# On Finsler manifolds with hyperbolic geodesic flows

Yong Fang

► To cite this version:

Yong Fang. On Finsler manifolds with hyperbolic geodesic flows. Archiv der Mathematik, 2021, 117, pp.233 - 239. 10.1007/s00013-021-01607-y. hal-03819368

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# On Finsler manifolds with hyperbolic geodesic flows

April 5, 2021

## Abstract

Let $(M, F)$ be a closed $C^\infty$ Finsler manifold and $\varphi$ its geodesic flow. In the case that $\varphi$ is Anosov, we extend to the Finsler setting a Riemannian vanishing result of M. Gromov about the $L^\infty$-cohomology.

*Mathematics Subject Classification.* 37A35, 34D20, 37D35.

*Keywords.* Finsler metric, Kähler hyperbolicity, Anosov flow.

## 1 Introduction

Let $M$ be a closed $C^\infty$ manifold of dimension $n$ and $TM = TM - \{0\}$. A Finsler metric on $M$ is a function $F : TM \to \mathbb{R}^+$ satisfying:

(a) $F(tv) = tF(v)$ for any $v \in TM$ and $t > 0$;

(b) $F$ is strictly positive and $C^\infty$ on $TM_0$; and

(c) in standard local coordinates $(x_i, y_i)$ on $TM$, the matrix $(\frac{\partial^2 F^2}{\partial y_i \partial y_j})_{1\le i,j\le n}$ is positively definite. For simplicity, we are denoting $F(v)$ instead of $F(x, v)$, where $(x, v) \in TM$.

A Finsler metric $F$ on $M$ is said to be *Riemannian* if for any $x \in M$, there exists an inner product $g_x$ on $T_x M$ such that $F_x^2(v) = g_x(v, v) \quad \forall v \in T_x M$. There exist lots of non-Riemannian Finsler metrics, for example the so-called Randers metrics (see [5]). Given a Finsler manifold $(M, F)$, let $c : [a, b] \to M$ be a piecewise $C^\infty$ curve in $M$. Since $F$ is positively homogeneous, the length of $c$ is well-defined as $L(c) = \int_a^b F(c'(t))dt$. For any $p, q \in M$, we define $d(p, q) = \inf_c L(c)$, where the infimum is taken over all piecewise $C^\infty$ curves $c$ issuing from $p$ to $q$. The geodesics of $(M, F)$ are characterized as the constant speed curves minimizing locally the length, which are then necessarily $C^\infty$ curves. The Finsler metric $F$ is said to be *reversible* if $F(-v) = F(v)$ for any $v \in TM$. In this article, we shall consider general Finsler metrics without the reversibility assumption. So generally speaking,
for any $p, q \in M$, 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]).

M. Gromov showed in [10] that if $M$ is a closed Riemannian manifold of negative sectional curvature, then every closed bounded form of degree $\ge 2$ on $\tilde{M}$ has a bounded primitive. This means that the $L^\infty$-cohomology of $\tilde{M}$ vanishes in degree $\ge 2$.

Given a closed Riemannian manifold $(M, g)$ of negative curvature, it is well-known 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, X. Cheng extended in [4] the vanishing result above to closed Riemannian manifolds with Anosov geodesic flows. In [2], K. Burns and G. 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:

**Theorem 1.1.** If $M$ is a closed Finsler manifold with Anosov geodesic flow, then every closed bounded form of degree $\ge 2$ on $\tilde{M}$ has a bounded primitive.

The vanishing of the $L^\infty$-cohomology in degree $\ge 2$ is closely related to the following Riemannian question of S. S. Chern: let $M$ be a $2m$-dimensional closed Riemannian manifold of negative sectional curvature. Is it true that the signals of the Euler number $\chi(M)$ and $(-1)^m$ are the same, i.e. $(-1)^m \cdot \chi(M) > 0$ ? M. Gromov proved in [10] this conjecture in the Kähler hyperbolic case.

**Definition 1.2.** A closed Kähler manifold $(M, g, J)$ is said to be *Kähler hyperbolic* if the lifted Kähler form $\tilde{\omega}$ to the universal covering space $\tilde{M}$ admits a bounded primitive.

More precisely, M. Gromov proved the following result:

**Theorem 1.3.** (M. Gromov [10]) Let $M$ be a $2m$-dimensional closed Kähler manifold. If $M$ is Kähler hyperbolic, then the Euler number $\chi(M)$ does not vanish and satisfies $(-1)^m \cdot \chi(M) > 0$.

Thus he obtained the following result: if $M$ is a $2m$-dimensional closed $C^\infty$ Riemannian manifold of negative sectional curvature, and $M$ is homotopy equivalent to a closed Kähler manifold, then the Euler number $\chi(M)$
## 2 SYMPLECTIC FORMULATION OF FINSLER GEODESIC FLOWS

does not vanish and satisfies $(-1)^m \cdot \chi(M) > 0$. We deduce from Theorems 1.1 and 1.3 the following:

**Corollary 1.** Let $M$ be a $2m$-dimensional closed $C^\infty$ Finsler manifold with Anosov geodesic flow. If $M$ is homotopy equivalent to a closed Kähler manifold, then the Euler number $\chi(M)$ does not vanish and satisfies $(-1)^m \cdot \chi(M) > 0$.

# 2 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 $C^\infty$ Finsler manifold and $\pi: TM_0 \to M$ be the canonical projection. The potential of $(M, F)$ is defined as

$$A_v(\eta) = \frac{1}{2} \frac{d}{dt} (F^2(v + d\pi_v(\eta))) \Big|_{t=0}, \quad \forall v \in TM_0, \eta \in T_v(TM_0),$$

which is a $C^\infty$ 1-form on $TM_0$. In the case of a Riemannian Finsler metric $\sqrt{g}$, we have

$$A_v(\eta) = g(d\pi_v(\eta), v),$$

which is the so-called Liouville 1-form of the Riemannian metric $g$.

Let $E: TM_0 \to \mathbb{R}$ be defined by $E(v) = \frac{1}{2}(F(v))^2$. Let $\omega$ be the symplectic form on $TM_0$ obtained by pulling back the canonical symplectic form of $T^*M_0$ via the Legendre transform, then we have $\omega = dA$ (see [11, Section 2]). Let $X$ be the Hamiltonian vector field of $E$ with respect to $\omega$, i.e.

$$i_X \omega = -dE.$$

The Hamiltonian flow $\Phi$ generated by $X$ preserves the map $E$ and the symplectic form $\omega$. The projections of the orbits of $\Phi$ to $M$ are just the geodesics of $(M, F)$. The unit sphere bundle with respect to $F$ is defined as

$$S_F M = E^{-1}(\frac{1}{2}) = \{v \in TM \mid F(v) = 1\}.$$

The restriction of the Hamiltonian flow on $S_F M$ is said to be the geodesic flow of $(M, F)$, denoted by $\varphi$. Since $\omega = dA$, $\varphi$ integrates the Reeb field of the contact 1-form $A|_{S_F M}$.
## 3 Anosov flows and transversality

Let us first recall some definitions: let $N$ be a closed $C^\infty$ manifold and $\psi: N \to N$ a $C^\infty$ flow generated by the vector field $X$. We say that $\psi$ is an *Anosov flow* if there exists a $D\psi$-invariant splitting of the tangent bundle

$$TN = E^{ss} \oplus \mathbb{R}X \oplus E^{su},$$

a Riemannian metric on $N$, and two positive numbers $a$ and $b$ such that

$$\| D\psi_t(v) \| \le a \cdot e^{-bt} \| v \| \quad \forall v \in E^{ss}, t \ge 0,$$

$$\| D\psi_{-t}(v) \| \le a \cdot e^{-bt} \| v \| \quad \forall v \in E^{su}, t \ge 0.$$

The vector bundles $E^{ss}$ and $E^{su}$ are called the *strong stable* and *strong unstable* distributions of $\psi$. They are both integrable to Hölder continuous foliations with $C^\infty$ leaves, denoted respectively by $\mathcal{F}^{ss}$ and $\mathcal{F}^{su}$. The vector bundles $E^{ss} \oplus \mathbb{R}X$ and $E^{su} \oplus \mathbb{R}X$ are called the *stable* and *unstable* distributions of $\psi$. They are also integrable to Hölder continuous foliations with $C^\infty$ leaves, denoted respectively by $\mathcal{F}^s$ and $\mathcal{F}^u$. For any $x \in N$, the leaves of $\mathcal{F}^{ss}$, $\mathcal{F}^{su}$, $\mathcal{F}^s$, and $\mathcal{F}^u$ 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 $C^\infty$ 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 $C^\infty$ Finsler manifold of dimension $n$ and $\varphi$ its geodesic flow defined on $S_F M$. If $\varphi$ is Anosov, then its stable foliation $\mathcal{F}^s$ is transverse to the fibers of the sphere bundle $\pi: S_F M \to M$.

The proof of the following proposition is given only for the convenience of the reader, since its arguments are known.

**Proposition 2.** Consider the *lifted stable foliation* $\tilde{\mathcal{F}}^s$ on the covering space $S_{\tilde{F}} \tilde{M}$. If the geodesic flow $\varphi$ is Anosov, then each leaf of $\tilde{\mathcal{F}}^s$ intersects each fiber of the sphere bundle $\tilde{\pi}: S_{\tilde{F}} \tilde{M} \to \tilde{M}$ exactly once. In addition, $\tilde{M}$ is *diffeomorphic to* $\mathbb{R}^n$, where $n = \dim(M)$.

*Proof.* Let $v \in S_{\tilde{F}} \tilde{M}$ and $\tilde{W}^s(v)$ the leaf of $\tilde{\mathcal{F}}^s$ containing $v$. We deduce from the above proposition that the foliation $\tilde{\mathcal{F}}^s$ is transverse to the fibers
of the sphere bundle $\tilde{\pi}: S_{\tilde{F}}\tilde{M} \to \tilde{M}$. Since the fibers are compact, a result of C. Ehresmann (see [3]) implies that the map

$$ \tilde{\pi}|_{\widetilde{W}^s(v)}: \widetilde{W}^s(v) \to \tilde{M} $$

is a covering map. Since $\tilde{M}$ is simply connected, $\tilde{\pi}|_{\widetilde{W}^s(v)}$ is a diffeomorphism. Therefore, $\widetilde{W}^s(v)$ intersects each fiber of the bundle $S_{\tilde{F}}\tilde{M} \to \tilde{M}$ at exactly one point.

Take $v \in S_{\tilde{F}}\tilde{M}$ such that $\widetilde{W}^s(v)$ contains no lifts of $\varphi$-periodic orbits. It is well-known that $\widetilde{W}^s(v)$ is diffeomorphic to $\mathbb{R}^n$. We deduce that $\tilde{M}$ is diffeomorphic to $\mathbb{R}^n$. $\square$

## 4 Proof of Theorem 1.1

Let $(M, F)$ be a closed $C^\infty$ Finsler manifold with Anosov geodesic flow $\varphi$. Let $v \in S_{\tilde{F}}\tilde{M}$ and $\widetilde{W}^s(v)$ be the leaf containing $v$ of the foliation $\tilde{\mathcal{F}}^s$. We denote by $\tilde{X}$ the tangent vector field of the lifted flow $\tilde{\varphi}$ over $S_{\tilde{F}}\tilde{M}$. By Proposition 2, $\tilde{\pi}$ sends $\widetilde{W}^s(v)$ diffeomorphically onto $\tilde{M}$. We define

$$ \hat{E}^{ss} = D\tilde{\pi}(\tilde{E}^{ss}|_{\widetilde{W}^s(v)}) \text{ and } \hat{X} = D\tilde{\pi}(\tilde{X}|_{\widetilde{W}^s(v)}). $$

Let $\tau$ be the $C^\infty$ flow on $\tilde{M}$ generated by $\hat{X}$. Therefore, for any $t \in \mathbb{R}$,

$$ \tau_t = \tilde{\pi} \circ \tilde{\varphi}_t \circ \tilde{\pi}^{-1} $$

and $\hat{E}^{ss}$ is $\tau$-invariant. Since by Proposition 1 the distribution $E^s$ is transverse to the vertical bundle $V(S_F M)$, there exists $C_1 > 0$ such that for any $\xi \in \hat{E}^s(v)$,

$$ \frac{1}{C_1} \tilde{F}(D\tilde{\pi}(\xi)) \le |\xi| \le C_1 \tilde{F}(D\tilde{\pi}(\xi)), $$

where $|\xi|$ 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 $x \in \tilde{M}$ and any $u \in \hat{E}^{ss}(x)$, we have that for any $t \ge 0$,

$$ \tilde{F}(D\tau_t(u)) \le C e^{-bt} \tilde{F}(u). $$
Proof. Since $\varphi$ is Anosov, there exist $a, b > 0$ such that for any $\xi \in \tilde{E}^{ss}$,

$$|D\tilde{\varphi}_t(\xi)| \leq ae^{-bt} |\xi|$$

for any $t \geq 0$. Let $\xi \in \tilde{E}^{ss}$ such that $d\tilde{\pi}(\xi) = u$. We have

$$\begin{aligned} \tilde{F}(D\tau_t(u)) &= \tilde{F}(D\tilde{\pi}(D\tilde{\varphi}_t(\xi))) \leq C_1 |D\tilde{\varphi}_t(\xi)| \\ &\leq C_1 ae^{-bt} |\xi| \leq C_1^2 ae^{-bt} \tilde{F}(u) \end{aligned}$$

for any $t \geq 0$. We set $C = C_1^2 a$. $\square$

<figure><img src="image_2.png" /></figure>

**Definition 4.1.** A $C^\infty$ k-form $\alpha$ on $\tilde{M}$ is said to be *bounded* if

$$\| \alpha \|_{\infty} = \sup_{x \in \tilde{M}} \| \alpha \|_{x} < +\infty,$$

where $\| \alpha \|_x = \sup_{e_1, \dots, e_k \in T_x \tilde{M} \setminus \{0\}} \frac{|\alpha(e_1, \dots, e_k)|}{\tilde{F}(e_1) \dots \tilde{F}(e_k)}$.

Now let us prove Theorem 1.1: we suppose that $k \geq 2$, let $\alpha$ be a closed bounded $C^\infty$ k-form on $\tilde{M}$. Since $\tilde{M}$ is diffeomorphic to $\mathbb{R}^n$, $\alpha$ is exact. Let us prove that there exists a bounded 1-form $\beta$ such that $\alpha = d\beta$. Let $T$ be an arbitrary, fixed real number, and $\tau$ be the flow on $\tilde{M}$ generated by the vector field $\hat{X}$. Then we define

$$\beta_T = - \int_0^T \tau_s^*(i_{\hat{X}} \alpha) ds.$$

Since $\alpha$ is closed, we have

$$\alpha - \tau_T^* \alpha = - \int_0^T \frac{d}{ds} \tau_s^* \alpha ds = - \int_0^T \tau_s^* \mathcal{L}_{\hat{X}} \alpha ds = - \int_0^T \tau_s^* d i_{\hat{X}} \alpha ds = d \beta_T.$$

As we have seen above, $\tilde{\pi} : \tilde{W}^s(v) \to \tilde{M}$ is a $C^\infty$ diffeomorphism. Moreover, $\hat{E}^{ss} = D\tilde{\pi}(\hat{E}^{ss}|_{\tilde{W}^s(v)})$ and $\hat{X} = D\tilde{\pi}(\hat{X}|_{\tilde{W}^s(v)})$. We deduce that

$$T\tilde{M} = \hat{E}^{ss} \oplus \mathbb{R} \cdot \hat{X}.$$

Now let $x \in \tilde{M}$ and $w \in T_x \tilde{M}$. Thus there exist $w' \in \hat{E}^{ss}(x)$ and $w''$ colinear with $\hat{X}(x)$ such that $w = w' + w''$. Since the distribution $E^{ss}$ on $S_F M$ is transverse to the vector field $X$, there exists a positive constant $C_2$ such that

$$\tilde{F}(w') \leq C_2 \tilde{F}(w).$$
Therefore, for any $w_1, \dots, w_{k-1} \in T_x \tilde{M}$, we have, by Proposition 3, the
following:

$$
\begin{align*}
| (\tau_s^*(i_{\hat{X}}\alpha))_x(w_1, \dots, w_{k-1}) | &= | \alpha_{\tau_s(x)}(\hat{X}(\tau_s(x)), D_x\tau_s(w_1), \dots, D_x\tau_s(w_{k-1})) | \\
&= | \alpha_{\tau_s(x)}(\hat{X}(\tau_s(x)), D_x\tau_s(w'_1), \dots, D_x\tau_s(w'_{k-1})) | \\
&\le \| \alpha \|_{\infty} \tilde{F}(\hat{X}(\tau_s(x))) \tilde{F}(D_x\tau_s(w'_1)) \dots \tilde{F}(D_x\tau_s(w'_{k-1})) \\
&\le \| \alpha \|_{\infty} C_1 \| X \|_{\infty} C^{k-1} e^{-(k-1)bs} \tilde{F}(w'_1) \dots \tilde{F}(w'_{k-1}) \\
&\le \| \alpha \|_{\infty} C_1 \| X \|_{\infty} C^{k-1} e^{-(k-1)bs} C_2^{k-1} \tilde{F}(w_1) \dots \tilde{F}(w_{k-1}),
\end{align*}
$$

where $\| X \|_{\infty}$ is calculated with respect to an arbitrarily chosen Riemannian
metric on $S_F M$, and we are considering, without loss of generality, that all
$k-1$ terms of type $C_2$ concerning $w'_i$ are equal, $i = 1, \dots, k-1$.

Hence the form $\beta_T$ converges as $T \to +\infty$ to a form $\beta$ such that

$$
| \beta_x(w_1, \dots, w_{k-1}) | \le \frac{\| \alpha \|_{\infty} \| X \|_{\infty} C_1 C^{k-1} C_2^{k-1}}{(k-1)b} \tilde{F}(w_1) \dots \tilde{F}(w_{k-1}),
$$

thus $\beta$ is bounded. Morever, since $\tau_T^*\alpha$ tends to zero, $\alpha = d\beta$. The proof of Theorem 1.1 is complete.

The proof of Corollary 1 is given for the convenience of the reader (see
[4]): suppose that $\Psi : M \to M_1$ is a homotopy equivalence of $M$ to a closed
Kähler manifold $M_1$. By an approximation if necessary, we can assume that
$\Psi$ is $C^\infty$. Let $\pi : \tilde{M} \to M$ and $\pi_1 : \tilde{M}_1 \to M_1$ be the universal covering maps.
Let $\omega$ be the Kähler form of $M_1$, which is certainly bounded. Therefore, with
respect to the lifted Finsler metric $\tilde{F}$, the pull back $(\Psi \circ \pi)^*(\omega) = \pi^*(\Psi^*(\omega))$
is also a bounded 2-form on $\tilde{M}$. Thus by Theorem 1.1, there exists a bounded
1-form $\beta$ on $\tilde{M}$ such that

$$
(\Psi \circ \pi)^*(\omega) = d\beta.
$$

Then we can deduce as in [6] that $\pi_1^*(\omega)$ also admits a bounded primitive, i.e.
$M_1$ is Kähler hyperbolic. So by Theorem 1.3, we have $(-1)^m \cdot \chi(M_1) > 0$.
Since the Euler number is a homotopy invariant, we get $(-1)^m \cdot \chi(M) > 0$.

**Acknowledgement:** I am grateful to the referee for valuable comments and suggestions.
References

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Yong Fang

Département de Mathématiques, CY Cergy Paris Université,
2, avenue Adolphe Chauvin 95302 Cergy-Pontoise Cedex, France
(e-mail : yfang@cyu.fr)