On Finsler manifolds with hyperbolic geodesic flows

Let (M, F) be a closed C∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^\infty $$\end{document} Finsler manifold and φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi $$\end{document} its geodesic flow. In the case that φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi $$\end{document} is Anosov, we extend to the Finsler setting a Riemannian vanishing result of M. Gromov about the L∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^\infty $$\end{document}-cohomology.

For simplicity, we write F (v) instead of F (x, v), where (x, v) ∈ T M. A Finsler metric F on M is said to be Riemannian if for any x ∈ M, there exists an inner product g x on T x M such that F 2 There exist lots of non-Riemannian Finsler metrics for example the so-called Randers metrics (see [5]). Given a Finsler manifold (M, F ), let c : [a, b] → M be a piecewise C ∞ curve in M . Since F is positively homogeneous, the length of c is well-defined as L(c) = b a F (c (t))dt. For any p, q ∈ M, we define d(p, q) = inf c L(c), where the infimum is taken over all piecewise C ∞ curves c issuing from p to q. The geodesics of (M, F ) are characterized as the constant speed curves locally minimizing the length, which are then necessarily C ∞ curves. The Finsler metric F is said to be reversible if F (−v) = F (v) for any v ∈ T M. In this article, we shall consider general Finsler metrics without the reversibility assumption. So generally speaking, for any p, q ∈ M, we do not necessarily have d(p, q) = d(q, p), i.e. the distance function d is asymmetric.
This kind of asymmetry occurs naturally in many applications and offers one of the most interesting aspects in Finsler geometry (see [1]).
Gromov showed in [10] that if M is a closed Riemannian manifold of negative sectional curvature, then every closed bounded form of degree ≥ 2 on M has a bounded primitive. This means that the L ∞ -cohomology of M vanishes in degree ≥ 2.
Given a closed Riemannian manifold (M, g) of negative curvature, it is wellknown that its geometric hyperbolicity resulting from the negative curvature assumption implies that its geodesic flow defined on the sphere bundle SM is Anosov, i.e. dynamically hyperbolic. Therefore, the geometric hyperbolicity can be considered as stronger than the dynamical hyperbolicity. In this perspective, Cheng extended in [4] the vanishing result above to closed Riemannian manifolds with Anosov geodesic flows. In [2], Burns and Paternain obtained the same vanishing result for Riemannian manifolds with Anosov magnetic flows. In this paper, we generalize this Riemannian result to the Finsler setting:

Symplectic formulation of Finsler geodesic flows.
In this section, we recall some definitions concerning Finsler geodesic flows. See, for instance, [11] or [9] for more details. Let (M, F ) be a closed C ∞ Finsler manifold and π : T M 0 → Vol. 117 (2021) On Finsler manifolds with hyperbolic geodesic flows 235 M be the canonical projection. The potential of (M, F ) is defined as which is a C ∞ 1-form on T M 0 . In the case of a Riemannian Finsler metric √ g, we have which is the so-called Liouville 1-form of the Riemannian metric g.
Let ω be the symplectic form on T M 0 obtained by pulling back the canonical symplectic form of T * M 0 via the Legendre transform, then we have ω = dA (see [11,Section 2]). Let X be the Hamiltonian vector field of E with respect to ω, i.e.
The Hamiltonian flow Φ generated by X preserves the map E and the symplectic form ω. The projections of the orbits of Φ to M are just the geodesics of (M, F ). The unit sphere bundle with respect to F is defined as The restriction of the Hamiltonian flow on S F M is said to be the geodesic flow of (M, F ), denoted by ϕ. Since ω = dA, ϕ integrates the Reeb field of the contact 1-form A | SF M .

Anosov flows and transversality.
Let us first recall some definitions: let N be a closed C ∞ manifold and ψ : N → N a C ∞ flow generated by the vector field X. We say that ψ is an Anosov flow if there exists a Dψ-invariant splitting of the tangent bundle a Riemannian metric on N , and two positive numbers a and b such that The vector bundles E ss and E su are called the strong stable and strong unstable distributions of ψ. They are both integrable to Hölder continuous foliations with C ∞ leaves, denoted respectively by F ss and F su . The vector bundles E ss ⊕ RX and E su ⊕ RX are called the stable and unstable distributions of ψ. They are also integrable to Hölder continuous foliations with C ∞ leaves, denoted respectively by F s and F u . For any x ∈ N , the leaves of F ss , F su , F s , and F u containing x are denoted respectively by W ss (x), W su (x), W s (x), and W u (x). Let (M, F ) be a closed C ∞ Finsler manifold of negative flag curvature (see [5]), it is well-known that its geodesic flow defined on S F M is Anosov [8]. See, for instance, [7] for a general study of Finsler geodesic flows in negative flag curvature. According to Theorem 2 and the comments concerning the Anosov hypothesis in [12], we have the following proposition: Proposition 1 ([12]). Let (M, F ) be a closed C ∞ Finsler manifold of dimension n and ϕ its geodesic flow defined on S F M . If ϕ is Anosov, then its stable foliation F s is transverse to the fibers of the sphere bundle π : S F M → M .
The proof of the following proposition is given only for the convenience of the reader since its arguments are known. Proof. Let v ∈ S F M and W s (v) the leaf of F s containing v. We deduce from the above proposition that the foliation F s is transverse to the fibers of the sphere bundle π : S F M → M . Since the fibers are compact, a result of C. Ehresmann (see [3]) implies that the map Therefore, W s (v) intersects each fiber of the bundle S F M → M at exactly one point.
Take v ∈ S F M such that W s (v) contains no lifts of ϕ-periodic orbits. It is well-known that W s (v) is diffeomorphic to R n . We deduce that M is diffeomorphic to R n . Let v ∈ S F M and W s (v) be the leaf containing v of the foliation F s . We denote by X the tangent vector field of the lifted flow ϕ over S F M . By Proposition 2, π sends W s (v) diffeomorphically onto M . We defineÊ ss = D π( E ss | W s (v) ) andX = D π( X | W s (v) ).
Let τ be the C ∞ flow on M generated byX. Therefore, for any t ∈ R, where | ξ | is calculated with respect to the lift of an arbitrarily chosen Riemannian metric on S F M .

Proposition 3.
There exist positive constants C and b such that for any x ∈ M and any u ∈Ê ss (x), we have that for any t ≥ 0, Vol. 117 (2021) On Finsler manifolds with hyperbolic geodesic flows 237 Proof. Since ϕ is Anosov, there exist a, b > 0 such that for any ξ ∈ E ss , for any t ≥ 0. Let ξ ∈ E ss such that d π(ξ) = u. We have . Now let us prove Theorem 1.1: we suppose that k ≥ 2, let α be a closed bounded C ∞ k-form on M . Since M is diffeomorphic to R n , α is exact. Let us prove that there exists a bounded 1-form β such that α = dβ. Let T be an arbitrary, fixed real number, and τ be the flow on M generated by the vector fieldX. Then we define Since α is closed, we have As we have seen above, π : W s (v) → M is a C ∞ diffeomorphism. Moreover, E ss = D π( E ss | W s (v) ) andX = D π( X | W s (v) ). We deduce that T M =Ê ss ⊕ R ·X.