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On Finsler Manifolds of Negative Flag Curvature On Finsler manifolds of negative flag curvature Yong Fang and Patrick Foulon D´epartement de Math´ematiques,Universit´ede Cergy-Pontoise, 2, avenue Adolphe Chauvin 95302 Cergy-Pontoise Cedex, France (e-mail : [email protected]) CIRM, Luminy, case 916, 13288 Marseille Cedex 9, France (e-mail: [email protected]) Abstract One of the key differences between Finsler metrics and Rieman- nian metrics is the non-reversibility, i.e. given two points p and q, the Finsler distance d(p; q) is not necessarily equal to d(q; p). In this pa- per, we build the main tools to investigate the non-reversibility in the context of large-scale geometry of uniform Finsler Cartan-Hadamard manifolds. In the second part of this paper, we use the large-scale geometry to prove the following dynamical theorem: Let ' be the geodesic flow of a closed negatively curved Finsler manifold. If its Anosov splitting is C2, then its cohomological pressure is equal to its Liouville metric entropy. This result generalizes a previous Riemannian result of U. Hamenst¨adt. 1 Introduction Let M be a C1 manifold and TM be its tangent bundle. A C1-Finsler + metric on M is a function F : TM ! R satisfying: (a) F (tu) = tF (u) for any u 2 TM and t ≥ 0; (b) F is strictly positive and C1 over TM − f0g; (c) In standard local coordinates (xi; yi) on TM, the matrix of partial deriva- 2 2 tives @ F is positive definite. @yi@yj A Finsler metric F is said to be Riemannian if there exists a C1- p Riemannian metric g such that F = g. We say that F is reversible if 1 1 INTRODUCTION 2 for any u 2 TM, F (−u) = F (u): If it is not the case, F is said to be non-reversible. Given a Finsler manifold (M; F ) and p; q 2 M, by taking the infimum of the F -length of piecewise C1 curves joining p to q, we obtain the Finsler distance d(p; q). Since F is generally non-reversible, d(p; q) is not necessarily equal to d(q; p). In Finsler geometry, the so-called flag curvature generalizes the Rieman- nian sectional curvature (see Section 2). We say that (M; F ) is a Finsler Cartan-Hadamard manifold if the following conditions are satisfied: (a) M is connected and simply connected; (b) There existsa ¯ > 0 such that the flag curvatures of (M; F ) are less than or equal to −a¯2; (c) The Finsler distance function d is forward complete, which means that forward Cauchy sequences converge (see Section 2). For any s 2 M , v 2 TxM − f0g and u 2 TxM we define in standard 2 2 local coordinates g (u; u) = 1 P @ F (v)uiuj, which is a positive definite v 2 @yi@yj p quadratic form. The associated norm is denoted by k · kv= gv(·; ·). Ac- cording to D. Egloff (see [3]), the uniformity constant of (M; F ) is defined as k u kv1 C = supx2M fsupv1;v2;u2TxM−{0g g: k u kv2 If C < +1 then the Finsler Cartan-Hadamard manifold (M; F ) is said to be uniform. Principal examples of uniform Finsler Cartan-Hadamard man- ifolds are universal coverings of closed negatively curved Finsler spaces. We should mention that there exist quite a few interesting non-uniform Finsler Cartan-hadamard manifolds (see [1]). Non-reversible Finsler metrics are quite different from Riemmanian or re- versible Finsler metrics. For example, let γ be an element in the fundamental group π1(M). The lengths of the closed geodesics representing respectively γ and γ−1 are not necessarily equal. Therefore the marked length spectrum is quite different from the Riemannian or reversible Finsler cases. Similarly, the notions such as cross ratio and geodesic currents must be reformulated due to the non-reversibility. It is important for the whole theory of Finlser manifolds that we do not symmetrize or reduce the general Finsler metric to the special Riemannian or reversible Finsler case. The non-reversibility is essential, which should be preserved. Here is the organization of the first part of this paper on the large-scale 1 INTRODUCTION 3 geometry of uniform Finsler Cartan-Hadamard manifolds: Given a uniform Finsler Cartan-Hadamard manifold (M; F ), we construct two boundaries at infinity: the stable boundary @sM and the unstable boundary @uM. We prove a Finlser Morse lemma which is used to construct a geometric u s flip homeomorphism σus : @ M ! @ M. A direct consequence is that the u s geodesic space of (M; F ) is naturally parametrized by @ M ×@ M −Gr(σus), where Gr(σus) denotes the graph of σus. By using our geometric flip, we define the so-called unstable-stable angle. Then we use it to formulate and prove the uniformly visible property in our context. Finally we prove the visibility axiom. In the second part of this paper, we study the geodesic flows of closed neg- atively curved Finsler manifolds. In this part, we denote by (M; F ) a closed Finsler manifold of negative flag curvature. Its universal covering (M;f Fe) is a uniform Finsler Cartan-Hadamard manifold. Let ' be the geodesic flow of (M; F ) over the spherebundle SM = fu 2 TM : F (u) = 1g. Let X be the generating vector field of '. It is well-known that ' is Anosov, i.e. the tangent bundle T (SM) admits a '-invariant continuous splitting into subbundles ss su T (SM) = RX ⊕ E ⊕ E ; ss su such that vectors of E ( resp. E ) are contracted exponentially by D't in the positive (resp. negative) sense. In general, the subbundles Ess and Esu are only H¨oldercontinuous and rarely more regular. For any k ≥ 1, we say that the Anosov splitting of ' is Ck if Ess and Esu are both Ck subbundles. Recall that the topological entropy htop('), the cohomological pressure P (') and the Liouville metric entropy hL(') are three important dynamical invariants of '. In general we have htop(') ≥ P (') ≥ hL('). The celebrated entropy conjecture asserts the following: Entropy conjecture. Let be the geodesic flow of a closed negatively curved Riemannian manifold. If htop( ) = hL( ) then the Riemannian metric is locally symmetric of rank one. For general Finsler geodesic flows, according to [12] and [6], the coincidence between htop(') and hL(') is not the appropriate hypothesis in Entropy conjecture (see Subsection 4.4), which should be replaced by the coincidence between P (') and hL('). 2 GEOMETRIC PRELIMINARIES 4 With the help of large-scale geometry, we prove the following result relat- ing the regularity of Anosov splitting with the coincidence between P (') and hL('), which generalizes a previous Riemannian result of U. Hamenst¨adt: Theorem. Let ' be the geodesic flow of a closed negatively curved Finsler manifold. If its Anosov splitting is C2, then the cohomological pressure of ' is equal to its Liouville metric entropy. 2 Geometric preliminaries 2.1 Non-reversible distance Let (M; F ) be a Finsler manifold and c :[a; b] ! M be a piecewise C1 curve. R b 0 The length of c is defined as l(c) = a F (c (t))dt. For p; q 2 M the Finsler distance between p and q is defined as d(p; q) = infcl(c) where the infimum is taken over all piecewise C1 curves joining p to q. The reversed curve of c is defined asc ¯ :[−b; −a] ! M such thatc ¯(t) = c(−t): Since F is generally non-reversible, the length l(¯c) is not necessarily equal to l(c), which implies that the Finsler distance d(p; q) is not necessarily equal to d(q; p). The uniformity constant C of (M; F ) is defined as in Section 1. It is clear that closed Finsler manifolds are uniform. Therefore their universal coverings with lifted Finsler metrics are also uniform. We define the conjugate Finsler metric of F as follows: F ∗(u) = F (−u); 8 u 2 T M: The non-reversible distance of F ∗ is denoted by d∗. We have the following proposition: Proposition 2.1. Let (M; F ) be a uniform Finsler manifold with uniformity constant C. For any v 2 TM and any p; q 2 M, 1 1 F (v) ≤ F ∗(v) ≤ CF (v); d(p; q) ≤ d∗(p; q) = d(q; p) ≤ Cd(p; q): C C Proof. Let v 2 TxM − f0g. Since the Finsler metric F is positively 2 homogeneous of degree one, then gv(v; v) = (F (v)) , i.e. k v kv= F (v): Similarly we have 2 ∗ 2 g−v(v; v) = g−v(−v; −v) = (F (−v)) = (F (v)) ; ∗ i.e. k v k−v= F (v): Thus by the definition of the uniformity constant, we 1 F ∗(v) 1 ∗ obtain C ≤ F (v) ≤ C; which implies that C d(p; q) ≤ d (p; q) ≤ Cd(p; q): 2 GEOMETRIC PRELIMINARIES 5 Let c be a piecewise C1 curve in M andc ¯ its reversed curve. By the definition of F ∗ it is clear that l∗(¯c) = l(c). Therefore d∗(p; q) = d(q; p): The proof is complete. Let c(t) be a C1 curve in M. We say that c is a geodesic if it has con- stant speed and verifies the following: For 8 t1 < t2 close enough; l(c j[t1;t2]) = d(c(t1); c(t2)): The Finsler manifold (M; F ) is said to be forward complete if any geodesic defined over [a; b[ can be extended to a geodesic over [a; +1[. Similarly (M; F ) is said to be backward complete if any geodesic defined over ]a; b] can be extended to a geodesic over ] − 1; b].
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