Are geodesic flows on surfaces with negative curvature Anosov?

Question

I'm just going through the original book by Anosov, where he tries to proof this result. I don't quite understand it.

So let $\phi_t:TM\rightarrow TM$ be a geodesic flow on a compact surface $M$ of negative curvature. Let $\theta=(x,v)\in TM$, then there exists an isomorphism $T_\theta TM\cong T_pM\oplus T_pM$

So how can we find stable and unstable directions and show that vectors in this direction contract or expand exponentially, as it is required by an Anosov flow?

First of all, I understand that the direction of the flow itself is invariant and the phase velocity stays constant there. Then I have also understood that a vector $\xi=(\xi_h,\xi_ v)\in T_pM\oplus T_pM$, that is transversal to the flow direction will have a length that is convex for all $t$, i.e. \begin{equation} \frac{d^2}{dt^2}\Big|\;\xi\;\Big|^2>0 \end{equation}

But how do we conclude the existence of contracting and expanding directions from this?

What is your approach? Sometimes I have seen people using Jacobi fields. But I don't know how to do that?

Thanks in advance

btw. do you know how to show that the geodesic flow on $M$, as above, has no conjugate points?

Answer 1

I have some vague understanding as following.

First, The isomorphism $\phi_{v}: T_{\theta} TM \longrightarrow T_{p}M \oplus T_{p}M $ is a bridge identifing the splitting of tangent bundle with Jacobi field. Consider curve $\theta(s)=(x(s),v(s))$ in $TM$ with initial vaule $\theta(0)=(x,v)$. So $\frac{d\theta(0)}{ds} \in T_{\theta}TM$. On the other hand,let's denote $J(0):=\frac{dx(0)}{ds}\in T_{p}M$, $D_{t}J(0):=\frac{dv(0)}{ds}\in T_{v}TM \cong T_{p}M$ as the initial value and initial derivatie of a Jacobi field. Now travel $\frac{d\theta(0)}{ds} $ along the differential of the geodesic flow $Dg_{t}$. This yields an isomorphism $ \phi_{g_{t} v}$ taking $Dg_{t}\frac{d\theta(0)}{ds}$ to $(J(t), D_tJ(t))$ which is the value at $t$ of the Jacobi field alone the geodesic corresponding to $g_{t}$ with initial data $J(0), D_{t}J(0)$.

With this identification, it's possible to talk about contracting and expanding of $(J(t),D_tJ(t))$ along $Dg_t$ instead of subbundle of $T_{\theta} TM$. With some effort, one can construct a nice norm $|| \cdot ||$ on $T_{p}M\oplus T_{p}M $ and show $$\frac{\frac{d||J,D_{t}J||}{dt}}{||J,D_{t}J||}>C$$ for some cone of $(J,D_{t}J)$ and some constant $C$.

The key here is to replace $D_{tt}J$ that appears on the upper part of the fraction by curvature which has some restriction in this setting. This suffices to explain expanding. Contracting is similar for some other cone.

For more details, please refer to Katok's Introduction to the modern theory of dynamical systems.