopf fibration S 1 → S 7 → C P 3 . Making use of the quaternionic an alogue of the Hopf map S 3 → S 7 → S 4 , w e presen t another 2-st ep b rac k et generating distribution that has the rank 4. W e conclude that the sphere S 7 admits tw o principally different s u b-Riemannian structures. The tangen t cone, in the first case, is isomorphic to the 3-dimensional Heisen- b erg group, and in the second case it has a structure of the qu aternionic Heisen b er g-type group w ith 3-dimensional cen ter [7]. In b oth cases we present the basis of the horizont al 4 MAURICIO GODOY M.,IRINA MARKINA distribution that is v ery u seful in future studies of geo desics and hypo elliptic operators related to the spherical sub -Riemann ian manifolds. 2. S 3 as a s ub-Riemannian manifold Throughout this pap er H will denote th e qu aternions, that is, H = ( R 4 , + , ◦ ) w here + stands for the usual co ord inate-wise addition in R 4 and ◦ is a non-comm u tativ e pr o duct giv en b y the formula ( x 0 + x 1 i + x 2 j + x 3 k ) ◦ ( y 0 + y 1 i + y 2 j + y 3 k ) = = ( x 0 y 0 − x 1 y 1 − x 2 y 2 − x 3 y 3 ) + ( x 1 y 0 + x 0 y 1 − x 3 y 2 + x 2 y 3 ) i + +( x 2 y 0 + x 3 y 1 + x 0 y 2 − x 1 y 3 ) j + ( x 3 y 0 − x 2 y 1 + x 1 y 2 + x 0 y 3 ) k . It is important to recall that H is a non-comm utativ e, a sso ciativ e and normed real division algebra. Let q = t + ai + bj + ck ∈ H , th en the conjugate of q , is giv en by ¯ q = t − ai − bj − ck . W e define th e norm | q | of q ∈ H , as for the complex num b ers, b y | q | 2 = q ¯ q . Giv en the fact that the sph er e S 3 can b e realized as th e s et of u nit quaternions, it has a non ab elian Lie group structur e induced by quaternion m ultiplication. The multiplicatio n ru le in H in duces a r ight translation R y ( x ) of an elemen t x = x 0 + x 1 i + x 2 j + x 3 k by the elemen t y = y 0 + y 1 i + y 2 j + y 3 k . The right in v arian t basis ve ctor fields are defined as Y ( y ) = Y (0)( R y ( x )) ∗ , where Y (0) are the basis vecto rs at the unit y of the group. The matrix corresp onding to the tangen t map ( R y ( x )) ∗ , obtained by the m ultiplicatio n rule, b ecomes ( R y ( x )) ∗ =     y 0 y 1 y 2 y 3 − y 1 y 0 − y 3 y 2 − y 2 y 3 y 0 − y 1 − y 3 − y 2 y 1 y 0     . Calculating the action of ( R y ( x )) ∗ in the basis of unit vecto rs of R 4 w e get the four v ecto r fields N ( y ) = y 0 ∂ y 0 + y 1 ∂ y 1 + y 2 ∂ y 2 + y 3 ∂ y 3 V ( y ) = − y 1 ∂ y 0 + y 0 ∂ y 1 − y 3 ∂ y 2 + y 2 ∂ y 3 X ( y ) = − y 2 ∂ y 0 + y 3 ∂ y 1 + y 0 ∂ y 2 − y 1 ∂ y 3 Y ( y ) = − y 3 ∂ y 0 − y 2 ∂ y 1 + y 1 ∂ y 2 + y 0 ∂ y 3 . It is easy to see th at N ( y ) is the unit normal to S 3 at y ∈ S 3 with resp ect to the usual Riemannian structure h· , ·i in T R 4 . Moreo v er, for an y y ∈ S 3 h N ( y ) , V ( y ) i y = h N ( y ) , X ( y ) i y = h N ( y ) , Y ( y ) i y = 0 and h N ( y ) , N ( y ) i y = h V ( y ) , V ( y ) i y = h X ( y ) , X ( y ) i y = h Y ( y ) , Y ( y ) i y = 1 . SUB-RIEMANNIAN GEOME T R Y OF P ARALLELIZABLE SPHERES 5 Since the matrix   − y 1 y 0 − y 3 y 2 − y 2 y 3 y 0 − y 1 − y 3 − y 2 y 1 y 0   has r an k three, we conclude that the v ector fi elds { V ( y ) , X ( y ) , Y ( y ) } form an orthonormal basis of T y S 3 with resp ect to h· , ·i y . Observing that [ X , Y ] = 2 V , w e see that the d istribution span { X, Y } is brac k et gener- ating, therefore it satisfies the h yp othesis of Th eorem 1 . The geo desics of the left inv ariant sub-Riemannian stru cture of S 3 are d etermined in [8], while in [15] the same results are ac h iev ed by considering the right in v arian t structur e of S 3 . Notice that the distribution span { X, Y } can also b e defined as the kernel of th e conta ct one form ω = − y 1 dy 0 + y 0 dy 1 − y 3 dy 2 + y 2 dy 3 . Remark: It is easy to see that [ V , Y ] = 2 X and [ X , V ] = 2 Y , therefore the distr ibutions span { Y , V } and sp an { X, V } are also brack et generating. The corresp onding con tac t forms are θ = − y 2 dy 0 + y 3 dy 1 + y 0 dy 2 − y 1 dy 3 and η = − y 3 dy 0 − y 2 dy 1 + y 1 dy 2 + y 0 dy 3 resp ectiv ely . This means that there is a priori no natural c hoice o f a sub-Riemannian structure on S 3 generated by the Lie group action of multiplicatio n of quaternions. Any c hoice that can b e made, will p ro duce essen tially the same geometry . 3. S 3 as a CR manifold Consider S 3 as the b ound ary of the un it ball B 4 on C 2 , that is, as the h yp ersur face S 3 := { ( z , w ) ∈ C 2 : z ¯ z + w ¯ w = 1 } . The sp here S 3 cannot b e en d o w ed with a complex stru cture, but n ev ertheless it p ossess a differentia ble structure compatible with the natural complex structur e of the b all B 4 as an op en set in C 2 . W e will show that this differen tiable structure o ver the sp h ere S 3 (CR structure) is equiv alen t to the sub-Riemannian one considered in the previous section. W e b egin by recalli ng the definition of a CR structur e, according to [4 ]. Definition 3. L et W b e a r e al ve ctor sp ac e. A line ar map J : W → W is c al le d an almost c omplex structur e map if J ◦ J = − I , wher e I : W → W is the identity map. In the case W = T p R 2 n , p = ( x 1 , y 1 , . . . , x n , y n ) ∈ R 2 n , we sa y that the standard almost complex structur e f or W is defin ed b y setting J n ( ∂ x j ) = ∂ y j , J n ( ∂ y j ) = − ∂ x j , 1 ≤ j ≤ n. F or a smooth real submanifold M of C n and a p oin t p ∈ M , in general the tangent space T p M is not inv ariant un der the almost complex structure map J n for T p ( C n ). W e are in terested in th e largest sub space in v arian t und er the action of J n . 6 MAURICIO GODOY M.,IRINA MARKINA Definition 4. F or a p oint p ∈ M , th e c omplex or holo morph ic tangent sp ac e of M at p is the ve ctor sp ac e H p M = T p M ∩ J n ( T p M ) . In this setting, the follo wing r esult tak es place (see [4]). Lemma 1. L et M b e a r e al submanifold of C n of r e al dimension 2 n − d . Then 2 n − 2 d ≤ dim R H p M ≤ 2 n − d, and dim R H p M is an ev en numb er. A r eal submanifold M of C n is sa id to ha v e a CR structur e if dim R H p M do es not dep end on p ∈ M . In particular, by the pr evious lemma, ev ery smo oth real hypersu rface S embedd ed in C n satisfies dim R M p S = 2 n − 2, therefore S is a CR manifold. This fact is applied to every o dd d im en sional sphere. The question addressed no w is to d escrib e H p S 3 . By t he discussion in th e p revious paragraph, H p S 3 is a complex vecto r space of complex dimension one. This description can b e ac hiev ed by considering the differen tial form ω = ¯ z dz + ¯ w dw and observing that k er ω is p recisely the set w e are lo oking for. Straigh tforw ard calcula- tions sho w that k er ω = span { ¯ w ∂ z − ¯ z ∂ w } . In real co ord inates this corresp onds to ¯ w ∂ z − ¯ z ∂ w = 1 2 ( − X + iY ) , where X and Y w ere defin ed in Section 2. It is imp ortant to remark that this is pr ecisely the maximal inv ariant J 2 − subsp ace of T p S 3 , namely J 2 ( X ) = Y , J 2 ( Y ) = − X , then J 2 (span { X, Y } ) = span { X , Y } , but J 2 ( V ) = − N / ∈ T p S 3 for an y p oint p ∈ S 3 . Therefore, the d istribution corr esp onding to the right in v arian t action of S 3 o v er itself is the same to its one dimensional (complex) CR structure. Remark: The distribu tion asso ciated to the anti -holomorphic form ¯ ω = z d ¯ z + w d ¯ w is the same as the p revious one. More explicitly: k er ω = sp an {− w∂ ¯ z + z ∂ ¯ w } and in real co ordin ates this corresp onds to − w ∂ ¯ z + z ∂ ¯ w = 1 2 ( X + iY ) . The same almost complex structure as the previously d escrib ed can b e obtained b y means of the co v ariant d eriv ativ e of S 3 considered as a smo oth Riemann ian manifold em b edded in R 4 . Namely , in [15] it is introduced the mapping J ( X ) = D X V , were D SUB-RIEMANNIAN GEOME T R Y OF P ARALLELIZABLE SPHERES 7 denotes the L evi-Civit´ a connection on ta ngen t bundle to S 3 and V is the v ector field defined in Section 2. 4. S 3 as princip al b u ndle In this section we describ e h o w the str u cture of a p r incipal S 1 − bund le o v er S 3 induces a brac k et generating distribution on S 3 . Namely , it is p ossible to consider S 3 as a S 1 − space, according to the action λ · ( z , w ) = ( λz , λw ) , where λ ∈ S 1 = { v ∈ C : | v | 2 = 1 } and ( z , w ) ∈ S 3 = { ( z , w ) ∈ C 2 : | z | 2 + | w | 2 = 1 } . Consider the Hopf map h : S 3 → S 2 as a principal S 1 − bund le, see [14], giv en explicitly b y h ( z , w ) = ( | z | 2 − | w | 2 , 2 z ¯ w ) , where S 2 = { ( x, ζ ) ∈ R × C : x 2 + | ζ | 2 = 1 } . Clearly , h is a submersion of S 3 on to S 2 , and it is a bijection b et w een S 3 /S 1 and S 2 , where S 3 /S 1 is un dersto o d as the orb it space of the S 1 − action o v er S 3 , pr eviously defined. Let p = ( x 0 , ζ 0 ) ∈ S 2 . It is easy to v erify that h − 1 ( p ) = ( z 0 , w 0 ) mo d S 1 , wh ere ( z 0 , w 0 ) is one pr eimage of p und er h . Consider the great circle γ p ( t ) = e 2 π it ( z 0 , w 0 ) , t ∈ [0 , 1] , in S 3 , that pro jects to p un d er the Hopf map. Now co nsider the follo wing ve ctor field asso ciated to γ p , defin ed b y the tangen t vect ors ˙ γ p ( t ) = 2 π ie 2 π it ( z 0 , w 0 ) ∈ T γ p ( t ) S 3 . W e w r ite the Hopf map in real co ordin ates, wher e, in particular, γ p ( t ) = ( z ( t ) , w ( t )) = ( x 0 ( t ) + ix 1 ( t ) , x 2 ( t ) + ix 3 ( t )) = ( x 0 ( t ) , x 1 ( t ) , x 2 ( t ) , x 3 ( t )). It is easy to see that [ d γ p ( t ) h ] = 2   x 0 ( t ) x 1 ( t ) − x 2 ( t ) − x 3 ( t ) x 2 ( t ) x 3 ( t ) x 0 ( t ) x 1 ( t ) − x 3 ( t ) x 2 ( t ) x 1 ( t ) − x 0 ( t )   . Th us, the Hopf map induces the follo wing action ov er the vec tor fi eld ˙ γ p ( t ): [ d γ p ( t ) h ] ˙ γ p ( t ) = 4 π   x 0 ( t ) x 1 ( t ) − x 2 ( t ) − x 3 ( t ) x 2 ( t ) x 3 ( t ) x 0 ( t ) x 1 ( t ) − x 3 ( t ) x 2 ( t ) x 1 ( t ) − x 0 ( t )       ˙ x 0 ( t ) ˙ x 1 ( t ) ˙ x 2 ( t ) ˙ x 3 ( t )     =     0 0 0 0     . Therefore, if [ d γ p ( t ) h ] is a full rank matrix, we would ha v e charac terized the kernel of it, b y (2) k er d γ p ( t ) h = span { ˙ γ p ( t ) } . Notice that, usin g the notation of Section 2, th e follo wing ident it y holds ˙ γ p ( t ) = 2 π V ( γ p ( t )) . (3) There is a simp le wa y to see that the matrix [ d γ p ( t ) h ] is fu ll rank. Denote b y D i the 3 × 3 matrix obtained by deleting the i − th column of [ d γ p ( t ) h ], i = 1 , 2 , 3 or 4, then det( D 1 ) 2 + det( D 2 ) 2 + det( D 3 ) 2 + det ( D 4 ) 2 = 8 MAURICIO GODOY M.,IRINA MARKINA =  ( x 0 ( t )) 2 + ( x 1 ( t )) 2 + ( x 2 ( t )) 2 + ( x 3 ( t )) 2  3 = 1 implies that [ d γ p ( t ) h ] is full r ank. Before describing ho w the Hopf map ind uces a h orizon tal distribution, it is necessary to present s ome definitions. Definition 5 (Ehresmann Connection [19], c hapter 11) . L et M and Q b e two differ e ntiable manifolds, and let π : Q → M b e a submersion. Denoting by Q m = π − 1 ( m ) the fib er thr ough m ∈ M , the ve rtical s p ace at q is the tangent sp ac e at the fib er Q π ( q ) and it is denote d by V q . An Ehresmann conn ection for the submersion π : Q → M is a distribution H ⊂ T Q which is eve rywher e tr ansversal to the vertic al, that is: V q ⊕ H q = T q Q. W e apply Defin ition 5 to S 3 in order to define the Ehresmann connectio n. Sin ce w e kno w that ke r d p h = span { V ( p ) } , for eve ry p ∈ S 3 b y (2) and (3), and moreo v er, h X ( p ) , V ( p ) i p = h Y ( p ) , V ( p ) i p = h X ( p ) , Y ( p ) i p = 0 where h· , ·i p stands for the usual Riemann ian structure defi n ed at p ∈ S 3 , we see that H p = sp an { X ( p ) , Y ( p ) } (4) is an Ehr esmann conn ection for the sub mersion h : S 3 → S 2 with V ( p ) as a v ertical space. Definition 6. L et G b e a Lie gr oup acting on Q and π : Q → M a submersion, with Ehr esmann c onne ction H , which is a fib er bund le with fib er G . The submersion π i s c al le d a princip al G − bund le with c onne c tion, if the fol lowing c onditions hold: • G acts fr e ely and tr ansitively, • the gr oup orbits ar e the fib ers of π : Q → M (thus M is i somorp hic to Q/G and π is the c anonic al pr oje ction) and • the G − action on Q pr eserves the horizontal distribution H ; W e conclude th at the Hopf fibration is a principal S 1 − bund le w ith connection H , d efined in (4). Definition 7. A sub-Riemannian metric ( H , h· , ·i ) on the princip al G -bu nd le π : G → M is c al le d a metric of b und le typ e if the inner pr o duct h· , ·i on the horizontal distribution H is induc e d fr om a R iemannian metric on M . The su b-Riemannian metric h· , ·i| H , obtained b y restricting th e u s ual Riemannian metric of S 3 to the d istribution H is, by constru ction, a metric of bu ndle t yp e. Th us, the Hopf map indicates in a very natural top ological w a y h o w to make a natural c hoice of the h orizonta l distribution H th at wa s not ob vious when w e considered th e righ t action of S 3 o v er itself. Remark: Observe that th e considered ve ctor fi elds coincide with the right in v arian t v ector fields. This p h enomenon do es not app ear when we c hange the right action to the left action of S 3 o v er itself. SUB-RIEMANNIAN GEOME T R Y OF P ARALLELIZABLE SPHERES 9 5. T angent vector fields for S 7 In Sections 5 - 7 w e study different sub-Riemannian structures o ver the sphere S 7 , making use of ideas of Sections 2 - 4. As a result, we obtain t w o p rincipally differen t types of horizonta l d istributions. One of them of r ank 6 and other of the rank 4. Moreo v er, as w e shall s ee, the sub -Riemannian structure induced b y the CR-structure and quaternionic analogue of the Hopf m ap are essentia lly d ifferen t. W e start fr om the construction of the tangen t v ecto r fields to S 7 . The m ultiplicat ion of u n it o ctonions is n ot associativ e, therefore S 7 is not a group in a con trast with S 3 . Neve rtheless, w e still able to use the multiplicat ion la w in order to fugue out global tangen t v ector fi elds . T o do this, we present a m ultiplication table for the basis v ectors of R 8 . The non-asso ciativ e multiplica tion giv es r ise to the division alge bra of the o ctonions O = span { e 0 , e 1 , e 2 , e 3 , e 4 , e 5 , e 6 , e 7 } . e 0 e 1 e 2 e 3 e 4 e 5 e 6 e 7 e 0 e 0 e 1 e 2 e 3 e 4 e 5 e 6 e 7 e 1 e 1 − e 0 e 3 − e 2 e 5 − e 4 − e 7 e 6 e 2 e 2 − e 3 − e 0 e 1 e 6 e 7 − e 4 − e 5 e 3 e 3 e 2 − e 1 − e 0 e 7 − e 6 e 5 − e 4 e 4 e 4 − e 5 − e 6 − e 7 − e 0 e 1 e 2 e 3 e 5 e 5 e 4 − e 7 e 6 − e 1 − e 0 − e 3 e 2 e 6 e 6 e 7 e 4 − e 5 − e 2 e 3 − e 0 − e 1 e 7 e 7 − e 6 e 5 e 4 − e 3 − e 2 e 1 − e 0 T able 1. Mu ltiplication table for the b asis of O . According to the T able 1, the formula for the p ro duct of t w o o ctonions is pr esented in the App endix, su bsection 8.1. This m ultiplicat ion ru le induces a matrix rep r esen tatio n of the righ t action of o ctonion multiplicatio n, giv en explicitly by: R ∗ =             y 0 − y 1 − y 2 − y 3 − y 4 − y 5 − y 6 − y 7 y 1 y 0 y 3 − y 2 y 5 − y 4 − y 7 y 6 y 2 − y 3 y 0 y 1 y 6 y 7 − y 4 − y 5 y 3 y 2 − y 1 y 0 y 7 − y 6 y 5 − y 4 y 4 − y 5 − y 6 − y 7 y 0 y 1 y 2 y 3 y 5 y 4 − y 7 y 6 − y 1 y 0 − y 3 y 2 y 6 y 7 y 4 − y 5 − y 2 y 3 y 0 − y 1 y 7 − y 6 y 5 y 4 − y 3 − y 2 y 1 y 0             . W e are able to find globall y defined tangen t v ect or fields which are in v arian t und er this acti on. W e pro ceed b y the analogy with Section 2. The explicit formulae are giv en in subsection 8.2 of the App endix. The v ector fields { Y 0 , . . . , Y 7 } form a frame. More explicitly , we ha v e that the follo wing identit y holds h Y i ( y ) , Y j ( y ) i y = δ ij , y ∈ S 7 , i, j ∈ { 0 , 1 , . . . , 7 } , 10 MAURICIO GODOY M.,IRINA MARKINA where h· , ·i is the stand ard Riemannian stru cture o v er R 8 . 6. C R -structure and the Ho pf map on S 7 In the b o ok [19] it is br iefly discu ssed the general idea of studying a sub-Riemannian geometry for od d dimen s ional sph eres via the higher Hopf fibrations. Namely , consider S 2 n +1 = { z ∈ C n +1 : k z k 2 = 1 } , then the S 1 − action on S 2 n +1 giv en by λ · ( z 0 , . . . , z n ) = ( λz 0 , . . . , λz n ) , for λ ∈ S 1 and ( z 0 , . . . , z n ) ∈ S 2 n +1 , induces the w ell- kno wn principal S 1 − bund le S 1 → S 2 n +1 → C P n giv en explicitly b y S 2 n +1 ∋ ( z 0 , . . . , z n ) 7→ [ z 0 : . . . : z n ] ∈ C P n where [ z 0 : . . . : z n ] denote h omogeneous co ord inates. This map is calle d higher Hopf fibration. The k er n el of the map h : S 2 n +1 → C P n pro du ces the v ertical space and a transv ersal to the v ertical space distribution giv es the Eh resmann connection. W e sho w that the v ertical s p ace is alwa ys giv en b y an action of standard almost complex structure on th e normal vect or field to S 2 n +1 , and the Eh r esmann connectio n coincides with the holomorphic tangen t sp ace at eac h p oint of S 2 n +1 . Theorem 2 asserts that an y odd dimensional sphere has at least one globally defin ed non v anish ing tangen t v ect or field. If th e dimension of the sphere is of the form 4 n + 1, then it has only one globally defin ed non v anishin g tangen t v ector fi eld. In the case that the dimens ion of the sphere is of th e f orm 4 n + 3, then the sphere adm its more than one v ecto r field. T he sphere S 2 n +1 p ossess the vect or field V n +1 ( y ) = − y 1 ∂ y 0 + y 0 ∂ y 1 − y 3 ∂ y 2 + . . . − y 2 n +2 ∂ y 2 n +1 + y 2 n +1 ∂ y 2 n +2 . Observe that this vecto r field has app eared already in tw o opp ortunities: the v ector field V in Sections 2 , 3 and 4 corresp onds to V 2 ; and the ve ctor field Y 1 in Subsection 8.2 of the App end ix corresp onds to V 4 . The v ector field V n +1 has the remark able prop erty that it encloses v aluable informa- tion concerning the CR structure of S 2 n +1 . W e kno w b y Lemma 1 that, as a smo oth h yp ersur face in C n +1 the sphere S 2 n +1 admits a holomorphic tangen t sp ace of dimension dim R H p S 2 n +1 = 2 n for any p oin t p ∈ S 2 n +1 . T he follo wing lemma implies the description of H p S 2 n +1 as the orthogonal complemen t to V n +1 . Lemma 2. L et W b e an E u clide an sp ac e of dimension n + 2 , n ≥ 1 , and inner pr o duct h· , ·i W . Consider an or tho g onal de c omp osition W = span { X, Y } ⊕ ⊥ ˜ W with r esp e ct to h· , ·i W and an ortho gonal endomorph ism A : W → W such that A (span { X, Y } ) = span { X, Y } , then ˜ W is an invariant sp ac e under the action of A , i.e. A ( ˜ W ) = ˜ W . SUB-RIEMANNIAN GEOME T R Y OF P ARALLELIZABLE SPHERES 11 Pr o of. Let v ∈ ˜ W , then for any α, β ∈ R it is clear th at h Av , αX + β Y i W = h v , A t ( αX + β Y ) i W = h v , A − 1 ( αX + β Y ) i W . Since A (span { X, Y } ) = span { X, Y } , there exist real n um b ers a, b suc h that A − 1 ( αX + β Y ) = aX + bY , and therefore h Av , αX + β Y i W = h v , aX + bY i W = 0 whic h implies that Av ∈ ˜ W .  As an app lication of Lemma 2, it is p ossible to obtain an explicit c h aracterizati on of H p S 2 n +1 . Lemma 3. F or any p ∈ S 2 n +1 , the ve ctor sp ac e H p S 2 n +1 is the ortho gonal c omplement to the ve ctor V n +1 ( p ) in T p S 2 n +1 . Pr o of. Consid er the v ector space W p = sp an { N n +1 ( p ) } ⊕ ⊥ T p S 2 n +1 ∼ = T p R 2 n +2 , where N n +1 ( p ) is the n orm al vect or to S 2 n +1 at p . The standard almost complex structure map J n +1 : W p → W p is clearly orthogonal. Moreo ve r J n +1 ( V n +1 ( p )) = − N n +1 ( p ) , J n +1 ( N n +1 ( p )) = V n +1 ( p ) . Using the decomp osition W p = ˜ W p ⊕ ⊥ span { V n +1 ( p ) , N n +1 ( p ) } it is p ossible to apply Lemma 2 in order to conclude th at ˜ W p , whic h is the orthogonal complemen t to V n +1 ( p ) in T p S 2 n +1 , is in v arian t under J n +1 . Since dim R ˜ W p = 2 n , we conclude that ˜ W p = H p S 2 n +1 .  Remark: The space H S 2 n +1 can also b e describ ed as th e kernel of one form ω = ¯ z 0 dz 0 + . . . + ¯ z n dz n . Indeed, tak e X ∈ H S 2 n +1 , then by straigh tforw ard calculations we ha v e ω ( X ) = h X, N n +1 i + i h X, V n +1 i = 0 . Lemma 3 p ro vides a horizon tal distribu tion of r ank 2 n for the spheres S 2 n +1 , by con- sidering the holomorph ic tangen t space. The question n o w is whether this distribution is br ack et generating. The b rac k et generating co ndition for S 3 is a lready d iscussed in Section 2. Here w e presen t a basis f or b rac k et generating distrib ution of rank 6 for S 7 . The follo wing idea s w ere in trod u ced to the authors by Prof. K. F urutani in a p riv ate comm unication. Theorem 3. The su b bund le H = span { Y 2 , . . . , Y 7 } of T S 7 is a b r acket gener ating distri- bution of r ank 6 and step 2. Pr o of. Define the follo wing v ector fields v 41 ( y ) = − y 4 ∂ y 0 + y 5 ∂ y 1 + y 0 ∂ y 4 − y 1 ∂ y 5 , v 42 ( y ) = y 6 ∂ y 2 − y 7 ∂ y 3 − y 2 ∂ y 6 + y 3 ∂ y 7 , v 51 ( y ) = − y 5 ∂ y 0 − y 4 ∂ y 1 + y 1 ∂ y 4 + y 0 ∂ y 5 , v 52 ( y ) = − y 7 ∂ y 2 − y 6 ∂ y 3 + y 3 ∂ y 6 + y 0 ∂ y 7 , 12 MAURICIO GODOY M.,IRINA MARKINA and observe that v 41 + v 42 = Y 4 and v 51 + v 52 = Y 5 . By s traigh tforw ard calculations w e see that h v 41 ( y ) , Y 0 ( y ) i y = h v 42 ( y ) , Y 0 ( y ) i y = h v 51 ( y ) , Y 0 ( y ) i y = h v 52 ( y ) , Y 0 ( y ) i y = 0 h v 41 ( y ) , Y 1 ( y ) i y = h v 42 ( y ) , Y 1 ( y ) i y = h v 51 ( y ) , Y 1 ( y ) i y = h v 52 ( y ) , Y 1 ( y ) i y = 0 whic h implies that v 41 , v 42 , v 51 , v 52 ∈ sp an { Y 2 , . . . , Y 7 } . The follo wing comm utation rela- tion [ v 41 , v 51 ] + [ v 42 , v 52 ] = − 2 Y 1 implies that the distribution H is b rac k et generating of step 2.  Remark: It is p ossible to rep eat the pr evious argumen t with other p airs of v ector fields. F or example, if instead of Y 4 and Y 5 w e emp lo y Y 2 and Y 3 , we can consider th e v ector fields v 21 ( y ) = − y 4 ∂ y 0 + y 5 ∂ y 1 + y 0 ∂ y 4 − y 1 ∂ y 5 , v 22 ( y ) = y 6 ∂ y 2 − y 7 ∂ y 3 − y 2 ∂ y 6 + y 3 ∂ y 7 , v 31 ( y ) = − y 5 ∂ y 0 − y 4 ∂ y 1 + y 1 ∂ y 4 + y 0 ∂ y 5 , v 32 ( y ) = − y 7 ∂ y 2 − y 6 ∂ y 3 + y 3 ∂ y 6 + y 0 ∂ y 7 . W e can p ro ceed in a s imilar wa y if w e use Y 6 and Y 7 . W e conclude this sectio n by pro ving that the line bu ndle span { V n +1 } is the v ertical space for the su bmersion given by the Hopf fibration S 1 ֒ → S 7 h − → C P n . This implies that the distribution H defined in Theorem 3 is a n Ehresmann connection for h . T o ac h iev e this, w e recall that the charts defin ing the holomorphic structure of C P n are given b y the op en sets U k = { [ z 0 : . . . : z n ] : z k 6 = 0 } , together with th e homeomorphisms ϕ k : U k → C n [ z 0 : . . . : z n ] 7→ ( z 0 z k , . . . , z k − 1 z k , z k +1 z k , . . . , z n z k ) . Then, without loss of generalit y w e w ill assume that n = 3 and w e will dev elop the exp licit calculatio ns for k = 0. The other cases can b e treated sim ilarly . Using the chart ( U 0 , ϕ 0 ) defined ab ov e, we ha ve the map ϕ 0 ◦ h : S 7 → C 3 ( z 0 , z 1 , z 2 , z 3 ) 7→ ( z 1 z 0 , z 2 z 0 , z 3 z 0 ) , whic h in real co ord inates can b e written as ϕ 0 ◦ h ( x 0 , . . . , x 7 ) =  x 0 x 2 + x 1 x 3 x 2 0 + x 2 1 , x 0 x 3 − x 1 x 2 x 2 0 + x 2 1 , x 0 x 4 + x 1 x 5 x 2 0 + x 2 1 , x 0 x 5 − x 1 x 4 x 2 0 + x 2 1 , x 0 x 6 + x 1 x 7 x 2 0 + x 2 1 , x 0 x 7 − x 1 x 6 x 2 0 + x 2 1  . The differen tial of th is mapping is give n by th e matrix d ( ϕ 0 ◦ h ) = SUB-RIEMANNIAN GEOME T R Y OF P ARALLELIZABLE SPHERES 13 0 B B B B B B B B B B B B @ ( x 2 1 − x 2 0 ) x 2 − 2 x 0 x 1 x 3 ( x 2 0 + x 2 1 ) 2 ( x 2 0 − x 2 1 ) x 3 − 2 x 0 x 1 x 2 ( x 2 0 + x 2 1 ) 2 x 0 x 2 0 + x 2 1 x 1 x 2 0 + x 2 1 0 0 0 0 ( x 2 1 − x 2 0 ) x 3 +2 x 0 x 1 x 2 ( x 2 0 + x 2 1 ) 2 ( x 2 1 − x 2 0 ) x 2 − 2 x 0 x 1 x 3 ( x 2 0 + x 2 1 ) 2 − x 1 x 2 0 + x 2 1 x 0 x 2 0 + x 2 1 0 0 0 0 ( x 2 1 − x 2 0 ) x 4 − 2 x 0 x 1 x 5 ( x 2 0 + x 2 1 ) 2 ( x 2 0 − x 2 1 ) x 5 − 2 x 0 x 1 x 4 ( x 2 0 + x 2 1 ) 2 0 0 x 0 x 2 0 + x 2 1 x 1 x 2 0 + x 2 1 0 0 ( x 2 1 − x 2 0 ) x 5 +2 x 0 x 1 x 4 ( x 2 0 + x 2 1 ) 2 ( x 2 1 − x 2 0 ) x 4 − 2 x 0 x 1 x 5 ( x 2 0 + x 2 1 ) 2 0 0 − x 1 x 2 0 + x 2 1 x 0 x 2 0 + x 2 1 0 0 ( x 2 1 − x 2 0 ) x 6 − 2 x 0 x 1 x 7 ( x 2 0 + x 2 1 ) 2 ( x 2 0 − x 2 1 ) x 7 − 2 x 0 x 1 x 6 ( x 2 0 + x 2 1 ) 2 0 0 0 0 x 0 x 2 0 + x 2 1 x 1 x 2 0 + x 2 1 ( x 2 1 − x 2 0 ) x 7 +2 x 0 x 1 x 6 ( x 2 0 + x 2 1 ) 2 ( x 2 1 − x 2 0 ) x 6 − 2 x 0 x 1 x 7 ( x 2 0 + x 2 1 ) 2 0 0 0 0 − x 1 x 2 0 + x 2 1 x 0 x 2 0 + x 2 1 1 C C C C C C C C C C C C A . By straigh tforw ard calculations, we kno w that det([ d ( ϕ 0 ◦ H )][ d ( ϕ 0 ◦ H )] t ) = ( x 2 0 + x 2 1 ) − 8 , therefore, the matrix d ( ϕ 0 ◦ H ) has rank 6 and the dimension of its k ernel is 2: dim R k er d ( ϕ 0 ◦ H ) = 2 . Moreo ver, since d ( ϕ 0 ◦ H )( N n +1 ) = d ( ϕ 0 ◦ H )( V n +1 ) = 0 , b y direct calculations, we conclude k er d ( ϕ 0 ◦ H ) = span { N n +1 , V n +1 } . This implies that k er dH = span { V n +1 } . 7. Applica t ion of t h e first qua t ernionic Hopf fibra tion T rying to imitate the work already done for S 3 , w e find through the quaternionic Hopf bund le S 3 → S 7 → S 4 one of the n atural c hoice of horizon tal distribu tions. W e consider the quaternionic Hopf m ap giv en by (5) h : S 7 → S 4 ( z , w ) 7→ ( | z | 2 − | w | 2 , 2 z ¯ w ) . It can b e written in r eal co ordinates: h ( x 0 , . . . , x 7 ) = ( x 2 0 + x 2 1 + x 2 2 + x 2 3 − x 2 4 − x 2 5 − x 2 6 − x 7 , (6) 2( x 0 x 4 + x 1 x 5 + x 2 x 6 + x 3 x 7 ) , 2( − x 0 x 5 + x 1 x 4 − x 2 x 7 + x 3 x 6 ) , 2( − x 0 x 6 + x 1 x 7 + x 2 x 4 − x 3 x 5 ) , 2( − x 0 x 7 − x 1 x 6 + x 2 x 5 + x 3 x 4 )) . The differen tial map dh of h is the follo wing: dh = 2       x 0 x 1 x 2 x 3 − x 4 − x 5 − x 6 − x 7 x 4 x 5 x 6 x 7 x 0 x 1 x 2 x 3 − x 5 x 4 − x 7 x 6 x 1 − x 0 x 3 − x 2 − x 6 x 7 x 4 − x 5 x 2 − x 3 − x 0 x 1 − x 7 − x 6 x 5 x 4 x 3 x 2 − x 1 − x 0       . Since no one of the commuta tors [ Y i , Y j ], i, j = 1 , . . . , 7 coincides with the Y k , k = 1 , . . . , 7, w e lo ok for th e k ernel of d h among the co mm utators Y ij , i, j = 1 , . . . , 7. W e found that [ dh ] Y 45 = [ dh ] Y 46 = [ dh ] Y 56 = 0. Define V = { Y 45 , Y 46 , Y 56 } . 14 MAURICIO GODOY M.,IRINA MARKINA Our n ext step is to find the h orizon tal distribution span {H} that is transv erse to span { V } and brac k et generating: span {H} p ⊕ span { V } p = T p S 7 for all p ∈ S 7 . T o b egin w e d efine fi v e basis for horizont al distribu tions, that we sh all work w ith. The n umeration is v alid only for this section. H 0 = { Y 47 , Y 57 , Y 67 , W } , H 1 = { Y 34 , Y 35 , Y 36 , Y 37 } , H 2 = { Y 24 , Y 25 , Y 26 , Y 27 } , H 3 = { Y 14 , Y 15 , Y 16 , Y 17 } , H 4 = { Y 04 , Y 05 , Y 06 , Y 07 } , where the vec tor field W will b e defined late r and the n otation Y 0 k = Y k is c h osen for th e con v enience. W e collect s ome u seful information ab out sets H m , m = 0 , . . . , 4, that w e exploit later. 1. All vec tor fi elds insid e H m , m = 0 , 1 , 2 , 3 , 4 are orthonormal (w e do not count W b efore we precise it). 2. All of col lections H m , m = 0 , 1 , 2 , 3 , 4 are brac k et ge nerating with the follo wing comm utator relations: 1 2 [ Y j 4 , Y j 5 ] = Y 45 , 1 2 [ Y j 4 , Y j 6 ] = Y 46 , 1 2 [ Y j 5 , Y j 6 ] = Y 56 , j = 0 , 1 , 2 , 3 , 1 2 [ Y 47 , Y 57 ] = Y 45 , 1 2 [ Y 47 , Y 67 ] = Y 46 , 1 2 [ Y 57 , Y 67 ] = Y 56 . 3. W e aim to calculat e the angles b et w een the v ector fields fr om H m , m = 0 , 1 , 2 , 3 , 4 and b et w een vec tor fields from H m and V . Beforehand, w e introd uce the follo wing notations for the co ord inates on th e sph ere S 4 giv en by the Hopf map S 3 → S 7 → S 4 . a 00 = y 2 0 + y 2 1 + y 2 2 + y 2 3 − y 2 4 − y 2 5 − y 2 6 − y 2 7 , a 11 = 2( y 0 y 4 + y 1 y 5 + y 2 y 6 + y 3 y 7 ) , a 22 = 2( − y 0 y 5 + y 1 y 4 − y 2 y 7 + y 3 y 6 ) , (7) a 33 = 2( − y 0 y 6 + y 1 y 7 + y 2 y 4 − y 3 y 5 ) , a 44 = 2( − y 0 y 7 − y 1 y 6 + y 2 y 5 + y 3 y 4 ) . The first index of a mk reflects the n um b er of the collection H m , where they will app ear and the second one is related to the num b er of the co ordinate on S 4 . W e start fr om H 0 and calculate the inner pro d ucts: h Y 45 , Y 67 i = −h Y 46 , Y 57 i = h Y 56 , Y 47 i = a 00 . All other ve ctor fi elds are orthogonal. W e con tin ue for H 1 . (8) h Y 45 , Y 36 i = −h Y 46 , Y 35 i = h Y 56 , Y 34 i = a 11 h Y 45 , Y 37 i = 2( − y 0 y 5 + y 1 y 4 + y 2 y 7 − y 3 y 6 ) = a 12 h Y 46 , Y 37 i = 2( − y 0 y 6 − y 1 y 7 + y 2 y 4 + y 3 y 5 ) = a 13 h Y 56 , Y 37 i = 2( y 0 y 7 − y 1 y 6 + y 2 y 5 − y 3 y 4 ) = a 14 . SUB-RIEMANNIAN GEOME T R Y OF P ARALLELIZABLE SPHERES 15 All other ve ctor fi elds in H 1 ∪ V are orthogonal. F o r th e s et H 2 w e see the follo wing: (9) −h Y 45 , Y 26 i = h Y 46 , Y 25 i = −h Y 56 , Y 24 i = a 22 h Y 45 , Y 27 i = 2( y 0 y 4 + y 1 y 5 − y 2 y 6 − y 3 y 7 ) = a 21 h Y 46 , Y 27 i = 2( − y 0 y 7 + y 1 y 6 + y 2 y 5 − y 3 y 4 ) = a 24 h Y 56 , Y 27 i = 2( − y 0 y 6 − y 1 y 7 − y 2 y 4 − y 3 y 5 ) = a 23 The other pr o ducts b et w een ve ctor fields from H 2 ∪ V v anish . F or H 3 the situation is very similar. (10) h Y 45 , Y 16 i = −h Y 46 , Y 15 i = h Y 56 , Y 14 i = a 33 h Y 45 , Y 17 i = 2( − y 0 y 7 − y 1 y 6 − y 2 y 5 − y 3 y 4 ) = a 34 h Y 46 , Y 17 i = 2( − y 0 y 4 + y 1 y 5 − y 2 y 6 + y 3 y 7 ) = a 31 h Y 56 , Y 17 i = 2( − y 0 y 5 − y 1 y 4 + y 2 y 7 + y 3 y 6 ) = a 32 All other ve ctor fi elds from H 3 ∪ V are orthogonal. F or the last collection H 4 w e obtain. (11) h Y 45 , Y 06 i = −h Y 46 , Y 05 i = h Y 56 , Y 04 i = a 44 h Y 45 , Y 07 i = 2( y 0 y 6 − y 1 y 7 + y 2 y 4 − y 3 y 5 ) = a 43 h Y 46 , Y 07 i = 2( − y 0 y 5 − y 1 y 4 − y 2 y 7 − y 3 y 6 ) = a 42 h Y 56 , Y 07 i = 2( y 0 y 4 − y 1 y 5 − y 2 y 6 + y 3 y 7 ) = a 41 with the rest of the p r o duct v anishing. W e notice some relations b et ween the co efficien ts a mk . The co ord in ates on S 4 p ossesses the equalit y (12) a 2 00 + a 2 11 + a 2 22 + a 2 33 + a 2 44 = 1 . The direct calculations also show a 2 00 + a 2 11 + a 2 12 + a 2 13 + a 2 14 = 1 a 2 00 + a 2 21 + a 2 22 + a 2 23 + a 2 24 = 1 (13) a 2 00 + a 2 31 + a 2 32 + a 2 33 + a 2 34 = 1 a 2 00 + a 2 41 + a 2 42 + a 2 43 + a 2 44 = 1 . In other words the sum of th e sq u ares of the cosines b et w een vect or fields from H m ∪ V , m = 1 , 2 , 3 , 4 is equal to 1 − a 2 00 . Let u s consider 2 cases: 0 < a 2 00 ≤ 1 and a 2 00 = 0. Case 0 < a 2 00 ≤ 1 . This case corresp onds to an y p oin t on S 4 except of the set (14) S 1 = { y 2 0 + y 2 1 + y 2 2 + y 2 3 = y 2 4 + y 2 5 + y 2 6 + y 2 7 = 1 / 2 } . W e observe that the sum of the square of the cosines fr om (13): 4 X k =1 a 2 mk = 1 − a 2 00 , m = 1 , 2 , 3 , 4 b elongs to the inte rv al (0 , 1) and no one of the cosines can b e equal to 1. W e conclude that eac h of H m , m = 1 , 2 , 3 , 4, is transv erse to V . P articularly , if a 2 00 = 1 then P 4 k =1 a 2 mk = 0 and H m ⊥ V . Th e latter situation o ccur s in th e antipo dal p oin ts ( ± 1 , 0 , 0 , 0 , 0) ∈ S 4 or is to sa y on the set S 2 = { y 2 0 + y 2 1 + y 2 2 + y 2 3 = 0 , y 2 4 + y 2 5 + y 2 6 + y 2 7 = 1 } ∪ 16 MAURICIO GODOY M.,IRINA MARKINA (15) { y 2 0 + y 2 1 + y 2 2 + y 2 3 = 1 , y 2 4 + y 2 5 + y 2 6 + y 2 7 = 0 } ∈ S 7 . W e also can consid er a colle ction H 0 , as a p ossible horizont al brac k et generating distri- bution, if we c h o ose an adequate ve ctor fi eld W . If a 00 ∈ (0 , 1) we ha v e 0 < a 2 41 + a 2 42 + a 2 43 + a 2 44 = 1 − a 2 00 < 1 and no one of the pro d ucts in (11) can giv e 1. W e conclude th at Y 07 can not b e collinear to V = { Y 45 , Y 46 , Y 56 } . Th erefore, w e choose W = Y 07 . By the same reason w e could tak e Y j 7 , j = 1 , 2 , 3. In the case wh en a 2 00 = 1 the vecto r fields Y j 7 , j = 0 , 1 , 2 , 3 are orthogonal to V but H 0 is collinea r to V and the collection H 0 with W = Y j 7 is not transv erse to V . Case a 2 00 = 0 . In this ca se the d istribution H 0 is nicely serve as a brac k et ge nerating if w e fin d a suitable v ector field W . Not ice that (12) b ecame (16) a 2 11 + a 2 22 + a 2 33 + a 2 44 = 1 . The a mm can not v anish sim ultaneously . Without lost of generalit y , w e can assume that a 44 6 = 0. Then a 2 41 + a 2 42 + a 2 43 = 1 − a 2 44 < 1 from (13) and the pro du cts (11) imp ly that Y 07 is transv erse to V and can b e used as a v ector fi eld W . In the case a 2 44 = 1 we get that Y 07 is orthogonal to V . S ince W ⊥ Y j 7 , j = 0 , . . . , 3 the collection H 0 with an y c h oice of Y j 7 , j = 0 , . . . , 3 will b e orthonormal. W e formulate the latter resu lt in the follo wing theorem Theorem 4. L et (5 ) b e the H opf map with the vertic al sp ac e V = { Y 45 , Y 46 , Y 56 } , S 1 and S 2 ar e given by (14) and (7) . Then the Hoph map pr o duc es the fol lowing Ehr es- mann Conne ction H p , p ∈ S 7 : (i) if p / ∈ S 1 then H p = ( H m ) p , for any choic e of m = 1 , 2 , 3 , 4 ; (ii) if p / ∈ S 2 then H p = ( H 0 ∪ Y j 7 ) p , j = 0 , 1 , 2 , 3 ; and we have span {H 0 , Y j 7 } p ⊕ span { V } p = T p S 7 , j = 0 , 1 , 2 , 3 for al l p ∈ S 7 . 8. Appendix 8.1. Multiplicat ion of o ctonions. Let o 1 = ( x 0 e 0 + x 1 e 1 + x 2 e 2 + x 3 e 3 + x 4 e 4 + x 5 e 5 + x 6 e 6 + x 7 e 7 ) and o 2 = ( y 0 e 0 + y 1 e 1 + y 2 e 2 + y 3 e 3 + y 4 e 4 + y 5 e 5 + y 6 e 6 + y 7 e 7 ) b e t w o o ctonions. Then w e ha v e according to T able 1 o 1 ◦ o 2 = ( x 0 e 0 + x 1 e 1 + x 2 e 2 + x 3 e 3 + x 4 e 4 + x 5 e 5 + x 6 e 6 + x 7 e 7 ) ◦ ( y 0 e 0 + y 1 e 1 + y 2 e 2 + y 3 e 3 + y 4 e 4 + y 5 e 5 + y 6 e 6 + y 7 e 7 ) = = ( x 0 y 0 − x 1 y 1 − x 2 y 2 − x 3 y 3 − x 4 y 4 − x 5 y 5 − x 6 y 6 − x 7 y 7 ) e 0 + +( x 1 y 0 + x 0 y 1 − x 3 y 2 + x 2 y 3 − x 5 y 4 + x 4 y 5 + x 7 y 6 − x 6 y 7 ) e 1 + +( x 2 y 0 + x 3 y 1 + x 0 y 2 − x 1 y 3 − x 6 y 4 − x 7 y 5 + x 4 y 6 + x 5 y 7 ) e 2 + SUB-RIEMANNIAN GEOME T R Y OF P ARALLELIZABLE SPHERES 17 +( x 3 y 0 − x 2 y 1 + x 1 y 2 + x 0 y 3 − x 7 y 4 + x 6 y 5 − x 5 y 6 + x 4 y 7 ) e 3 + +( x 4 y 0 + x 5 y 1 + x 6 y 2 + x 7 y 3 + x 0 y 4 − x 1 y 5 − x 2 y 6 − x 3 y 7 ) e 4 + +( x 5 y 0 − x 4 y 1 + x 7 y 2 − x 6 y 3 + x 1 y 4 + x 0 y 5 + x 3 y 6 − x 2 y 7 ) e 5 + +( x 6 y 0 − x 7 y 1 − x 4 y 2 + x 5 y 3 + x 2 y 4 − x 3 y 5 + x 0 y 6 + x 1 y 7 ) e 6 + +( x 7 y 0 + x 6 y 1 − x 5 y 2 − x 4 y 3 + x 3 y 4 + x 2 y 5 − x 1 y 6 + x 0 y 7 ) e 7 . 8.2. Right in v a ria n t v ector fields. According to the previous multiplic ation rule, we ha v e the follo wing un it v ector fields of R 8 arising as righ t in v arian ts v ect or fields under the o ctonion multiplica tion. Y 0 ( y ) = y 0 ∂ y 0 + y 1 ∂ y 1 + y 2 ∂ y 2 + y 3 ∂ y 3 + y 4 ∂ y 4 + y 5 ∂ y 5 + y 6 ∂ y 6 + y 7 ∂ y 7 Y 1 ( y ) = − y 1 ∂ y 0 + y 0 ∂ y 1 − y 3 ∂ y 2 + y 2 ∂ y 3 − y 5 ∂ y 4 + y 4 ∂ y 5 − y 7 ∂ y 6 + y 6 ∂ y 7 Y 2 ( y ) = − y 2 ∂ y 0 + y 3 ∂ y 1 + y 0 ∂ y 2 − y 1 ∂ y 3 − y 6 ∂ y 4 + y 7 ∂ y 5 + y 4 ∂ y 6 − y 5 ∂ y 7 Y 3 ( y ) = − y 3 ∂ y 0 − y 2 ∂ y 1 + y 1 ∂ y 2 + y 0 ∂ y 3 + y 7 ∂ y 4 + y 6 ∂ y 5 − y 5 ∂ y 6 − y 4 ∂ y 7 Y 4 ( y ) = − y 4 ∂ y 0 + y 5 ∂ y 1 + y 6 ∂ y 2 − y 7 ∂ y 3 + y 0 ∂ y 4 − y 1 ∂ y 5 − y 2 ∂ y 6 + y 3 ∂ y 7 Y 5 ( y ) = − y 5 ∂ y 0 − y 4 ∂ y 1 − y 7 ∂ y 2 − y 6 ∂ y 3 + y 1 ∂ y 4 + y 0 ∂ y 5 + y 3 ∂ y 6 + y 2 ∂ y 7 Y 6 ( y ) = − y 6 ∂ y 0 + y 7 ∂ y 1 − y 4 ∂ y 2 + y 5 ∂ y 3 + y 2 ∂ y 4 − y 3 ∂ y 5 + y 0 ∂ y 6 − y 1 ∂ y 7 Y 7 ( y ) = − y 7 ∂ y 0 − y 6 ∂ y 1 + y 5 ∂ y 2 + y 4 ∂ y 3 − y 3 ∂ y 4 − y 2 ∂ y 5 + y 1 ∂ y 6 + y 0 ∂ y 7 . The v ector fields Y i , i = 1 , . . . , 7 form an o rthonormal fr ame of T p S 7 , p ∈ S 7 , with resp ect to restriction of the inner pro duct h· , ·i from R 8 to the tangen t space T p S 7 at eac h p ∈ S 7 . 8.3. C omm utators b et w een rig h t in v ariant v ector fields. Denoting by Y ij ( y ) = 1 2 [ Y i ( y ) , Y j ( y )] the comm utator b et w een the r igh t inv ariant v ector fields, describ ed in the previous subsection, we ha v e the follo wing list: Y 12 ( y ) = y 3 ∂ y 0 + y 2 ∂ y 1 − y 1 ∂ y 2 − y 0 ∂ y 3 + y 7 ∂ y 4 + y 6 ∂ y 5 − y 5 ∂ y 6 − y 4 ∂ y 7 Y 13 ( y ) = − y 2 ∂ y 0 + y 3 ∂ y 1 + y 0 ∂ y 2 − y 1 ∂ y 3 + y 6 ∂ y 4 − y 7 ∂ y 5 − y 4 ∂ y 6 + y 5 ∂ y 7 Y 14 ( y ) = y 5 ∂ y 0 + y 4 ∂ y 1 − y 7 ∂ y 2 − y 6 ∂ y 3 − y 1 ∂ y 4 − y 0 ∂ y 5 + y 3 ∂ y 6 + y 2 ∂ y 7 Y 15 ( y ) = − y 4 ∂ y 0 + y 5 ∂ y 1 − y 6 ∂ y 2 + y 7 ∂ y 3 + y 0 ∂ y 4 − y 1 ∂ y 5 + y 2 ∂ y 6 − y 3 ∂ y 7 Y 16 ( y ) = y 7 ∂ y 0 + y 6 ∂ y 1 + y 5 ∂ y 2 + y 4 ∂ y 3 − y 3 ∂ y 4 − y 2 ∂ y 5 − y 1 ∂ y 6 − y 0 ∂ y 7 Y 17 ( y ) = − y 6 ∂ y 0 + y 7 ∂ y 1 + y 4 ∂ y 2 − y 5 ∂ y 3 − y 2 ∂ y 4 + y 3 ∂ y 5 + y 0 ∂ y 6 − y 1 ∂ y 7 Y 23 ( y ) = y 1 ∂ y 0 − y 0 ∂ y 1 + y 3 ∂ y 2 − y 2 ∂ y 3 − y 5 ∂ y 4 + y 4 ∂ y 5 − y 7 ∂ y 6 + y 6 ∂ y 7 Y 24 ( y ) = y 6 ∂ y 0 + y 7 ∂ y 1 + y 4 ∂ y 2 + y 5 ∂ y 3 − y 2 ∂ y 4 − y 3 ∂ y 5 − y 0 ∂ y 6 − y 1 ∂ y 7 Y 25 ( y ) = − y 7 ∂ y 0 + y 6 ∂ y 1 + y 5 ∂ y 2 − y 4 ∂ y 3 + y 3 ∂ y 4 − y 2 ∂ y 5 − y 1 ∂ y 6 + y 0 ∂ y 7 Y 26 ( y ) = − y 4 ∂ y 0 − y 5 ∂ y 1 + y 6 ∂ y 2 + y 7 ∂ y 3 + y 0 ∂ y 4 + y 1 ∂ y 5 − y 2 ∂ y 6 − y 3 ∂ y 7 Y 27 ( y ) = y 5 ∂ y 0 − y 4 ∂ y 1 + y 7 ∂ y 2 − y 6 ∂ y 3 + y 1 ∂ y 4 − y 0 ∂ y 5 + y 3 ∂ y 6 − y 2 ∂ y 7 Y 34 ( y ) = − y 7 ∂ y 0 + y 6 ∂ y 1 − y 5 ∂ y 2 + y 4 ∂ y 3 − y 3 ∂ y 4 + y 2 ∂ y 5 − y 1 ∂ y 6 + y 0 ∂ y 7 Y 35 ( y ) = − y 6 ∂ y 0 − y 7 ∂ y 1 + y 4 ∂ y 2 + y 5 ∂ y 3 − y 2 ∂ y 4 − y 3 ∂ y 5 + y 0 ∂ y 6 + y 1 ∂ y 7 Y 36 ( y ) = y 5 ∂ y 0 − y 4 ∂ y 1 − y 7 ∂ y 2 + y 6 ∂ y 3 + y 1 ∂ y 4 − y 0 ∂ y 5 − y 3 ∂ y 6 + y 2 ∂ y 7 Y 37 ( y ) = y 4 ∂ y 0 + y 5 ∂ y 1 + y 6 ∂ y 2 + y 7 ∂ y 3 − y 0 ∂ y 4 − y 1 ∂ y 5 − y 2 ∂ y 6 − y 3 ∂ y 7 Y 45 ( y ) = y 1 ∂ y 0 − y 0 ∂ y 1 − y 3 ∂ y 2 + y 2 ∂ y 3 + y 5 ∂ y 4 − y 4 ∂ y 5 − y 7 ∂ y 6 + y 6 ∂ y 7 18 MAURICIO GODOY M.,IRINA MARKINA Y 46 ( y ) = y 2 ∂ y 0 + y 3 ∂ y 1 − y 0 ∂ y 2 − y 1 ∂ y 3 + y 6 ∂ y 4 + y 7 ∂ y 5 − y 4 ∂ y 6 − y 5 ∂ y 7 Y 47 ( y ) = − y 3 ∂ y 0 + y 2 ∂ y 1 − y 1 ∂ y 2 + y 0 ∂ y 3 + y 7 ∂ y 4 − y 6 ∂ y 5 + y 5 ∂ y 6 − y 4 ∂ y 7 Y 56 ( y ) = − y 3 ∂ y 0 + y 2 ∂ y 1 − y 1 ∂ y 2 + y 0 ∂ y 3 − y 7 ∂ y 4 + y 6 ∂ y 5 − y 5 ∂ y 6 + y 4 ∂ y 7 Y 57 ( y ) = − y 2 ∂ y 0 − y 3 ∂ y 1 + y 0 ∂ y 2 + y 1 ∂ y 3 + y 6 ∂ y 4 + y 7 ∂ y 5 − y 4 ∂ y 6 − y 5 ∂ y 7 Y 67 ( y ) = y 1 ∂ y 0 − y 0 ∂ y 1 − y 3 ∂ y 2 + y 2 ∂ y 3 − y 5 ∂ y 4 + y 4 ∂ y 5 + y 7 ∂ y 6 − y 6 ∂ y 7 . Referen ces [1] J. F. Adams , V e ct or Fields on Spher es . Ann. of Math. (2) 75 (1962) 603–632. [2] A. Agra chev, U . Boscain, J.-P. Gauthier, F. Ros si , The intrinsic hyp o el liptic L aplacian and its he at kernel on unimo dular Lie gr oups . arXiv :0806.0 734 . [3] D. Aleksee vsky, Y. Kamishi m a , Pseudo-c onformal quaternionic CR st ructur e on (4 n + 3) - dimensional manifolds . Ann . Mat. Pura Appl. (4) 187 (2008), no. 3, 487–529. [4] A. Boggess , CR manifol ds and the tangential Cauchy-R iemann c omplex . Studies in Adva nced Math- ematics. CRC Press, Bo ca Raton, FL, 1991. 364 pp . [5] U. Boscain, F. Rossi , Invariant Carnot-Car athe o dory metrics on S 3 , S O (3) , S L (2) and lens sp ac es . SIAM J. Con trol Optim. 47 (2008), no. 4, 1851–1878. [6] R. B ott, J. M i lnor , On the p ar al lelizability of the spher es. Bull. Amer. Math. So c. 64 (1958) 87–89. [7] D.-C. Cha ng, I. Markina , Ge ometric analysis on quat ernion H -typ e gr oups . J. Geom. Anal. 16 (2006), no. 2, 265–294. [8] D.-C. Chang, I. Markina, A. V asil ’ev , Sub-R iemannian ge o desics on the 3-D spher e . Complex Anal. Op er. Theory (2008). [9] W.-L. Chow , ¨ Ub er Systeme von li ne ar en p artiel len Di ffer entialgleichungen erster Or dnung . (German) Math. Ann. 117 (1939) 98–105. [10] S. Dragomir, G . Tomassini , Differ ential ge ometry and analysis on CR manifolds . Progress in Math- ematics, 246. Birkhuser Boston, Inc., Boston, MA, 2006. xv i+487 pp . [11] B. Ga veau , Pri ncip e de moindr e action, pr op agation de la chaleur et estim´ ees sous el li ptiques sur c ertains gr oup es ni lp otents , Acta Math. 139 (1977), n o. 1–2, 95–153. [12] V. Y a. Gershko vi ch, A. M. Vershik , Ge o desic flow on SL(2 , R ) with nonholonomic c onstr aints . (Russian) Zap. Nauchn. Sem. Leningrad. Otdel. Mat. In st. St eklo v. ( LOMI) 155 (1986), D ifferen tsial- nay a Geometriya, Gruppy Li i Mekh. VI I I, 7–17, 193; t ranslation in J. Soviet Math. 41 (1988), no. 2, 891-898 [13] M. Gro mo v , Carnot-Car ath´ eo dory sp ac es se en fr om within, sub-R iemannian ge ometry , Prog. Math. 144 (1996), 79–32 3. Birkh¨ auser, Basel. [14] H. Hopf , ¨ Ub er die Abbildungen von Sph¨ ar en auf Sph¨ ar en nie driger er Dimension . Math. Ann. 104 (1931), 637–665. [15] A. Hur t ado, C. Rosales , A r e a-stationary surfac es inside the sub-R iemannian thr e e-spher e . Math. Ann. 340 (2008), n o. 3, 675–708. [16] W. Liu, H. J. Sussmann , Shortest p aths for sub-R iemannian m etrics on r ank-two distributions. Mem. Amer. Math. So c. /bf 118 (1995), no. 564, 104 pp. [17] D. W. L yons , An elementary intr o duction to the Hopf fibr ation . Math. Mag. 76 (2003), no. 2, 87-98. [18] J. Mitche ll , On Carnot-Car ath´ eo dory metrics . J. Differential Geom. 21 (1985), 35-45. [19] R. Montgomer y , A tour of subriemannian ge ometries, their ge o desics and applic ations . Mathemat- ical Surveys and Monographs, 91 . American Mathematical So ciety , Pro vidence, RI, 2002. 259 pp. [20] P. K. Ra shevski ˘ ı , Ab out c onne cting two p oints of c omplete nonholonomic sp ac e by admissible curve , Uch. Zapiski Pe d. Inst. K. Liebknecht 2 (1938), 83–94. [21] R. Strichar tz , Sub-R iemannian Ge ometry. Journal of Differen tial Geometry , 24 (1986) 221–2 63. [22] H. Sussmann , Orbits of f am ilies of ve ctor fields and inte gr ability of distributions. T rans. Amer. Math. Soc. 180 (1973), 171–188. [23] H. K. Urbantke The Hopf fibr ation—se ven times in physics. J. Geom. Ph ys. 46 (2003), no. 2, 125–150 . SUB-RIEMANNIAN GEOME T R Y OF P ARALLELIZABLE SPHERES 19 [24] J. Vick , Homolo gy The ory: An Intr o duction to Algebr ai c T op olo gy . Graduate T ex ts in Mathematics, V ol 145. Sprin ger-V erla g N ew Y ork , 1994. Dep ar tment of Ma thema tics, University of Bergen, Nor w a y. E-mail addr ess : maurici o.godoy@m ath.uib.no Dep ar tment of Ma thema tics, University of Bergen, Nor w a y. E-mail addr ess : irina.m arkina@ui b.no