# Are geodesic flows on surfaces with negative curvature Anosov?

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. $$\frac{d^2}{dt^2}\Big|\;\xi\;\Big|^2>0$$

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?

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