Hyperbolic rank rigidity for manifolds of \frac14-pinched negative curvature
aa r X i v : . [ m a t h . DG ] M a y HYPERBOLIC RANK RIGIDITY FOR MANIFOLDS OF -PINCHEDNEGATIVE CURVATURE CHRIS CONNELL † , THANG NGUYEN, RALF SPATZIER ‡ Abstract.
A Riemannian manifold M has higher hyperbolic rank if every geodesic has aperpendicular Jacobi field making sectional curvature -1 with the geodesic. If in addition,the sectional curvatures of M lie in the interval [ − , − ] , and M is closed, we show that M is a locally symmetric space of rank one. This partially extends work by Constantineusing completely different methods. It is also a partial converse to Hamenstädt’s hyperbolicrank rigidity result for sectional curvatures ≤ − , and complements well-known results onEuclidean and spherical rank rigidity. Introduction
Given a closed Riemannian manifold M and a unit vector v ∈ SM , we define the hyperbolicrank rk h ( v ) of v as the dimension of the subspace of v ⊥ ⊂ T M which are the initial vectorsof a Jacobi field J ( t ) along g t v which spans a plane of sectional curvature − with g t v for all t ≥ (where J ( t ) = 0 ). The hyperbolic rank of M , rk h ( M ) , then is the infimum of rk h ( v ) over all unit vectors v . We also say that M has higher hyperbolic rank if rk h ( M ) > .Our notion of hyperbolic rank is a priori weaker than either the usual one which requiresthat the Jacobi fields in question make curvature − for t ∈ ( −∞ , ∞ ) or else the versionthat uses parallel fields in place of Jacobi fields. In strict negative curvature these distinctformulations turn out to coincide (see Corollary 2.8). Actually, the techniques of our proofsrequire us to introduce the notion of hyperbolic rank for positive time.The main goal of this paper is the following hyperbolic rank rigidity result. Theorem 1.1.
Let M be a closed Riemannian manifold of higher hyperbolic rank and sec-tional curvatures K between − ≤ K ≤ − . Then M is a rank one locally symmetric space.In particular, if the pinching is strict then M has constant curvature − . Constantine [Con08, Corollary 1] characterized constant curvature manifolds among thoseof nonpositive curvature under one of two conditions: odd dimension without further curva-ture restrictions, or even dimension provided the sectional curvatures are pinched between − ( . and − . He also showed that if one uses the stronger notion of parallel fields in placeof Jacobi fields then one may relax the lower curvature bound of − , though still requiringthe same pinching in even dimensions. His method is rather different from ours, drawing onergodicity results for the 2-frame flow of such manifolds. For -pinched manifolds of negativecurvature however, ergodicity of the frame flow has been conjectured now for over 30 years,with no avenue for an approach in sight [Bri82, Conjecture 2.6]. Mathematics Subject Classification.
Primary 53C24; Secondary 53C20,37D40. † Supported in part by Simons Foundation grant ‡ Supported in part by NSF grants DMS 1307164 and DMS 1607260. oth Constantine’s and our result are counterpoints to Hamenstädt’s hyperbolic rankrigidity theorem [Ham91b]: Theorem 1.2. (Hamenstädt)
Closed manifolds with sectional curvatures K ≤ − and higherhyperbolic rank are locally symmetric spaces of real rank 1. Compactness is truly essential in these results. Indeed, Connell found a counterexam-ple amongst homogeneous manifolds of negative curvature whilst proving hyperbolic rankrigidity for such spaces under an additional condition [Con02].Lin and Schmidt recently constructed non-compact manifolds of higher hyperbolic rank in[LS16] with both upper and lower curvature bounds − and curvatures arbitrarily pinched.In addition, their examples are not even locally homogeneous and every geodesic lies ina totally geodesic hyperbolic plane. In dimension three, Lin showed that finite volumemanifolds with higher hyperbolic rank always have constant curvature, without imposingany curvature properties [LS16].The notion of hyperbolic rank is analogous to that of strong Euclidean and spherical rank where we are looking for parallel vector fields (not just Jacobi fields) along geodesics thatmake curvature 0 or 1 respectively. When 0, 1 or -1 are also extremal as values of sectionalcurvature, various rigidity theorems have been proved. In particular we have the results ofBallmann and Burns-Spatzier in nonpositive curvature where higher rank Euclidean mani-folds are locally either products or symmetric spaces (cf. [Bal85, Bal95, BS87], Eberlein andHeber [EH90] for certain noncompact manifolds and Watkins [Wat13] for no-focal points).When the sectional curvatures are less than 1, and M has higher spherical rank, Shankar,Spatzier and Wilking showed that M is locally isometric to a compact rank one symmetricspace [SSW05]. Notably, there are counterexamples in the form of the Berger metrics forthe analogous statements replacing Jacobi fields by parallel fields in the definition of higherspherical rank (see [SSW05]).Thus the situation for closed manifolds is completely understood for upper curvaturebounds, and we have full rigidity. For lower curvature bounds, the situation is more com-plicated. For one, there are many closed manifolds of nonnegative curvature and higherEuclidean rank. The first examples were given by Heintze (private communication) andwere still homogeneous. More general and in particular inhomogeneous examples were con-structed by Spatzier and Strake in [SS90]. For higher spherical rank and lower bound on thesectional curvature by 1, Schmidt, Shankar and Spatzier again proved local isometry to asphere of curvature 1 if the spherical rank is at least n − > , n is odd or if n = 2 , and M is a sphere [SSS16]. No counterexamples are known. If M in addition is Kähler of dimensionat least 4, then M is locally isometric to complex projective space with the Fubini-Studymetric. In dimension 3, Bettiol and Schmidt showed that higher rank implies local splittingof the metric, without any conditions on the curvature [BS16].Let us outline our argument for Theorem 1.1 which occupies the remainder of this paper. Infact, all of our arguments hold for manifolds of with sectional curvature bounds − ≤ K < until Section 5. We show that we may assume that every geodesic c ( t ) has orthogonal parallelfields E with sectional curvature − . The dimension of the latter vector space is called the strong hyperbolic rank of c . Following Constantine in [Con08, Section 5], strong rank agreeswith the rank under lower sectional curvature bound − (cf. Proposition 2.5). Then we show n Section 2 that the regular set R of unit tangent vectors v for which rk h ( v ) = rk h ( M ) isdense and open. Additionally it has the property that if v ∈ R is recurrent then its stableand unstable manifolds also belong to R . Next in Section 3, we show that the distributionof parallel fields of curvature − is smooth on the regular set. Then, for bi-recurrent regularvectors, we characterize these parallel fields in Section 4 in terms of unstable Jacobi fields ofLyapunov exponent 1. We use this to show that the slow unstable distribution extends to asmooth distribution on R .In Section 5, we prove the result under the stronger assumption of strict -curvaturepinching as the technicalities are significantly simpler and avoid the use of measurable normalforms from Pesin theory. We are inspired here by arguments of Butler in [But15]. Weconstruct a Kanai like connection for which the slow and fast stable and unstable distributionsare parallel. The construction is much motivated by a similar one by Benoist, Foulon andLabourie in [BFL90]. We use this to prove integrability of the slow unstable distribution.This distribution is also invariant under stable holonomy by an argument of Feres and Katok[FK90], and hence defines a distribution on ∂ f M . As it is integrable and π ( M ) -invariant, weget a π ( M ) -invariant foliation on ∂ f M which is impossible thanks to an argument of Foulon[Fou94] (or the argument for Corollary 4.4 in [Ham91b].)Lastly, in Section 6 we treat the general case of non-strict -curvature pinching. By a resultof Connell [Con03] relying on Theorem 1.2, if M is not already a locally symmetric space, thenthere is no uniform resonance in the Lyapunov spectrum. Now we can use recent work ofMelnick [Mel16] on normal forms to obtain a suitably invariant connection (cf. also Kalinin-Sadovskaya [KS13]). This allows us to prove integrability of the slow unstable distributionon almost every unstable manifold. As before we can obtain a π -invariant foliation on ∂ f M and finish with the result of Foulon as before. This is technically more complicated, however,because we no longer have C holonomy maps. Instead we adapt an argument of Feres andKatok, to show that stable holonomy maps almost everywhere preserve the tangencies ofour slow unstable foliation. To this end, we show that the holonomy maps are differentiablewith bounded derivatives, though not necessarily C , between good unstable manifolds. Thisallows us to obtain the desired holonomy invariance as in the strict -pinching case to finishthe proof of the main theorem.In light of the above, in particular Theorem 1.1 as well as Constantine’s results, we makethe following Conjecture 1.3.
A closed manifold with sectional curvatures ≥ − and higher hyperbolicrank is isometric to a locally symmetric space of real rank 1. Let us point out that the starting point of the proofs for upper and lower curvature boundsare radically different, although they share some common features. In the hyperbolic rankcase in particular, for the upper curvature bound, we get control of the slow unstable foliationin terms of parallel fields. Hamenstädt used the latter to create Carnot metrics on theboundary with large conformal group leading to the models of the various hyperbolic spaces.The lower curvature bound in comparison gives us control of the fast unstable distributionwhich is integrable and does not apparently tell us anything about the slow directions. It s clear that the general case will be much more difficult, even if we assume that the metrichas negative or at least non-positive curvature.Finally let us note a consequence of Theorem 1.1 in terms of dynamics. Consider thegeodesic flow g t on the unit tangent bundle of a closed manifold M . For a geodesic c ⊂ M ,the maximal Lyapunov exponent λ max ( c ) , for c is the biggest exponential growth rate of thenorm of a Jacobi field J ( t ) along c : λ max ( c ) := max J Jacobi for c lim 1 t log k J ( t ) k . Note that λ max ( c ) ≤ if the sectional curvatures of M are bounded below by − , by Rauch’scomparison theorem.Given an ergodic g t -invariant measure µ on the unit tangent bundle SM , λ max ( c ) is con-stant µ -a.e.. In fact, it is just the maximal Lyapunov exponent in the sense of dynamicalsystems for g t and µ (cf. Section 4). Corollary 1.4.
Let M be a closed Riemannian manifold with sectional curvatures K between − ≤ K ≤ − . Let µ be a probability measure of full support on the unit tangent bundle SM which is invariant and ergodic under the geodesic flow g t . Suppose that the maximalLyapunov exponent for g t and µ is . Then M is a rank one locally symmetric space. We supply a proof in Section 6. In fact, the reduction to Theorem 1.1 is identical toConstantine’s in [Con08, Section 6] which in turn adapts an argument of Connell for uppercurvature bounds [Con03].
Acknowledgements:
The third author thanks the Department of Mathematics at IndianaUniversity for their hospitality while part of this work was completed.2.
Definitions, Semicontinuity and Invariance on Stable Manifolds
Let M be compact manifold of negative sectional curvature, and denote its unit tangentbundle by SM . We let g t : SM → SM be the geodesic flow, and denote by pt : SM → M the footpoint map, i.e. v ∈ T pt ( v ) M . For v ∈ SM , let c v be the geodesic determined by v and let v ⊥ denote the perpendicular complement of v in T pt ( v ) M . Recall that rk h ( v ) isthe dimension of the subspace of v ⊥ which are the initial vectors of Jacobi fields that makecurvature − with g t v for all t ≥ , and rk h ( M ) is the minimum of rk h ( v ) for v ∈ SM . Lemma 2.1.
Let v be a unit vector recurrent under the geodesic flow. Suppose that rk h ( v ) > . Then there is also an unstable or stable Jacobi field making curvature -1 with g t v for all t ∈ R .Proof. Since rk h ( v ) > , there is a Jacobi field J ( t ) making curvature -1 with g t v for all t ≥ . First assume that J ( t ) is not stable. Decompose J ( t ) into its stable and unstablecomponents J ( t ) = J s ( t ) + J u ( t ) . Suppose g t n v → v with t n → ∞ . Then, for a suitablesubsequence of t n , J ( t + t n ) k J ( t n ) k will converge to a Jacobi field Y ( t ) along c v ( t ) . Note then that g t + t n ( v ) → g t v as t n → ∞ . Moreover, for any t ∈ R , Y ( t ) is the limit of the vectors J ( t + t n ) which make curvature -1 with g t + t n ( v ) . Hence Y ( t ) also makes curvature -1 with g t v for any t . Also Y ( t ) is clearly unstable since J u ( t ) . f J ( t ) = J s ( t ) is stable, then the same procedure will produce a stable Jacobi field Y ( t ) along c ( t ) that makes curvature − with g t ( v ) for all t ∈ R . (cid:3) Lemma 2.2.
Suppose that rk h ( M ) > . Then along every geodesic c ( t ) , we have an unstableJacobi field that makes curvature − with c ( t ) for all t ∈ R . Similarly, there is a stable Jacobifield along c ( t ) that makes curvature − with c ( t ) for all t ∈ R .Proof. Since the geodesic flow for M preserves the Liouville measure µ , µ -a.e. unit tangentvector v is recurrent. By Lemma 2.1, the geodesics c v ( t ) have stable or unstable Jacobi fieldsalong them that make curvature − with the geodesic for all t ∈ R . As µ has full supportin SM , such geodesics are dense and the same is true for any geodesic by taking limits.Next we show that there are both stable and unstable Jacobi fields along any geodesic thatmake curvature − with the geodesic. Indeed, let A + ⊂ SM be the set of unit tangent vectors v that have an unstable Jacobi field along c v ( t ) that make curvature − with c v ( t ) . Similarly,define A − ⊂ SM as the set of unit tangent vectors v that have a stable Jacobi field along c v ( t ) that make curvature − with c v ( t ) . Note that A − = − A + , and that SM = A + ∪ A − by what we proved above. Hence neither A + nor A − can have measure 0 w.r.t. Liouvillemeasure µ . Also, both A + and A − are invariant under the geodesic flow g t . Since g t isergodic w.r.t. µ , both A + and A − must each have full measure. Now the claim is clear onceagain by taking limits. (cid:3) Denote by Λ( v, t ) w the unstable Jacobi field along g t v with initial value w ∈ v ⊥ . Then welet E ( v ) ⊂ v ⊥ be the subspace of v ⊥ defined as follows: w ∈ v ⊥ belongs to E ( v ) if Λ( v, t ) w makes curvature -1 with g t v for all t ≥ .We define R = { v | rk h ( v ) = rk h M } . We note that for v ∈ R and for all u ∈ SM , dim E ( v ) ≤ dim E ( u ) . Lemma 2.3.
Suppose v ∈ R and w ∈ E ( v ) . Then Λ( v, t ) w makes curvature -1 with c v ( t ) for all t ∈ R and R is invariant under the backward geodesic flow.Proof. First note that for t ∈ R , the unstable Jacobi field Λ( v, t ) : v ⊥ → ( g t v ) ⊥ , definedby w Λ( v, t ) w is an isomorphism. We have by definition that Λ( v, t ) E ( g − t v ) ⊂ E ( v ) for t > . Since v ∈ R , we have dim E ( v ) ≤ dim E ( g − t v ) . Thus Λ( v, t ) E ( g − t v ) = E ( v ) for t > .Therefore, for w ∈ E ( v ) , the Jacobi field Λ( v, t ) w along c v makes curvature -1 with g t v forall t ∈ R . This immediately implies the last statement. (cid:3) Next, define b E ( v ) ⊂ v ⊥ be the subspace of v ⊥ defined as follows: w ∈ v ⊥ belongs to b E ( v ) if the parallel vector field along c v ( t ) determined by w makes curvature -1 with g t v for all t ∈ R . We have that b E ( v ) ⊂ E ( v ) . Indeed if E ( t ) is a parallel vector field along a geodesic c ( t ) that makes curvature -1 with c ( t ) , then e t E ( t ) is an unstable Jacobi field that againmakes curvature -1 with c ( t ) . Definition 2.4.
The strong hyperbolic rank rk sh ( v ) of v is the dimension of b E ( v ) . The strong hyperbolic rank rk sh ( M ) of M is the minimum of the strong hyperbolic ranks rk sh ( v ) over all v ∈ SM . We use an argument of Constantine [Con08, Section 5] to prove: roposition 2.5. If M is a closed manifold with lower sectional curvature bound − , v ∈ R and w ∈ E ( v ) , then the parallel vector field determined by w along c v ( t ) makes curvature − for all t ∈ R . Thus for all v ∈ R , rk h ( v ) = rk sh ( v ) and b E ( v ) = E ( v ) .Proof. By Lemma 2.3, the unstable Jacobi field Λ( v, t ) w makes curvature − with c v ( t ) forall t ∈ R . Then Λ( v, t ) w is a stable Jacobi field along c − v ( t ) still making curvature -1 with c − v ( t ) . Hence the discussion in [Con08, Section 5] shows that Λ( v, t ) w = e t E where E isparallel along c v ( t ) for all t ∈ R . Clearly, E makes sectional curvature -1 with c v ( t ) aswell. (cid:3) Note that E and b E may not be continuous a priori. However, E and b E are semicontinuousin the following sense. Lemma 2.6. If v n , v ∈ SM and v n → v as n → ∞ , then(1) lim n →∞ E ( v n ) ⊂ E ( v ) and rk h ( v ) ≥ lim sup n →∞ rk h ( v n ) (2) lim n →∞ b E ( v n ) ⊂ b E ( v ) and rh sh ( v ) ≥ lim sup n →∞ rk sh ( v n ) .Here lim n →∞ E ( v n ) simply denotes the set of all possibly limit points of vectors in E ( v n ) , andsimilarly for b E .Proof. These claims are clear. (cid:3)
We now define b R = { v | rk sh ( v ) = rk sh M } . Lemma 2.7.
The sets R and b R are both open with full measure and hence dense. Moreover, b R is invariant under the geodesic flow.Proof. By Lemma 2.6, R is open. Since the geodesic flow is ergodic on SM w.r.t. Liouvillemeasure and R is invariant under backward geodesic flow by Lemma 2.3, R has full measure.By Lemma 2.6, b R is open and it is flow invariant by definition. Therefore the same argumentapplies. (cid:3) Corollary 2.8. If M is a closed manifold with lower sectional curvature bound − , then rk h ( M ) = rk sh ( M ) and R ⊂ b R .Proof. By Lemma 2.5, strong and weak rank agree on R which is an open dense set byLemma 2.7. By Lemma 2.6, both weak and strong ranks can only go up outside R . (cid:3) The next argument is well known and occurs in Constantine’s work for example.As usual we let W u ( v ) denote the (strong) unstable manifold of v under the geodesic flow,i.e. the vectors w ∈ SM such that d ( g t ( v ) , g t ( w )) → as t → −∞ . We define the (strong)stable manifold W s ( v ) similarly for t → ∞ . Lemma 2.9. If v ∈ b R is backward recurrent under g t , then W u ( v ) ⊂ b R . If v ∈ b R is forwardrecurrent under g t , then W s ( v ) ⊂ b R .Proof. Let w ∈ W u ( v ) , then g − t w approximates g − t v when t large. On the other hand, since b R is open, there is a neighborhood U of v in b R . Since v is backward recurrent, g − t v comesback to U and approximates v infinitely often. Thus there is t large that g − t w ∈ U ⊂ b R . Itfollows that w ∈ b R as b R is invariant under the geodesic flow (cf. Lemma 2.7). The argumentfor the forward recurrent case and stable leaf is similar. (cid:3) . Smoothness of Hyperbolic Rank
Assume now that M has sectional curvature -1 as an extremal value, that is, either thesectional curvature K ≤ − or K ≥ − . We want to prove smoothness of b E on the regularset b R . Our arguments below are inspired by Ballmann, Brin and Eberlein’s work [Bal85]and also [Wat13]. First let us recall a lemma from [SSS16, Lemma 2.1]: Lemma 3.1.
For v ∈ S p M , the Jacobi operator R v : v ⊥ → v ⊥ is defined by R v ( w ) = R ( v, w ) v . Then w is an eigenvector of R v with eigenvalue -1 if and only if K ( v, w ) = − . While we don’t use it, let us mention [SSS16, Lemma 2.9] where smoothness of theeigenspace distribution of eigenvalue -1 is proved on a similarly defined regular set. Oursituation is different as we characterize hyperbolic rank in terms of parallel transport of avector not just the vector. To this end, we define the following quadratic form: Let E ( t ) and W ( t ) be parallel fields along the geodesic c v ( t ) , and set Ω Tv ( E ( t ) , W ( t )) = Z T − T h− E ( t ) − R g t v E ( t ) , − W ( t ) − R g t v W ( t ) i . Lemma 3.2.
The parallel field E ( t ) belongs to the kernel of Ω Tv if and only if E ( t ) makescurvature -1 with c v ( t ) for t ∈ [ − T, T ] . In consequence, if S < T , then ker Ω Tv ⊂ ker Ω Sv .Proof. If E ( t ) makes curvature -1 with c v ( t ) for t ∈ [ − T, T ] , then − E ( t ) − R g t v E ( t ) = 0 byLemma 3.1, and hence E ( t ) is in the kernel of Ω Tv .Conversely, if E ( t ) is in the kernel of Ω Tv , let W ( t ) = E ( t ) . Since the integrand now is ≥ for all t ∈ [ − T, T ] , E ( t ) − R g t v E ( t ) = 0 and hence E ( t ) makes curvature -1 with c v ( t ) , asclaimed. (cid:3) Hence b E ( v ) consists of the initial vectors of ∩ T ker Ω Tv which is the intersection of thedescending set of vector subspaces ker Ω Tv as T increases. Hence there is a smallest number T ( v ) < ∞ such that b E ( v ) consist of the initial vectors of ker Ω Tv for all T > T ( v ) . Proposition 3.3. b E is smooth on b R . In particular, b E is smooth on W s ( v ) (resp. W u ( v ) )where v ∈ b R is forward (resp. backward) recurrent.Proof. Let v ∈ b R , and let v n → v . We may assume that v n ∈ b R since b R is open. Note that T ( v n ) < T ( v )+1 for all large enough n . Otherwise, we could find rk h M +1 many orthonormalparallel fields along c v n which make curvature -1 with c v n ( t ) for − T ( v ) − < t < T ( v ) + 1 .Taking limits, we find rk h M + 1 many orthonormal parallel fields along c v which makecurvature -1 with c v n ( t ) for − T ( v ) − < t < T ( v ) + 1 . Therefore there exists a neighborhood U ⊂ b R of v such that T ( u ) < T ( v ) + 1 for all u ∈ U . Since the quadratic forms Ω T ( v )+1 w aresmooth on the neighborhood U of v , we see that the distribution is smooth on b R .The last claim is immediate from smoothness on b R and Lemma 2.9. (cid:3) Maximal Lyapunov exponents and hyperbolic rank
The geodesic flow g t : SM → SM preserves the Liouville measure µ on SM , and isergodic. Hence Lyapunov exponents are defined and constant almost everywhere w.r.t. µ . e recall that they measure the exponential growth rate of tangent vectors to SM underthe derivative of g t . As is well-known, double tangent vectors to M correspond in a 1-1 waywith Jacobi fields J ( t ) , essentially since J ( t ) is uniquely determined by the initial condition J (0) , J ′ (0) . Moreover we have ( Dg t )( J (0) , J ′ (0)) = ( J ( t ) , J ′ ( t )) . Thus we can work with Jacobi fields rather than double tangent vectors whenever convenient.We note that stable (resp. unstable) vectors for g t correspond to Jacobi fields which tend to0 as t → ∞ (resp. as t → −∞ ).If − ≤ K ≤ , then all Lyapunov exponents of unstable Jacobi field along the geodesicflow for any invariant measure are between 0 and 1, cf. e.g. [Bal95, ch. IV,Prop. 2.9].Similarly, if K ≤ − , all Lyapunov exponents have absolute value at least 1. We want tounderstand the extremal case better. We suppose K ≥ − throughout. Lemma 4.1.
Let b E ( v ) ⊥ be the orthocomplement (with respect to the Riemannian metric on M ) of b E ( v ) . Then Λ( v, t ) sends b E ( v ) ⊥ to b E ( g t v ) ⊥ .Proof. Indeed, let E ( t ) , . . . , E n − ( t ) be a choice of parallel orthonormal fields along g t v andperpendicular to g t v such that { E ( t ) , . . . , E k ( t ) } forms a basis of b E ( g t v ) . For any w ∈ v ⊥ ,the formula for an unstable Jacobi field becomes Λ( v, t ) w = X i f i ( t ) E i ( t ) . Setting a ij = h R ( g t v, E i ( t )) g t v, E j ( t ) i , the Jacobi equation is equivalent to f ′′ j ( t ) + X i a ij ( t ) f i ( t ) = 0 . Since e t E i ( t ) is an unstable Jacobi field for i ≤ k and the { E i ( t ) } are orthonormal, h R ( g t v, E i ( t )) g t v, E j ( t ) i = − h E i ( t ) , E j ( t ) i = − δ ji , for all i ≤ k and any j ≤ n − . By the symmetries of the curvature tensor, a ij = a ji and sowe also have a ji ( t ) = a ij ( t ) = − δ ji for either i ≤ k or j ≤ k . It follows that for all t ∈ R andall i ≤ k f ′′ i ( t ) + X j a ij ( t ) f j ( t ) = f ′′ i ( t ) − f i ( t ) . Since Λ( v, t ) w is unstable, lim t →−∞ f i ( t ) = 0 for all i . If w ∈ b E ( v ) ⊥ , then f i (0) = 0 for all i ≤ k .These two conditions together imply f i ( t ) = 0 for all t ∈ R and i ≤ k . Hence, Λ( v, t ) leaves b E ⊥ invariant. (cid:3) Lemma 4.2. [Bal95, ch. IV,Prop. 2.9] [Con03, Lemma 2.3] k Λ( v, t ) w k ≤ k w k e t for all t ≥ . The equality holds at a time T ∈ R if and only if the sectional curvature of the planespanned by Λ( v, − t ) w and g t v is -1 for all ≤ t ≤ T if and only if Λ( v, t ) w = k w k e t W ( t ) where W ( t ) is parallel for all ≤ t ≤ T . roof. By the Rauch Comparison Theorem, k Λ( v, t ) w k ≤ k w k e t and k Λ ′ ( v, t ) w k ≤ k Λ( v, t ) w k for all t ≥ (cf. [Bal95, ch. IV, Prop. 2.9] which states a similar result for stable Jacobifields). If equality holds at time T > then k Λ( v, t ) w k = e t k w k for all ≤ t ≤ T . Indeed,should k Λ( v, t ) w k < e t k w k for some < t < T , then we get a contradiction since k Λ( v, T ) w k ≤ k Λ(Λ( v, t ) w, T − t ) k ≤ e T − t k Λ( v, t ) w k < e T − t e t k w k = e T k w k . Therefore the vector field W ( t ) for which Λ( v, t ) w = k w k e t W ( t ) is a field of norm .Hence h W ( t ) , W ′ ( t ) i = 0 and we have k w k e t (1 + k W ′ ( t ) k ) / = k w k (cid:13)(cid:13) ( e t W ( t ) + e t W ′ ( t )) (cid:13)(cid:13) = k Λ ′ ( v, t ) w k ≤ k Λ( v, t ) w k = e t k w k , by the estimate above on the derivative of the unstable Jacobi field. We see that W ′ = 0 ,i.e. W is parallel as desired. That the sectional curvature between W ( t ) and the geodesic is-1 now follows from the Jacobi equation. (cid:3) By covering the unit tangent bundle with countable base of open sets that generate thetopology, and applying the ergodic theorem to the Liouville measure, there is a full measureset of unit tangent vectors that comes back to all its neighborhoods with positive frequency.The argument in the next lemma is similar to that of Lemma 3.4 of [BBE85] and Propo-sition 1 of [Ham91a], but for the setting of a lower curvature bound of − . Lemma 4.3.
Suppose v ∈ R returns with positive frequency to all its neighborhoods under g t . Then for w ∈ E ( v ) ⊥ = b E ( v ) ⊥ , the unstable Jacobi field Λ( v, t ) w has Lyapunov exponentstrictly smaller than 1. We have a similar statement for stable Jacobi fields of Lyapunovexponent -1.Proof. Let
T > be such that the dimension of parallel vector fields making curvature -1with g t v for all ≤ t ≤ T is k = rk h ( M ) , i.e. k = dim E ( v ) since v ∈ R (2.8).Pick w ∈ E ( v ) ⊥ that minimizes { k w kk Λ( v,T ) w k : w ∈ E ( v ) ⊥ } . By Lemma 4.2, we havethat k Λ( v, T ) w k ≤ e T k w k . Suppose that we have the equality k Λ( v, T ) w k = e T k w k .Then by Lemma 4.2, the parallel field of w along g t v makes curvature -1 with g t v for all ≤ t ≤ T . Since w ∈ E ( v ) ⊥ , the space of parallel fields making curvature -1 with g t v for all ≤ t ≤ T has dimension at least dim E ( v ) + 1 = k + 1 , a contradiction. Therefore k Λ( v, T ) w k < e T k w k .Let ǫ > be such that k Λ( v, T ) w k = (1 − ǫ ) e T k w k . By continuity, we can choose aneighborhood U ⊂ R of v such that for all u ∈ U and w ∈ E ( u ) ⊥ , we have the estimate k Λ( v, T ) w k ≤ (1 − ǫ ) e T k w k .Since g t v visit U with a positive frequency, there are δ > and T > such that for all S > T |{ t ∈ [0 , S ] : g t v ∈ U }| > δS. Now suppose that w ∈ v ⊥ ∩ E ( v ) ⊥ . We note that by Lemma 4.1, since E ( v ) ⊥ = b E ( v ) ⊥ on R , Λ( v, S ) w ∈ E ( g S v ) ⊥ for all S > . Then k Λ( v, S ) w k ≤ e S (1 − ǫ ) [ δST ] k w k . It follows that Λ( v, t ) w has Lyapunov exponent strictly smaller than 1. (cid:3) We remark that the argument in the last proof only provides information that the unstableJacobi fields come from parallel fields in forward time. This forced us to introduce both sets R and b R and use the equality of E and b E on R . ecall that there is a contact form θ on SM invariant under the geodesic flow. Its exteriorderivative ω = dθ is a symplectic form on stable plus unstable distribution E s + E u . Also θ and hence ω are invariant under the geodesic flow, and thus every Oseledets space E λ withLyapunov exponent λ is ω -orthogonal to all E λ ′ unless λ ′ = − λ . Since ω is non-degenerate, ω restricted to E λ × E − λ is also non-degenerate for each λ . Note that b E gives rise to unstableJacobi fields with Lyapunov exponent 1.This immediately gives the following Corollary 4.4.
The maximal Lyapunov spaces E can be extended to be a C distribution E u on the regular set b R . The orthogonal complement ( E u ) ⊥ ∩ E s w.r.t. ω is defined and C on the same set b R and equals ⊕ − <λ< E λ almost everywhere. The analogous statementshold for E − (yielding E s ) and E ⊥− ∩ E u = ⊕ <λ< E λ a.e. as well. We will call the spaces E s< := ( E u ) ⊥ ∩ E s and E u< := ( E s ) ⊥ ∩ E u the extended slow stableand unstable subspaces. Similarly we call E s and E u the extended fast stable and unstable subspaces. Proof. On b R , b E is defined and smooth. On R , E is also defined and smooth. Moreover thedistribution E agrees with b E on R .By Lemma 4.3, on the set Ω = { v ∈ R , v is forward recurrent under g t v } , E agrees withthe lift to unstable Jacobi fields of E on T SM , i.e. w ∈ E ( v ) is identified with the Jacobi field Λ( v, t ) w . The set Ω has full measure. Hence E extends smoothly on b R to a distribution E u .Now take the orthogonal complement (w.r.t. the form ω ) to E u , ( E u ) ⊥ ∩ E s , on b R in thestable distribution. Since E s is C , and E u is even C ∞ , ( E u ) ⊥ ∩ E s is C .Since ω pairs Lyapunov spaces where defined a.e. on b R , ⊕ − <λ< E λ ⊂ ( E u ) ⊥ ∩ E s . Since ω is nondegenerate, the dimension of the latter subspace is exactly n − − rk h ( M ) everywhereon b R , and hence they agree.A similar argument applies to E − and its perpendicular complement w.r.t. ω in theunstable subspace where now we use forward recurrent vectors. (cid:3) Slow Stable Spaces and Integrability
In the tangent bundle
T SM of the unit tangent bundle, consider the subset T b R ⊂
T SM ,which is the union of tangent fibers of SM at points in b R . On T b R , there is a C decompo-sition E s + E s< + E + E u + E u< , where E s/u , E s/u< denote the extended stable/unstable fastand slow Lyapunov exponent distributions respectively defined in the last section.We will define a special connection for which this decomposition is parallel, and use thatto argue integrability of the slow unstable direction. Such connections were introduced byKanai to study geodesic flows with smooth stable and unstable foliations in [Kan88]. Ourparticular construction is motivated by that of Benoist, Foulon and Labourie in [BFL90]where they classify contact Anosov flows with smooth Oseledets’ decomposition. We referto [GHL04, Definition 2.49 and Proposition 2.58] for the basic facts on affine connections wewill need. e recall the formula for the contact 1-form θ : θ ( x,v ) ( ξ ) = < v, ξ > , where ( x, v ) ∈ SM and ξ ∈ T ( x,v ) SM , where ξ = d pt ( ξ ) . Then the 2-form dθ becomes dθ ( ξ, η ) = < ξ u , η s > + < ξ u< , η s< > − < ξ s , η u > − < ξ s< , η u< >, where the indices indicate appropriate components when we decompose ξ or η w.r.t. thedecomposition E s + E s< + E + E u + E u< .We let X denote the geodesic spray, i.e., the generator of the geodesic flow which isthe vector field belonging to E obtained by lifting unit tangent vectors of M to T SM horizontally.
Proposition 5.1.
There exists a unique connection ∇ on T b R such that(1) ∇ θ = 0 , ∇ dθ = 0 , and ∇ E ⊂ E , ∇ E s/ui ⊂ E s/ui for i ∈ { , < } .(2) For any sections Z s , Z s< , Z u , Z u< of E s , E s< , E u , E u< respectively, we have for i, j ∈{ , < } ∇ Z si Z uj = p E uj ([ Z si , Z uj ]) , ∇ Z ui Z sj = p E sj ([ Z ui , Z sj ]) , ∇ X Z s/ui = [ X , Z s/ui ] , where the p E s/uj are the projections to the E s/uj subspaces.In addition, ∇ is invariant under the geodesic flow g t .Proof. We note that ∇ dθ = 0 is equivalent with W dθ ( Y, Z ) = dθ ( ∇ W Y, Z ) + dθ ( Y, ∇ W Z ) forany vector fields W, Y, Z . And ∇ θ = 0 is equivalent with θ ( ∇ Y Z ) = Y θ ( Z ) for any vectorfields Y, Z . Thus θ ( ∇ Y X ) = 0 and dθ ( ∇ Y X , Z ) = 0 for any vector fields Y, Z . It followsthat ∇X = 0 . Furthermore, given a C function f : b R → R , we set ∇ Y ( f X ) = Y ( f ) X .Moreover, ∇ Z ui Z uj is uniquely determined by the condition ∇ E ui ⊂ E ui and the equality Z ui dθ ( Z uj , Z s ) = dθ ( ∇ Z ui Z uj , Z s ) + dθ ( Z uj , ∇ Z ui Z s ) , for arbitrary section Z s of E s and i, j ∈ { , < } . Similarly ∇ Z si Z sj is uniquely determined.By linearity, we have defined ∇ Y Z for all vector fields Y, Z .It is now easy to check that ∇ satisfies the properties of a connection on b R (cf. e.g.[GHL04, Definition 2.49]). That ∇ is invariant under the geodesic flow g t follows from theconstruction. Indeed, the slow and fast stable and unstable spaces are invariant under g t , ( g t ) ∗ ([ Y, Z ]) = [( g t ) ∗ Y, ( g t ) ∗ Z ] and X is invariant under g t by definition. (cid:3) The next lemma is basically well-known (cf. e.g. [BFL90, Lemma 2.5]). Since our connec-tion is only defined on a dense open set and not necessarily bounded we outline the proof.Since Liouville measure is ergodic for the geodesic flow g t on SM , the Lyapunov exponents γ i are defined and constant on a g t -invariant full measure set Σ in b R . We can assume inaddition that all v ∈ Σ are forward and backward recurrent for g t , and that the Oseledetsdecomposition T v R = ⊕ E γ i into Lyapunov subspaces E γ i is defined on Σ . Thus if Z i ∈ E γ i ,the forward and backward Lyapunov exponents are defined and equal to γ i . Lemma 5.2.
Let v ∈ Σ . If K is a geodesic flow invariant tensor and Z , . . . , Z k are vectors in T v b R with Z i ∈ E γ i , then K ( Z , . . . , Z k ) is either zero or has Lyapunov exponent γ + · · · + γ k . roof. There is a neighborhood U of v and C > such that k K ( Y , . . . , Y k ) k ≤ C k Y k · · · k Y k k ,for any vectors Y , . . . , Y k with footpoints in the neighborhood U . Suppose that K ( Z , . . . , Z k ) =0 . If g t ( v ) ∈ U for some t > then k D v g t K ( Z , . . . , Z k ) k = k K ( D v g t Z , . . . , D v g t Z k ) k ≤ C k D v g t Z k · · · k D v g t Z k k . Thus, t log( k D v g t K ( Z , . . . , Z k ) k ) ≤ t (log( C ) + log ( k D v g t Z k ) + · · · + log( k D v g t Z k k )) . Since v is forward recurrent, therre will be a sequence of times t → ∞ with g t ( v ) ∈ U .Thus the forward Lyapunov exponent of K ( Z , . . . , Z k ) is at most γ + · · · + γ k . Hence K ( Z , . . . , Z k ) cannot have nonzero components in E γ if γ > γ + · · · + γ k .Similarly, if g s ( v ) ∈ U for some s < then s log( k D v g s K ( Z , . . . , Z k ) k ) ≥ s (log( C ) + log ( k D v g s Z k ) + · · · + log( k D v g s Z k k )) . Since v is backward recurrent, arguing as above, the backward Lyapunov exponent of K ( Z , . . . , Z k ) is at least γ + · · · + γ k . Hence K ( Z , . . . , Z k ) cannot have nonzero com-ponents in E γ if γ < γ + · · · + γ k . (cid:3) Recall that the connection ∇ is only C , and only defined on b R . This means that thetorsion tensor is only a C -tensor, and the curvature tensor is not defined. However, slowand fast stable and unstable distributions are smooth on stable and unstable manifolds in b R .Hence the restriction of ∇ to stable or unstable manifolds is also smooth by the constructionof ∇ . In particular, the curvature tensor of ∇ restricted to stable or unstable manifolds iswell defined. Corollary 5.3.
The torsion and curvature tensors of ∇ restricted to the slow Lyapunovdistributions E s/u< and also each stable/unstable space E s/u are zero.Proof. Since ∇ is geodesic flow invariant, so are the torsion and curvature tensors. In strict -pinched manifolds, the ratio of any two Lyapunov exponents lies in ( , . Thus thiscorollary follows at points of Σ immediately from the previous lemma and the strict -pinching condition. Since Σ is of full measure, and therefore dense, the statements holdeverywhere on b R by continuity. (cid:3) Corollary 5.4.
The slow unstable Lyapunov distribution E u< is integrable.Proof. The slow unstable Lyapunov distribution E u< is invariant under the parallel transportby ∇ , by construction of ∇ . Since ∇ is flat, parallel transport is independent of path.Thus we can choose canonical local parallel C vector fields tangent to and spanning thedistribution. On the other hand, since the restriction of torsion on unstable leaves is zerowe have that the commutators of these vector fields are zero. By the Frobenius Theorem for C vector fields, [Lan95, Theorem 1.1 Chapter 6], the distribution is integrable. (cid:3) As usual we will consider the π ( M ) -lifts of the stable and unstable manifolds by the samenotation in S f M , and we will work in SM or S f M as appropriate without further comment. iven v ∈ S f M , the map π v : W u ( v ) → ∂ f M − { c v ( −∞ ) } , defined by π v ( w ) = c w ( ∞ ) , is a C diffeomorphism. For w ∈ W s ( v ) , the stable holonomy is defined as h v,w : W u ( v ) − { π − v ( c w ( −∞ ) } → W u ( w ) − { π − w ( c v ( −∞ )) } . Note that h v,w ( x ) is simply the intersection of the weak stable manifold of x with W u ( w ) .In particular the stable holonomy maps are C . Indeed, the sectional curvatures of M arestrictly -pinched and hence the weak stable foliation is C [HP75]. Moreover, the stableholonomy maps h a,b are C with derivative bounded uniformly in d S f M ( a, b ) for b ∈ ∪ t g t W s ( a ) .This follows from the fact that the unstable foliation is uniformly transversal to the stablefoliation, by compactness of SM . In fact, Hasselblatt [Has94, Corollary 1.7] showed that thederivative is even Hölder continuous.We call a distribution stable holonomy invariant if it is invariant under (the derivativemap of) all holonomies h v,w for all v ∈ S f M and w ∈ W s ( v ) . We will now adapt an argumentby Feres and Katok [FK90, Lemma 4]. Lemma 5.5.
The slow unstable spaces E u< ⊂ T b R are stable holonomy invariant.Proof. First consider v ∈ Σ ∩ R and w ∈ ∪ t g t W s ( v ) ∩ Σ ∩ R . The distance between g t v and g t w remains bounded in forward time. Hence the derivatives of the holonomy maps h g t v,g t w are uniformly bounded for all t ≥ . If u ∈ E u< ( v ) , then u has forward Lyapunov exponent λ < since v ∈ Σ . It follows that the image vector Dh v,w ( u ) also has forward Lyapunovexponent λ < and hence belongs to E u< ( w ) . In particular, Dh v,w E u< ( v ) ⊂ E u< ( w ) . Bycontinuity of Dh v,w and of the extended slow space on b R the same holds for all v, w ∈ b R . (cid:3) We follow ideas of Butler [But15] to derive:
Corollary 5.6.
The slow unstable distributions are trivial.Proof.
By the strict -pinching, the boundary ∂ f M of the universal cover admits a C struc-ture for which the projection maps from points or horospheres are C ([HP75]). By Lemma5.5, the projection of the lifts of the slow unstable distribution is independent of the projec-tion point on the horosphere. Using different horospheres we obtain a well-defined distribu-tion on all of ∂ f M . Note that this distribution is also invariant under π ( M ) .By Corollary 5.4, this distribution is integrable and yields a C foliation F on the boundary ∂ f M which is also π ( M ) -invariant. Since there is a hyperbolic element of π ( M ) which actswith North-South dynamics on ∂ f M , by Foulon [Fou94, Corollaire ], the foliation generatedby this distribution has to be trivial. (cid:3)
We are now ready to finish the proof of our main result.
Proof of Theorem 1.1. (strict -pinching case): Since the slow unstable distribution is trivial,all unstable Jacobi fields belong to E u . Hence all sectional curvatures are − on b R . Since b R is open dense in SM , it follows that all sectional curvatures are − . (cid:3) . Non-Strictly -Pinched Case In this section we extend the proof of the main theorem to the non-strictly -pinchedcurvature case.First, consider the set O ⊂ SM of vectors whose smallest positive Lyapunov exponentis . If O has positive Liouville measure then Theorem 1.3 of [Con03] implies M is locallysymmetric and our theorem holds. Hence we may assume O has measure and there is aflow invariant full measure set P and a ν > such that for all v ∈ P the unstable Lyapunovexponents satisfy + ν < χ + i ( v ) ≤ .Note that, unlike in the strict quarter-pinched case, we cannot immediately use the van-ishing of the torsion of the generalized Kanai connection established in Proposition 5.1.Indeed, the construction of the generalized Kanai connection used that both stable andunstable distributions are C on SM which we do not a priori know in our case.Instead, we replace the generalized Kanai connection with a similar one assembled fromthe flow invariant system of measurable affine connections on unstable manifolds constructedby Melnick in [Mel16]. The connections are defined on whole unstable manifolds but theyare only defined for unstable manifolds W u ( v ) for v in a set of full measure. Moreover,the transversal dependence is only measurable. Mark that we have switched from Melnick’susage of stable manifolds to unstable manifolds.Following the notation in [Mel16], let her E be the smooth tautological bundle over SM whose fiber at v is W u ( v ) . We consider the cocycle F tv which is g t restricted to W u ( v ) . Theratio of maximal to minimal positive Lyapunov exponents lies in [1 , , and hence the integer r appearing in Theorem 3.12 of [Mel16] is . This theorem then reads in our notation as: Lemma 6.1.
There is a full measure flow-invariant set
U ⊂ SM where there is a smoothflow-invariant flat connection ∇ on T W u ( v ) for v ∈ U . Now we build a connection on vector fields tangent to the slow unstable distribution E u< on W u ( v ) for v ∈ U . We emphasize that we do not assume integrability of the slowunstable distribution. We just construct a connection on sections of the vector bundle givenby the slow unstable distribution. More specifically on slow unstable distribution we havethe following. Lemma 6.2.
On each unstable leaf W u ( v ) for v in a full measure flow invariant subset Q ⊂ U ∩ P ∩ b R , there exists a torsion free and flow invariant connection ∇ < : T W u ( v ) × C ( W u ( v ) , E u< ) → C ( W u ( v ) , E u< ) on E u< . Moreover the restriction of the connection to E u< X is torsion free.Proof. Recall that the distribution E u< is smooth on W u ( v ) for v ∈ b R .Given X ∈ T W u ( v ) and Y ∈ C ( W u ( v ) , E u< ) we define the covariant derivative ∇ < X Y tobe the section in C ( W u ( v ) , E u< ) given by projection of the Melnick connection, ∇ < X Y := proj E u< ∇ X Y. ote that this operator is R -bilinear in X and Y since projections are linear, and for f ∈ C ( W u ( v )) since scalar functions commute with projection we have ∇ < fX Y = proj E u< f ∇ X Y = f proj E u< ∇ X Y = f ∇ < X Y ∇ < X f Y = proj E u< X ( f ) Y + f ∇ X Y = X ( f ) Y + f ∇ < X Y. Here we have used that proj E u< Y = Y . Hence ∇ < is C ( W u ( v )) -linear in X , and satisfiesthe derivation property of connections.For v ∈ P ∩ U , X, Y ∈ C ( W u ( v ) , E u< ) the torsion tensor T ( X, Y ) = ∇ < X Y − ∇ < Y X − [ X, Y ] is indeed a tensor due to the derivation property of the connection and bracket wherewe take the bracket of vector fields in W u ( v ) .Next we show that ∇ < is torsion free. Since [ X, Y ] and ∇ < are invariant under Dg t , sois T ( X, Y ) . Also, since v ∈ P the sum of any two Lyapunov exponents lies in (1 , . ByFubini, and absolute continuity of the W u foliation, we may choose Q ⊂ U ∩ P ∩ b R to bean invariant full measure set where for each v ∈ Q a.e. w ∈ W u ( v ) is forward and backwardrecurrent. Now we can apply Lemma 5.2 to almost every w ∈ W u ( v ) to obtain that ∇ < istorsion free on a dense subset and hence on all of W u ( v ) . (cid:3) Corollary 6.3.
The slow unstable Lyapunov distribution E u< is integrable on every leaf W u ( v ) for v ∈ Q .Proof. For v ∈ Q , and X, Y ∈ C ( W u ( v ) , E u< ) the vanishing of the torsion tensor implies T ( X, Y ) = 0 = ∇ < X Y − ∇ < Y X − [ X, Y ] . However, by definition ∇ < X Y and ∇ < Y X belong to E u< , and therefore so does [ X, Y ] . In particular, E u< is integrable. (cid:3) The above corollary gives us well defined slow unstable foliations on almost every W u ( v ) .Next we will show that these foliations are invariant under stable holonomy. This is sub-stantially more difficult in the non-strict -pinched case since the unstable holonomy mapsa priori are not known to be C .To simplify notation, we use Dg t,v for the derivative of g t at v restricted to E u ( v ) . Since M is strictly + δ -pinched for any δ > , Corollary 1.7 of [Has94] implies the following. Lemma 6.4.
The foliations W s and W u are α -Hölder for all α < . Now choose an α > − ν . As in Kalinin-Sadovskaya [KS13, Section 2.2] we have locallinear identifications I vw : E u ( v ) → E u ( w ) which vary in an α -Hölder way on a neighborhoodof the diagonal in SM × SM . We also have that Dg t is an α -Hölder cocycle, since it therestriction of the smooth Dg t to an α -Hölder bundle. In other words, with respect to theseidentifications, we have (cid:13)(cid:13) Dg t,v − I − g t v,g t w ◦ Dg t,w ◦ I v,w (cid:13)(cid:13) ≤ C ( T ) d ( v, w ) α (6.1)for any T > and all t ≤ T .We aim to show convergence of Dg − t,w ◦ I g t v,g t w ◦ Dg t,v for w ∈ W s ( v ) locally and both arein some good set.For a.e. v ∈ SM , we let T ( v ) = inf { s > t log k Dg t,v ζ k > + ν for all t >s and for all ζ ∈ E u ( v ) } . he following lemma can be found in Kalinin-Sadovskaya [KS13] under a hypothesis ofuniform bunching. We have a similar statement under a nonuniform hypothesis. Thisprovides the morale for Lemma 6.6, the result we will actually use. Lemma 6.5.
Let v ∈ SM with all Lyapunov exponents satisfying + ν < χ + i ( v ) ≤ , andlet w ∈ W s ( v ) . Then the limit H v,w = lim t → + ∞ Dg − t,w ◦ I g t v,g t w ◦ Dg t,v converges.Proof. Denote H tvw = Dg − t,w ◦ I g t v,g t w ◦ Dg t,v . We show the convergence of ( H tvw ) − = H twv instead.Consider t > T ( v ) such that the identifications I g t v,g t w are defined and have Hölderdependence. Then for all such t sufficiently large and t < T , we claim k H t + twv − H t wv k isexponentially small in terms of t . Indeed, H t + twv − H t wv = Dg − t ,v ◦ ( Dg − t,g t v ◦ I g t t w,g t t v ◦ Dg t,g t w − I g t w,g t v ) ◦ Dg t ,w . Thus, k H t + twv − H t wv k ≤ k Dg − t ,v k . k Dg t ,w k . k Dg − t,g t v ◦ I g t t w,g t t v ◦ Dg t,g t w − I g t w,g t v k . We have k Dg − t ,v k . k Dg t ,w k < e ( − − ν ) t e t = e ( − ν ) t . On the other hand, by (6.1) appliedto v = g t w and w = g t v , k Dg − t,g t v ◦ I g t t w,g t t v ◦ Dg t,g t w − I g t w,g t v k≤ k Dg − t,g t v ◦ I g t t w,g t t v kk Dg t,g t w − I − g t t w,g t t v ◦ Dg t,g t v ◦ I g t w,g t v k≤ C ( T ) C ( T ) d ( g t w, g t v ) α ≤ C ( T ) d ( w, v ) α e − αt . The last inequality holds because of the curvature condition and w ∈ W s ( v ) . Here wehave absorbed C ( T ) into the generic constant C ( T ) . Combining inequalities, and since wechoose α > − ν , we get the estimate: k H t + twv − H t wv k ≤ C ( T ) d ( w, v ) α e ( − ν ) t e − αt ≤ C ( T ) d ( w, v ) α e ( (1 − α ) − ν ) t ≤ C ( T ) d ( w, v ) α e − ν t . This shows the convergence of H twv . (cid:3) Let v, w ∈ b R be backward recurrent under g t . It follows that W u ( v ) ⊂ b R and W u ( w ) ⊂ b R .Recall that for any η in the weak unstable manifold of w , every vector in E u ( η ) has paralleltranslate making curvature -1 with g t η for all time t ∈ R . Denote by proj E u ( w ) the orthogonalprojection from E u ( w ) onto E u ( w ) . We define H t, vw = Dg − t,w ◦ proj E u ( w ) ◦ I g t v,g t w ◦ Dg t,v (thesuperscript “1” indicates the fast subspace E u as before). Lemma 6.6.
We have the following(1) H t, v,w converges to a limit, denoted H v,w for every v, w with w ∈ W s ( v ) .(2) If ξ ∈ E u ( v ) with forward Lyapunov exponent χ ( v, ξ ) < then H vw ( ξ ) = 0 .(3) The operator norm k H vw k is locally bounded as v and w , in the same weak stable leaf,vary in a sufficiently small neighborhood in b R × b R . roof. For (1), we follow the mode of proof of Lemma 6.5. Let ξ ∈ E u ( v ) . For t large and t < T consider ( H t + t, vw − H t , vw )( ξ ) = Dg − t ,w ◦ ( Dg − t,g t w ◦ proj E u ( g t t w ) ◦ I g t t v,g t t w ◦ Dg t,g t v − proj E u ( g t w ) ◦ I g t v,g t w ) ◦ Dg t ,v ( ξ ) . Since ( Dg − t,g t w ◦ proj E u ( g t t w ) ◦ I g t t v,g t t w ◦ Dg t,g t v − proj E u ( g t w ) ◦ I g t v,g t w ) ◦ Dg t ,v ( ξ ) ∈ E u ( g t v ) , we have that k ( H t + t, vw − H t , vw )( ξ ) k = e − t k ( Dg − t,g t w ◦ proj E u ( g t t w ) ◦ I g t t v,g t t w ◦ Dg t,g t v − proj E u ( g t w ) ◦ I g t v,g t w ) ◦ Dg t ,v ( ξ ) k≤ e − t k ( Dg − t,g t w ◦ proj E u ( g t t w ) ◦ I g t t v,g t t w ◦ Dg t,g t v − proj E u ( g t w ) ◦ I g t v,g t w ) k . k Dg t ,v ( ξ ) k≤ k ξ k . k ( Dg − t,g t w ◦ proj E u ( g t t w ) ◦ I g t t v,g t t w ◦ Dg t,g t v − proj E u ( g t w ) ◦ I g t v,g t w ) k . Since projection and I v,w are as regular as the underlying vector bundle, by Lemma 6.4 wehave a Hölder estimate k ( Dg − t,g t w ◦ proj E u ( g t t w ) ◦ I g t t v,g t t w ◦ Dg t,g t v − proj E u ( g t w ) ◦ I g t v,g t w ) k≤ C ( T ) d ( g t w, g t v ) α ≤ C ( T ) d ( w, v ) α e − αt . It follows that we get the convergence.Next for (2), assuming χ ( v, ξ ) = 1 − δ for some δ > we have k H t, v,w ( ξ ) k = e − t . k proj E u ( g t w ) ◦ I g t v,g t w ◦ Dg t,v ( ξ ) k≤ e − t e (1 − δ ) t . k proj E u ( g t w ) ◦ I g t v,g t w k . k ξ k ≤ Ce − δt k ξ k , where C is a constant that bounds norms of identifications between close enough points. Itfollows that H v,w ( ξ ) = 0 .Finally for (3), let η ∈ W u ( v ) and η ∈ W u ( w ) be vectors in a small neighborhood of v and w such that η is in the weak stable manifold of η . For some sufficiently small neighborhoodof v and w , there is small time t η ∈ ( − δ, δ ) for some δ > such that g t η η ∈ W s ( η ) . If ξ ∈ E u ( η ) has forward Lyapunov exponent 1 then k H t, η ,η ( ξ ) k = k Dg − t + t η ,η ◦ proj E u ( η ) ◦ I g t η ,g t + tη η ◦ Dg t,η ( ξ ) k≤ e δ k proj E u ( η ) ◦ I g t η ,g t + tη η k k ξ k ≤ Ce δ k ξ k , for t large enough. The claim then follows. (cid:3) We define a map I vw := exp w | E u ( w ) ◦ I vw ◦ (exp v | E u ( v ) ) − from a neighborhood V of v in W u ( v ) into a neighborhood W of w in W u ( w ) , where exp w | E u ( v ) and exp w | E u ( w ) denoteexponential maps into neighborhoods of v in W u ( v ) and w in W u ( w ) .Let h tvw = g − t ◦ I vw ◦ g t : V → W . The maps h tvw converges to h vw locally at v as t → + ∞ .The map h tvw is smooth and Dh tvw = Dg − t,w ◦ I g t v,g t w ◦ Dg t,v . emma 6.7. Let v, w ∈ Q be backward recurrent under g t . Assume further that v and w arechosen that almost every vector of W u ( v ) and W u ( w ) are in R and forward recurrent under g t . Then the holonomy h v,w maps slow unstable leaves in W u ( v ) to slow unstable leaves in W u ( w ) .Proof. The image of the C slow unstable foliation in W u ( v ) under h v,w is a C foliation in W u ( w ) . We will first show that h v,w maps slow unstable leaves to locally.Choose foliation charts for the slow unstable foliations around v and w . Let H be theconnected component containing v of the intersection with the slow unstable leaf through v with the chart, i.e. the plaque of v . Similarly, let V be the plaque of the fast unstableleaf containing w . Assume first that almost every vector in H is forward recurrent. Aftershrinking H if necessary, let f : H → V be defined by choosing f ( η ) ∈ V to be the intersectionof the slow unstable leaf containing h v,w ( η ) and V . We show that f is differentiable on H ,and its derivative is exactly H v,w . Indeed, since projection to V along the slow unstable leaf,denoted by p V , commutes with g t we have that p V ◦ h tvw | H = p V ◦ g − t ◦ I vw ◦ g t | H = g − t ◦ p V ◦ I vw ◦ g t | H converges uniformly to f in a neighborhood of v . And since D ( p V ◦ h tvw ) = H t, vw converges,we have that Df = H vw .If η ∈ H is forward recurrent then for any slow Lyapunov vector ξ ∈ E u< ( η ) we havethat χ ( η, ξ ) < . By Lemma 6.6, Df ( η ) = 0 . Since almost every vector in H is recurrent, Df = 0 almost everywhere. Moreover, Df is locally bounded, also by Lemma 6.6. Thus Df equals the zero map in the sense of distributions, and similarly the same holds for all ofits higher derivatives. Hence, by the Sobolev embedding theorem, f is smooth and Df = 0 everywhere. It follows that f is constant, i.e. the leaf H locally maps to one leaf.By connectedness of the leaves, h v,w preserves the entire leaf. Now consider the collectionof slow unstable leaves with almost every vector being recurrent. By the same argument,such leaves map to slow unstable leaves. Since the foliation by slow unstable leaves is C in W u ( v ) , such leaves are generic by Fubini. Hence every leaf maps to a leaf by continuity. (cid:3) Corollary 6.8.
The slow unstable distributions are trivial.Proof.
For any v, w ∈ Q with w
6∈ ∪ t g t W u ( v ) , the projections of the slow unstable foliationson W u ( v ) and W u ( w ) to ∂ f M agree off of the backward endpoints of the geodesics through v and w in ∂ f M by Theorem 6.7.Hence we obtain a common C foliation of ∂ f M . Moreover, this foliation is invariant under π M since the E u< distributions are π ( M ) invariant. Again by Foulon [Fou94, Corollaire ]this foliation is trivial. (cid:3)
The last step in the proof of our main result is now essentially the same as in the strictpinching case.
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Department of Mathematics, Indiana University, Bloomington, IN 47405
E-mail address : [email protected] Courant Institute of Mathematical Sciences, New York University, New York, NY 10012
E-mail address : [email protected] Department of Mathematics, University of Michigan, Ann Arbor, MI, 48109.
E-mail address : [email protected]@umich.edu