Some Rigidity Theorems for Anosov Geodesic Flows in Manifolds of Finite Volume

. In this paper, we prove that if the geodesic ﬂow of a complete manifold without conjugate points with sectional curvatures bounded below by − c 2 is of Anosov type, then the constant of contraction of the ﬂow is ≥ e − c . Moreover, if M has a ﬁnite volume, the equality holds if and only if the sectional curvature is constant. We also apply this result to get a certain rigidity for bi-Lipschitz, and consequently, for C 1 -conjugacy between two geodesic ﬂows.


Introduction and Main Results
Geodesic flows appear naturally when we have a Riemannian metric on a complete manifold.Its properties are closely linked with the geometry of the Riemannian metric.
From Hadamard's work on cutting sequences and Hopf's work, which proved the ergodicity of geodesic flows on surfaces of negative curvature [19], we know that geometry influences the chaotic properties of geodesic flow.This property was further explored by Anosov in his seminal work [1], where he proved that geodesic flows of negative curvature manifolds are uniformly hyperbolic systems.Currently, we simply call them "Anosov " systems.The dynamic properties of the geodetic flow of negative curvature manifolds have been studied for many decades.To date, the study of the ergodic and dynamic properties of geodesic flow is a very active topic.In [1], Anosov began the task of knowing which types of geometries exhibit the Anosov property.
In [13], Eberlein determined geometric conditions that are equivalent to the Anosov property of the geodesic flow when M is compact or compactly homogeneous i.e., the isometry group of its universal cover acts co-compactly.More specifically, Eberlein's conditions give information related to the divergence of Jacobi fields or about the geodesics passing through points of negative sectional curvature.In [4], Bolton observed that the condition of compactly homogeneous by Eberlein at [13] is unnecessary.
In the opposite direction, we can ask the following question: What geometric properties impose the geodesic flow to be Anosov?
A satisfactory answer to this question was given by Klingenberg, who proved that a compact Riemannian manifold with the geodesic flow of Anosov type has no conjugate points (cf.[23]).This result was generalized by Mañé in [26] for the case of finite volume.Recently, the infinite volume case was proved by the authors in [28].
The results of Klingenberg at [23], Mañé at [26], and Melo-Romaña at [28] show that some geometric properties of manifolds with Anosov geodesic flow are consequences of dynamical properties.In the same spirit, in this paper, we will show some results which give an important relation between the geometry and dynamical behavior of the geodesic flow.Moreover, we will obtain that a dynamical rigidity implies geometric rigidity (cf.Theorem 1.1), which will be used to show rigidity in the level of conjugacy between two geodesic flows (cf.Theorem 1.2 and Theorem 1.4).
To state our results, we start with the precise definition of Anosov geodesic flow for general case, even non-compact manifold: Let M be a complete Riemannian manifold and SM the unitary tangent bundle, endowed with the Sakaki metric (see Section 2.1).Let φ t M : SM → SM be the geodesic flow, and suppose that φ t M is Anosov, this means that the tangent bundle of SM , T (SM ), has a splitting for all t ≥ 0 with C > 0 and 0 < λ < 1, where the constant λ is called a constant of contraction of the flow and G is the geodesic vector field.In our first result, we give a lower bound for the possible values of λ depending on the lower bound of the curvature.Furthermore, we prove that if λ reaches its minimum value, then we have a rigidity of the geometry.More specifically The first part of Theorem 1.1 implies that for Anosov geodesic flows the dynamics is controlled, to some extent, by the curvature.The second part is more general: it says that rigidity in the dynamical sense provides rigidity in the geometrical sense.We emphasize that our result does not require the compactness of the manifold.
It is worth noting that the first part of Theorem 1.1 follows from the proof of the corollary of Theorem A by Mañe in [26], this result can also be proved using the Rauch comparison theorem, for more details see [24].However, Mañe's proof and the approach using the Rauch comparison theorem do not provide any information concerning rigidity.Here we will give a different proof of this theorem in which, if the equality holds, the additional information obtained in the proof provides a new understanding of the rigidity problem.
Another important thing is to observe that in the compact case, the rigidity of Theorem 1.1 can be proved using Pesin's formula for the entropy and Lyapunov exponents and some techniques of [16], but it is worth noting that, in general, Pesin's formula is not valid in non-compact manifolds (cf.[32]), so as some theorems in [16].As we are interested also in the non-compact case, our proof does not use Pesin's formula.
In the proof of the first part of Theorem 1.1, the hypothesis of finite volume is required only to ensure the non-existence of conjugate points (see [26]).Therefore, when the volume is infinite and without conjugate points, the first part of this theorem is also valid, see Lemma 3.1.It is worth mentioning that the condition of no conjugate points is redundant due to the authors' recent preprint (see [28]).
To state the other results concerning the regularity of conjugacy between two geodesic flows, we need the following definition.
We say that the two flows ϕ t : N → N and η t : S → S are equivalents if there is a continuous map f : N → S such that f • ϕ t = η t • f .The map f is called an equivalence.Equivalences between two systems are very important in dynamical systems because relevant information about one system is transferred to another by the equivalences.
When an equivalence is a homeomorphism it is called a conjugacy.An equivalence (conjugacy) f is called 1-equivalence (1-conjugacy) if f is bi-lipschitz 1 , i.e., there are two constants C 1 and C 2 such that (1.1) where d and D are the distances in N and S, respectively.
In the next theorem, we obtain a rigidity result between two 1-equivalent flows, where one of them is an Anosov geodesic flow.
Theorem 1.2.Let M , N be two complete Riemannian manifolds such that sectional curvatures satisfy inf It is important to note that in Theorem 1.2 the compactness of M and N is not required.But, when M is a compact manifold, then Theorem 1.2 has an important consequence.We say that a map f : N → S is an immersion if for all x ∈ N it holds that The above result imposes a certain rigidity for regular equivalences of two Anosov geodesic flows.Observe that the only condition about the dimension of M and N is dimM ≤ dimN , since f is an immersion.Therefore, if dimM = dimN , then f is a C 1diffeomorphism and N should be compact.However, if N is a compact manifold and dim M = dim N we have a more general result by mixing the proofs of Theorem 1.1 and Theorem 1.2.
Theorem 1.4.Let M , N be two compact Riemannian manifolds with the same dimension such that inf K M ≥ sup K N , and M has no conjugate points.If φ t M and φ t N are 1-conjugate, then Since compact manifolds with Anosov geodesic flow have no conjugate points (cf.[23]), an immediate consequence is Corollary 1.5.Let M , N be two compact Riemannian manifolds with the same dimension such that inf The previous results are related to the following conjecture (cf.Conjecture 5.2.1 at [6]): Conjecture: Compact Riemannian manifolds with negative curvature must be isometric if they have C 0 -conjugate geodesic flows.
Note that, in particular, the hypothesis of Theorem 1.4 assures that there is no 1-conjugacy unless the curvature of both manifolds is constant, which proves the conjecture for 1conjugacy (consequently for C 1 -conjugacy) and some relation of the curvatures.
The latter results are in the direction of recent research related to the smoothness of conjugacy for geodesic flows in compact manifolds.Matters related to rigidity in the conjugacy of geodesic flows have been and are being much studied recently.Let us cite some related articles.The first is an article by three authors, Gallot, Besson, and Courtois (see [3]), which among other things, proves that if the geodesic flow of a compact manifold locally 1 We can think in α-equivalence (α-conjugacy), when there are two constants C 1 and C 2 such that but such definition do not make any sense for C 1 flows.symmetric and negative curvature is C 1 -conjugate to the geodesic flow of another manifold, then the metrics are isometric (see also [2] for other results about the regularity of conjugacy).In [10] the conjecture was shown for manifolds of rank ≥ 2. Also, J. Feldman and D. Ornstein showed that the conjugacy between two geodesic flows of surfaces of negative curvature is C 1 , which was generalized by Pollicott (see [31]) showing that the conjugacy is C ∞ .Note that the manifold M in Theorem 1.4 does not have necessarily negative curvature, therefore that result of Feldman and Ornstein does not apply.Feres and Katok (see [15]), proved that when the horospheric foliation is smooth and the curvature is negative, then the flow is C 1 -conjugate to a geodesic flow of a manifold of constant negative curvature (see [21] for the smooth case).To finish our references, there is a result of C. Connell (see [8]) which assumes a relation of curvatures between two compact manifolds one of them being locally symmetric, then he shows that Lipschitz semi-conjugacy between their geodesic flows makes the other manifold also locally symmetric.In other words, in the locally symmetrical context, we have rigidity for conjugacy.Here we observe that, in this work, all manifolds involved are not necessarily locally symmetric, which makes our results interesting (see [9] for more results about rigidity in Riemannian manifolds).
In Section 4, we will prove the Theorems 1.2 and 1.4 and their corollaries.We conclude this introduction by showing the relations of the techniques developed in this work with the scenario of Lyapunov exponents.
In [7], Butler studied the rigidity results of Lyapunov exponents for geodesic flows on a compact negatively curved manifold.He proved that if each periodic orbit of the geodesic flow has the same Lyapunov exponent over the unstable bundle, then the manifold has constant negative sectional curvature.
Since the geodesic flow on compact manifolds of negative curvature is of Anosov type, then by Butler's result we can ask the following: Conjecture 1: Let M be a complete Riemannian manifold with finite volume and Anosov geodesic flow.If the unstable Lyapunov exponents are constant in all periodic orbits, then M has constant negative sectional curvature.
Let M be a manifold with curvature bounded below.For each x ∈ M , let T x M be the tangent space at x. So, for each plane P ⊂ T x M , we denote by K x (P ) the sectional curvature of the plane P .If the geodesic flow φ t M is Anosov then, from Corollary 3.2, we define the positive and finite number c inf as follows, Therefore, using the techniques developed to prove Theorem 1.1, we reduce Conjecture 1 to the following conjecture (see Corollary 3.7).
Conjecture 2: Let M be a complete Riemannian manifold with finite volume, sectional curvature bounded below, and Anosov geodesic flow.If the unstable Lyapunov exponents in all periodic orbits are constant equal to a, then a = c inf .
It was an open problem that the Anosov condition in the case of infinite volume does not imply the condition of no conjugate points (cf.[24]).However, the authors in [28], recently gave a positive answer to this fact.Thus, the proof of Theorem 1.1 will be carried out in two steps.In the first phase, we will prove the inequality relating any constant of contraction of the flow with the exponential of the bound of the curvature, assuming no conjugate points and Anosov condition, without any assumption about the volume of the manifold, see Subsection 3.1 (or from [28] only assuming the Anosov condition).In the second phase, we will prove that the Lyapunov exponents are constants for every point and, finally, using geometric arguments associated with geodesic flows in manifold without conjugate points, we will conclude that the curvature is constant.For the proofs of Theorem 1.2 and Theorem 1.4, we will use the equivalence to transfer all information of the hyperbolicity from φ t N to φ t M and thus use the techniques of Theorem 1.1.

Notation and Preliminaries
Throughout this paper, M = (M, g) will denote a boundaryless complete Riemannian manifold of dimension n ≥ 2, T M its tangent bundle, SM its unit tangent bundle.The mapping π : T M → M will denote the canonical projection and µ the Liouville measure of SM .All settings of this section can be found in [30] (see also [24]).
2.1.Geodesic flow.Let θ = (p, v) be a point of SM and let γ θ (t) be the unique geodesic with initial conditions γ θ (0 Recall that this family is a flow (called the geodesic flow ) in the sense that φ t+s Recall that, α is defined as follows: For each θ, the maps Using the decomposition T θ T M = H(θ) ⊕ V (θ), we can identify a vector in ξ ∈ T θ T M with the pair of vectors in T p M , (D θ π(ξ), K θ (ξ)) and define naturally a Riemannian metric on T M that makes H(θ) and V (θ) orthogonal.This metric is called the Sasaki metric and is given by . From now on, we consider the Sasaki metric restricted to the unit tangent bundle SM .It is easy to prove that the geodesic flow preserves the volume measure generated by this Riemannian metric in SM .Furthermore, this volume measure in SM coincides with the Liouville measure m up to a constant.When M has finite volume the Liouville measure is finite.
Consider the one-form β in SM defined for θ = (p, v) by Furthermore, β is a contact form invariant by the geodesic flow whose Reeb vector field is the geodesic vector field G. Furthermore, the sub-bundle S = ker β is the orthogonal complement of the subspace spanned by G. Since β is invariant by the geodesic flow, then the sub-bundle S is invariant by φ t M , i.e., φ t M (S(θ)) = S(φ t M (θ)) for all θ ∈ SM and for all t ∈ R. To understand the behavior of dφ t M let us introduce the definition of a Jacobi field.A vector field J along of a geodesic γ θ is called the Jacobi field if it satisfies the equation (2.1) where R is the Riemann curvature tensor of M and " ′ " denotes the covariant derivative along γ θ .Note that, for ξ = (w 1 , w 2 ) ∈ T θ SM , with w 1 , w 2 ∈ T p M and v, w 2 = 0, it is known that d(φ t M ) θ (ξ) = (J ξ (t), J ′ ξ (t)), where J ξ denotes the unique Jacobi vector field along γ θ such that J ξ (0) = w 1 and J ′ ξ (0) = w 2 .For more details see [30].2.2.No conjugate points and Ricatti equation.Suppose that p and q are points on a Riemannian manifold, and γ is a geodesic that connects p and q.Then p and q are conjugate points along γ if there exists a non-zero Jacobi field along γ that vanishes at p and q.When neither two points in M are conjugate, we say the manifold M has no conjugate points.Another important type of manifold for this paper are the manifolds without focal points, we say that a manifold M has no focal point, if for any unit speed geodesic γ in M and for any Jacobi vector field Y on γ such that Y (0) = 0 and Y ′ (0) = 0 we have (||Y || 2 ) ′ (t) > 0, for any t > 0. It is clear that if a manifold has no focal points, then also has no conjugate points.
The more classical example of manifolds without focal and therefore without conjugate points are the manifolds of non-positive curvature.It is possible to construct a manifold having positive curvature somewhere, and without conjugate points (cf.[12]) .Manifolds without conjugate points are connected with manifolds with Anosov geodesic flow.In [26], Mañé proved that, for manifolds of finite volume, the Anosov property of the geodesic flow implies no conjugate points.This had been proved early by Klingenberg (cf.[23]) in the compact case.Recently, in [28], the case of infinite volume was proved by the authors.
The last fact showed that, if we would like to work with the geodesic flows of Anosov type, we assume then that our manifolds have no conjugate points (condition superfluous for manifolds of finite volume via Mañe result).Therefore, from now on, we can assume that the manifold M has no conjugate points.Now suppose that M has no conjugate points and its sectional curvatures are bounded below by −c 2 .In this case, if the geodesic flow φ t M : SM → SM is Anosov, then in [4], Bolton showed that E s (θ) ∩ V (θ) = {0} and E u (θ) ∩ V (θ) = {0} for all θ ∈ SM .This last property will be very useful for the proof of Theorem 1.1.
There are important subbundles of T SM to study the dynamic behavior of the geodesic flow of manifolds without conjugate points, the Green subbundles are defined as follows: , where G(θ) is the geodesic vector field, which are called the stable and unstable Green subbundle, respectively.For a manifold without conjugate points and dimension n, then the dimension of Green subbundles is n − 1.Moreover, if the curvature is nonpositive, then the Green subbundles depend continuously on θ ∈ SM .In the general case, so characterizes Anosov geodesic flows, if the Green subbundles depend continuously on θ and G s θ ∩G u θ = {0}, then the geodesic flow is Anosov (see [13] for more details on Green subbundles).
For θ = (p, v) ∈ SM , we denote by N (θ) := {w ∈ T x M : w, v = 0}.By the identification of the Subsection 2.1 we can write S(θ) Hence, there exists a unique linear map T : H(θ) ∩ S(θ) → V (θ) ∩ S(θ) such that E is the graph of T .In other words, there exists a unique linear map T : Furthermore, the linear map T is symmetric if and only if E is Lagrangian (see [30]).
It is known that if the geodesic flow is Anosov, then for each θ ∈ SM , the sub-bundles E s (θ) and E u (θ) are Lagrangian and E s (θ) ⊕ E u (θ) = S(θ).Therefore, for each t ∈ R, Now we describe a useful method of L. Green (cf.[17]), to see what properties the maps U s (t) and U u (t) satisfy.
Let γ θ be a geodesic, and consider V 1 , . . ., V n a system of parallel orthonormal vector fields along γ θ with V n (t) = γ ′ θ (t).If Z(t) is a perpendicular vector field along γ θ (t), we can write Note that Z(s) can be identified with the curve α(s) = (y 1 (s), . . ., y n−1 (s)) and Z ′ (s) can be identified with the curve α ′ (s) = (y ′ 1 (s), . . ., y ′ n−1 (s)).Conversely, any curve in R n−1 can be identified with a perpendicular vector field on γ θ (t), so we can identify N (φ t M (θ)) with R n−1 and consider the maps associated to stable and unstable subbundles defined in R n−1 .Now for each t ∈ R, consider the symmetric matrix R(t) = (R i,j (t)), where ) and R is the curvature tensor of M .The family of operators U s (t) : R n−1 → R n−1 and U u (t) : R n−1 → R n−1 satisfies the Ricatti equation where {v, v 1 , v 2 , . . ., v n−1 } is an orthonormal basis of T x M (see [11] for more details).Moreover, using the previous discussion, we have that the Ricci curvature along γ θ (t) is given by where trR(t) denotes trace of R(t).Now consider the (n − 1) × (n − 1) matrix Jacobi equation x corresponds to a Jacobi perpendicular vector on γ θ (t).For θ ∈ SM , s ∈ R, we consider Y θ,s (t) to be the unique solution of (2.4) satisfying Y θ,s (0) = I and Y θ,s (s) = 0.In [17], Green proves that lim s→−∞ Y θ,s (t) exists for all θ ∈ SM (see also [13,Sect. 2]).He also shows that if we define: (2.5) we obtain a solution of Jacobi equation (2.4) such that det Y + θ (t) = 0.Moreover, it is proved in [17] (see also [16] and [13]) that And it can be proved easily that (see [16]) for every t ∈ R. It follows that U + is a symmetric solution of the Ricatti equation (2.2).Analogously, taking the limit when s → +∞, we have defined U − (θ), which also satisfies the Ricatti equation (2.2).Furthermore, in [17], Green also showed that in the case of curvature bounded below by −c 2 , symmetric solutions of the Ricatti equation which are defined for all t ∈ R are bounded by c.In particular, When the geodesic flow is Anosov,

Proof of Theorem 1.1
In this section, we show several lemmas that will be used to obtain the proof of Theorem 1.1.Our first lemma (Lemma 3.1) proves the first part of Theorem 1.1 in a more general setting using only the no conjugate points condition (or only the Anosov condition by [28]), without any condition about the volume.Since any manifold of finite volume with Anosov geodesic flow has no conjugate points (cf.[26]), Lemma 3.1 implies in fact the proof of the of item (a) of Theorem 1.1.Proof.Since the curvature is bounded below by −c 2 , then by Lemma 2.16 at [24] (see also [24,Lemma 2.17]), we have Using the definition of Anosov flow in (3.1) (similar argument using (3.2)), we have If we write λ = e −κ , the last inequality becomes which is valid if and only if c ≥ κ or equivalently λ = e −κ ≥ e −c as we desired.
Corollary 3.2.No manifold M of finite volume and non-negative curvature has the geodesic flow of the Anosov type.
Proof.By contradiction, if such manifold M exists, then M has no conjugate points (cf.[26]).Since the curvature K of M is non-negative, then for all ǫ > 0, K ≥ −ǫ 2 .Let 0 < λ < 1 be the constant of contraction of φ t M , then by Theorem 1.1 we have λ ≥ e −ǫ for all ǫ > 0, which implies that λ ≥ 1 and this is a contradiction.

Rigidity and Lyapunov exponents.
In this subsection, we shall prove that the map U + (θ) (from Subsection 2.2) has all information about Lyapunov exponents.The following Lemma was proved by Freire and Mañé ([16, Theorem II]) in compact manifolds without conjugate points.Here we do the proof in the non-compact case for Anosov geodesic flows.
Lemma 3.3.Let M be a complete manifold with curvature bounded below by −c 2 without conjugate points and whose geodesic flow is Anosov.Then ) ds holds for every θ ∈ SM .
Proof.For each θ = (p, v) ∈ SM , we denote by N (θ) the subspace of T p M orthogonal to v. Then by construction, for all x ∈ N (θ) the Jacobi field Y + θ (t)x is an unstable Jacobi field.Thus, it follows that U + (θ) satisfies that Let π θ : E u (θ) → N (θ) the projection in the first coordinate.Then Moreover, for (v, w) ∈ E u (θ) Thus, by the equation (3.4) From Linear Algebra we know where A is an invertible linear map of vector spaces of dimension m.
As dim E u (θ) = n − 1, using the inequality (3.6) and the inequalities Therefore, the inequalities (3.5) and (3.7) provide us with Since det Y + θ (t) = 0, then is easy to prove that (3.9) From (3.8) and (3.9) The following lemma shows the rigidity of Lyapunov exponent.Lemma 3.4 (Rigidity of Lyapunov Exponent).In the same conditions as Theorem 1.1.
Proof.From Lemma 3.4, we have lim . The proof for E s (θ) is analogous.
To conclude this section, we prove the following Lemma, which provides the final step in proving the item (b) of the Theorem 1.1.Lemma 3.6 (Rigidity).In the same conditions as Theorem 1.1, λ = e −c if and only if the sectional curvature of the manifold M is constant, equal to −c 2 .
Proof.Note that if K = −c 2 , then the classical proof that the geodesic flow is Anosov implies that λ = e −c .So our task is to prove the other side.In fact: from Lemma 3.3 and Corollary 3.5, we have that lim 2 ) Since the sectional curvature satisfies K ≥ −c 2 , then taking trace and integrating from 0 to t the Ricatti equation (2.2) and taking into account (2.3), from the Cauchy-Schwarz inequality we have ) − tr((U + )(θ)) = 0. Thus, taking limit as t → +∞ in the inequality (3.12), we get (3.13)lim To conclude our argument, let us use the Birkhoff Ergodic theorem.First, we note that as K ≥ −c 2 , then the negative part of the Ricci curvature is integrable on SM .Thus, from a result of Guimarães in [18], we have that the Ricci curvature is integrable on SM .Therefore, as M has finite volume, from Birkhoff ergodic theorem and equation (3.13) we have Since the sectional curvature satisfies K ≥ −c 2 , then the previous equation implies that Ric(θ) ≡ −c 2 .Thus, we conclude that As an immediate consequence of Lemma 3.6 we get the following corollary, which proves that Conjecture 2 implies Conjecture 1.
Corollary 3.7.Let M be a complete Riemannian manifold with finite volume, sectional curvature bounded below, and Anosov geodesic flow.If the unstable Lyapunov exponents of all periodic orbits are constants equal to c inf , then M has constant negative sectional curvature equal to −c 2 inf .

Rigidity of Conjugacy
The main purpose of this section is to prove Theorem 1.2, Theorem 1.4, and its corollaries.The strategy of the proof of Theorem 1.2 is to show that, if λ N is a constant of contraction of φ t N and λ M is a constant of contraction of φ t M , then λ M = λ N and apply the Theorem 1.1 to conclude our result.For this sake, we need the following lemma.Lemma 4.1.Let (S, g) be a Riemannian manifold and d its Riemannian distance.If α(t) is a curve on S, differentiable at t = 0, then Proof.The exponential map exp α(0) defines a diffeomorphism in a neighborhood of 0 ∈ T α(0) S on a neighborhood of α(0) with D exp α(0) 0 = Id.Then, for a small t we have that Proof of Theorem 1.2.We note that as M has finite volume and φ t M is Anosov, then Corollary 3.2 implies that inf K M < 0, thus sup K N < 0 and φ t N is also Anosov.We put −a 2 := sup K N .Now assume that f : SM → SN is a 1-equivalence between φ t M and φ t N with constant C 1 and C 2 (see (1.1)).Without loss of generality, we denote by d the metric of SM and SN .Denote by Γ the set of points in SM where there exists df θ , then as f is a Lipschitz map, the set Γ has full Liouville measure.Consider θ ∈ Γ, then as f is bi-Lipschitz with the help of Lemma 4.1 we have Consequently, df θ is an injective linear map.So we can define the subspace F where E s(u) are the stable and unstable subbundle from the definition of Anosov flow of φ t M .It is easy to see that Γ is φ t M -invariant and since Claim: For all θ ∈ Γ holds that where Proof of Claim.Since f is a 1-equivalence, then (4.1) ), for all t ∈ R and all x, y ∈ SM .Consider θ ∈ Γ and ξ ∈ E s θ \ {0} and β(r) ⊂ SM a C 1 -curve, such that β(0) = θ and β ′ (0) = ξ.From Lemma 4.1, for t ∈ R we have The last two equalities and inequality (4.1) implies that Consequently, for all t ∈ R . Thus, the last characterization and (4.3) allows us to complete the proof of claim in the stable case.The proof of the unstable case is completely analogous.

Since a constant of contraction of φ t
N is e − √ − sup KN = e −a (cf.[24]), then the inequality (4.3) and Claim provide for all t ≥ 0 and all θ ∈ Γ.However, since Γ has full measure, then the last inequality holds for all θ ∈ SM .In other words, λ M = e −a .However, by hypotheses K M ≥ inf K M ≥ sup K M = −a 2 , then Theorem 1.1 implies that K M ≡ −a 2 , thus we conclude the proof of the theorem.
Proof of Corollary 1.3.The strategy here is to show that C 1 -equivalences that are injective immersions are actually 1-equivalences, at least for the compact case.Let f : SM → SN be a C 1 equivalence, which is a injective immersion.Then f : SM → f (SM ) is a diffeomorphism and f (SM ) is a compact submanifold of SN .If we consider f (SM ) with the extrinsic distance of SN and let d f be such distance, it is easy to see that for all θ, θ ∈ SM From (4.4) and Corollary A.2, there is Γ > 1 such that Thus, f is a 1-equivalence.Thus, Theorem 1.2 allows us to conclude that K M ≡ sup K N as we wish.
Proof of Theorem 1.4.Observe that since M has no conjugate points, then inf K M ≤ 0 (cf.[17] and [20]).Therefore the hypotheses on curvatures imply that sup K N ≤ inf K M ≤ 0. To prove the theorem, we consider two cases: The proof in each case is slightly different because in case 1 the behavior of the geodesic flow is hyperbolic and case 2 is not necessarily hyperbolic.
Case 1: In this case, the proof is quite similar to the proof Theorem 1.2, but we will use the hyperbolic splitting of φ t N to construct a hyperbolic splitting for φ t M .Put −a 2 := sup K N < 0, then φ t N is Anosov and denoted by E s(u) the stable and unstable bundle, respectively.Let f be a 1-conjugacy and denote by Γ the set of points in SM where there for all t ∈ R.
Since the constant of contraction of φ t N is e − √ − sup KN = e −a (cf.[24]), then the inequality (4.6) provides that for all t ≥ 0 The two last inequalities together with the equation (4.5) provide a hyperbolic behavior of φ t M along φ t M (θ).Therefore, as M has no conjugate points and K M ≥ sup K N = −a 2 , then the same arguments of proof of Theorem 1.1 provide that (4.where Ric is the Ricci curvature.Since the sectional curvature satisfies K M ≥ −a 2 , then the previous equation implies that Ric(θ) ≡ −a 2 for Liouville almost every point θ ∈ SM .Thus, we conclude that K M ≡ −a 2 and therefore the splitting, given by equation (4.5), coincides with its hyperbolic splitting.To conclude the proof of case 1, we need only to prove that K N ≡ −a 2 .For this sake, note that since K M ≡ −a 2 and for ξ ∈ F u θ (cf.[24]) we have that ||(dφ t M ) θ (ξ)|| = 1 + a From Theorem 1.1 at [7] we have that K N ≡ −a 2 , which completes the proof of case 1.
Case 2: In this case sup K N = 0. So, our main goal is to prove that K M = K N = sup K N = 0. Since sup K N = 0, then 0 ≤ sup K N ≤ inf K M , in other words, 0 ≤ K M .By hypothesis M has no conjugate points, then K M ≡ 0 (cf.[17] and [20]).So, our goal has been reduced to prove that K N ≡ 0. By contradiction, assume that N is not flat and assume that there is c > 0 such that −c 2 ≤ K N ≤ 0. We have the following claim: Claim: There is w ∈ SN such that lim sup Finally, as K N ≤ 0, then Ric(θ) = 0 for all θ ∈ SM and by definition of Ric we conclude that K N = 0 and N is flat, so we have a contradiction and the claim is proven.
Using the last claim, since N is a compact manifold with non-positive curvature, in particular, N has no focal points, then with similar arguments of Proposition 1 and Corollary 7 at [27] we have that the unstable Green subbundle G u of T SN (see Section 2.2 and [13] for more details) satisfies lim sup for each pair (w, η), with w ∈ SN and η ∈ T w SN .Thus, the last equation is a contradiction with the inequality (4.13), which ends the proof of Case 2 and, consequently, the proof of Theorem.

Appendix A. Comparing distances between a manifold and its submanifolds
This appendix is devoted to prove the Lemma A.1.This a general lemma that can be in other context., t = 0 1, t = 0, where exp P x denotes the exponential map of P .It is easy to see that H is continuous for all ((x, v), t) with t = 0. We state that H is continuous at any point ((x, v), 0).To prove that, we used the compactness of P and the following claim: Claim: For each x ∈ P the function H x : T 1 x P × [0, r P ] → R defined by H x (v, t) := H((x, v), t) is uniformly continuous function in T 1 x P ×[0, r P ], where T 1 x P is the set of unitary tangent vectors of P at x. Proof of Claim.By definition of H x and compactness of T 1 x P × [0, r P ], we only need to prove that lim n→+∞ H x (v n , t n ) = H x (v, 0) = 1, for any sequence (v n , t n ) ∈ T 1 x P × [0, r P ], which converges to (v, 0).In fact: first note that if (x, v) ∈ SP and α(t) = exp P x tv, then from Lemma 4.1 we have that  As H x (v n , 0) = 1 then we can assume, without loss of generality, that t n = 0 for all n.
For n large enough, consider the family of C The last claim and compactness of P allow us to conclude the continuity of H in every point ((x, v), 0) and consequently the uniformly continuity in SP × [0, r P ].
The last inequality implies that Consequently, if x ∈ B P δ (x), then there is t with | t| < δ, v ∈ T 1 x P such that exp P x tv = x.Thus, (A.1) provides the result of the lemma. .We state that there is Γ > 1 such that Γ(x) ≤ Γ for all x ∈ P .In fact, by contradiction assume that for each n ∈ N there is x n such that Γ n ≥ n.By definition of Γ(x n ) there is y n = x n with d P (x n , y n ) d(x n , y n ) ≥ n − 1.Since P is a compact submanifold, then we can assume that x n converges to x and y n converges to y.From the last inequality, we have that x = y.Therefore, from Lemma A.1, for n large enough y n ∈ B δ (x n ) and which provides a contradiction.

Theorem 1 . 1 .
Let M be a complete Riemannian manifold with sectional curvature bounded below by −c 2 and Anosov geodesic flow φ t M .(a) Any constant of contraction λ of φ t M satisfies λ ≥ e −c .(b) If M has finite volume, then λ = e −c if and only if the sectional curvature of M is constant, equal to −c 2 .

Lemma 3 . 1 .
Let M be a complete manifold with curvature bounded below by −c 2 without conjugate points and whose geodesic flow is Anosov.If the constant of contraction of the geodesic flow is λ then λ ≥ e −c .

Lemma A. 1 .
Let Q be a complete Riemannian manifold and P a compact submanifold of Q.Consider d the natural distance in Q and d P the extrinsic distance in P .Then there is δ > 0 such that for each x ∈ Pd P (x, y) ≤ 3 2 d(x, y) for all y ∈ B P δ (x),where B P δ (x) is the ball of center x and radius ǫ in P .Proof.Since P is a compact submanifold of Q, then the injectivity radius r P of P with extrinsic metric is a positive number.Denote by SP the unitary bundle of P and consider the non-negative real functionH : SP × [0, r p ] → R defined by H((x, v), t) =          d P (exp Px tv, x) d(exp P x tv, x) lim ||α ′ (0)|| P is the norm with restrict metric of P ⊂ Q which is equal to ||α ′ (0)||.

Corollary A. 2 .
In the same condition of the Lemma A.1, there is a constant Γ > 1 such that d P (x, y) ≤ Γ • d(x, y) for all x, y ∈ P. Proof.For each x ∈ P consider the function Γ(x) = sup y =x d P (x, y) d(x, y) 2• e at ||π 1 (ξ)||, where π 1 (•) is the projection on the first coordinate in the horizontal and vertical decomposition of T SM (see Section 2.1).Proof of Claim.Let w ∈ SN and η ∈ E u w , then by the density of Λ and continuity of unstable bundle E u , we have that there are w n ∈ Λ and η n ∈ E u w n such that (w n , η n ) converges to (w, η).Note that for all t ∈ R Moreover, by continuity of dφ t N , for each t ≥ t 0 there is n(t) such that 1 t log w (η)|| .By uniformity in the convergence of inequality (4.8), given ǫ > 0 there is t 0 such that a − 1 t log ||(dφ t N ) wn (η n ) < ǫ 2 for each t ≥ t 0 and all n.w (η)|| < ǫ for each t ≥ t 0 .w= a(n − 1).Let w ∈ SN be a periodic point of φ t N of period τ and r 1 , . . ., r n−1 the set of eigenvalues of uniformly in bounded regions of pair (θ, ξ) for each θ ∈ SM and ξ ∈ T θ SM .Proceeding in the same way as in Case 1, we have To finish our arguments, observe that asK M ≡ 0, then lim t→+∞ 1 t log ||(dφ t M ) θ (ξ)|| = 0 1-curves γ n (t) = exp P x (t n v + tt n (v n − v)), t ∈ [0, 1], and note thatd(exp P x t n v, exp P x t n v n ) ≤ d P (exp P x t n v, exp P x t n v n ) ≤ ||D(exp P x ) v ||.Hence, since lim n→+∞ H x (v, t n ) = 1, then d(exp P x t n v n , x) ≥ d(exp P x t n v, x) − Kt n ||v n − v|| = t n H x (v, t n )) −1 − K||v n − v|| > 0.In particular, we have1 ≤ H x (v n , t n )) = t n d(exp P x t n v n , x) ≤ 1 (H x (v, t n ))) −1 − K||v n − v|| .