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

Abstract

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 $\ge e^{-c}$. 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.

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.

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 $\delta >0$ such that for each $x\in P$

$$\begin{aligned} d_{P}(x, y)\le \frac{3}{2}d(x,y) \, \, \, \text {for all} \, \, \, y\in B^{P}_{\delta }(x), \end{aligned}$$

where $B^{P}_{\delta }(x)$ is the ball of center x and radius $\epsilon$ 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 function ${\mathcal {H}}:SP\times [0,r_p] \rightarrow {\mathbb {R}}$ defined by

$$\begin{aligned} {\mathcal {H}}((x,v),t) = {\left\{ \begin{array}{ll} \dfrac{d_{P}(\text {exp}^P_{x}\,tv, x)}{d(\text {exp}^P_{x}\, tv, x)}, &{} t\ne 0 \\ \, \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\ 1, &{} t=0, \end{array}\right. } \end{aligned}$$

where $\text {exp}^P_{x}$ denotes the exponential map of P.

It is easy to see that ${\mathcal {H}}$ is continuous for all ((x, v), t) with $t\ne 0$. We state that ${\mathcal {H}}$ is continuous at any point ((x, v), 0). To prove that, we used the compactness of P and the following claim:

Claim: For each $x\in P$ the function ${\mathcal {H}}_{x}:T^{1}_{x}P\times [0,r_P] \rightarrow {\mathbb {R}}$ defined by ${\mathcal {H}}_{x}(v,t):={\mathcal {H}}((x,v),t)$ is uniformly continuous function in $T^{1}_{x}P\times [0,r_P]$, where $T^{1}_{x}P$ is the set of unitary tangent vectors of P at x.

Proof of Claim

By definition of ${\mathcal {H}}_{x}$ and compactness of $T^{1}_{x}P\times [0,r_P]$, we only need to prove that $\displaystyle \lim \nolimits _{n\rightarrow +\infty }{\mathcal {H}}_{x}(v_n,t_n)={\mathcal {H}}_{x}(v,0)=1$, for any sequence $(v_n, t_n)\in T^{1}_{x}P\times [0,r_P]$, which converges to (v, 0). In fact: first note that if $(x,v)\in SP$ and $\alpha (t)=\text {exp}^{P}_{x}\,tv$, then from Lemma 4.1 we have that

$$\begin{aligned} \displaystyle \lim _{t\rightarrow 0^+}{\mathcal {H}}((x,v),t))=\displaystyle \lim _{t\rightarrow 0^+}\dfrac{\dfrac{d_P(\text {exp}^{P}_{x}\,tv, x)}{t}}{\dfrac{d(\text {exp}^{P}_{x}\,tv, x)}{t}}=\dfrac{\displaystyle \lim _{t\rightarrow 0^+}\dfrac{d_{P}(\alpha (t),\alpha (0))}{t}}{\displaystyle \lim _{t\rightarrow 0^+}\dfrac{d(\alpha (t),\alpha (0))}{t}}{=}\frac{||\alpha '(0)||_{P}}{||\alpha '(0)||}{=}1, \end{aligned}$$

since $||\alpha '(0)||_{P}$ is the norm with restrict metric of $P\subset Q$ which is equal to $||\alpha '(0)||$.

As ${\mathcal {H}}_{x}(v_n,0)=1$ then we can assume, without loss of generality, that $t_n\ne 0$ for all n.

For n large enough, consider the family of $C^1$-curves $\gamma _{n}(t) = \text {exp}^{P}_{x} (t_nv + t t_n(v_n - v))$, $t\in [0,1]$, and note that

$$\begin{aligned} d(\text {exp}^{P}_{x}\,t_nv, \text {exp}^{P}_{x}\,t_nv_n)\le & {} d_{P}(\text {exp}^{P}_{x}\,t_nv, \text {exp}^{P}_{x}\,t_nv_n) \le \displaystyle \int _{0}^{1} ||\gamma '_{n}(s)||_{P} \, ds \\\le & {} Kt_n||v_n-v||, \end{aligned}$$

where $K:=\displaystyle \sup \nolimits _{\{v\in T_{x}P: ||v||\le 2r_P\}} || D (\text {exp}^{P}_{x})_{v}||$. Hence, since $\displaystyle \lim \nolimits _{n \rightarrow +\infty }{\mathcal {H}}_{x}(v,t_n)=1$, then

$$\begin{aligned} d(\text {exp}^{P}_{x}\,t_nv_n, x)\ge & {} d(\text {exp}^{P}_{x}\,t_nv, x) - Kt_n||v_n-v||\\= & {} t_n\left( {\mathcal {H}}_{x}(v,t_n))^{-1}-K||v_n-v||\right) >0. \end{aligned}$$

In particular, we have

$$\begin{aligned} 1 \le {\mathcal {H}}_{x}(v_n,t_n)) = \dfrac{t_n}{d(\text {exp}^{P}_{x}\, t_nv_n, x)} \le \dfrac{1}{({\mathcal {H}}_{x}(v,t_n)))^{-1} - K||v_n-v||}. \end{aligned}$$

Therefore $\displaystyle \lim \nolimits _{ n \rightarrow + \infty }{\mathcal {H}}_{x}(v_n,t_n) = 1$ as desired. $\square$

The last claim and compactness of P allow us to conclude the continuity of ${\mathcal {H}}$ in every point ((x, v), 0) and consequently the uniformly continuity in $SP\times [0,r_P]$.

To conclude the proof of the lemma, from the uniformly continuity of ${\mathcal {H}}$, given $\epsilon =\frac{1}{2}$ there is $\delta$ such that

$$\begin{aligned} |t|=d(((x,v),t), ((x,v),0))<\delta \,\, \text {implies} \, \, |{\mathcal {H}}((x,v),t)-1|< \frac{1}{2}. \end{aligned}$$

The last inequality implies that

$$\begin{aligned} |t|< \delta \, \, \, \text {implies}\, \, \, d^s(\text {exp}^s_{x}\,tv, x) \le \frac{3}{2}d(\text {exp}_{x}\, tv, x). \end{aligned}$$

(A.1)

Consequently, if $\tilde{x}\in B^P_{\delta }(x)$, then there is $\tilde{t}$ with $|\tilde{t}|<\delta$, ${v}\in T^1_{x}P$ such that $\text {exp}^P_{x}\, \tilde{t}{v}=\tilde{x}$. Thus, (A.1) provides the result of the lemma. $\square$

Corollary A.2

In the same condition of the Lemma A.1, there is a constant $\Gamma >1$ such that

$$\begin{aligned} d_{P}(x, y)\le \Gamma \cdot d(x,y) \, \, \, \text {for all} \, \, \, x,y\in P. \end{aligned}$$

Proof

For each $x\in P$ consider the function $\Gamma (x)=\displaystyle \sup \nolimits _{y\ne x}\dfrac{d_{P}(x, y)}{d(x, y)}$. We state that there is $\Gamma >1$ such that $\Gamma (x)\le \Gamma$ for all $x\in P$. In fact, by contradiction assume that for each $n\in {\mathbb {N}}$ there is $x_n$ such that $\Gamma _{n}\ge n$. By definition of $\Gamma (x_n)$ there is $y_n\ne x_n$ with $\dfrac{d_{P}(x_n, y_n)}{d(x_n, y_n)}\ge n-1$. 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 $y_n\in B_{\delta }(x_n)$ and

$$\begin{aligned} n-1\le \dfrac{d_{P}(x_n, y_n)}{d(x_n, y_n)}\le \frac{3}{2}, \end{aligned}$$

which provides a contradiction. $\square$