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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?

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I am stuck on the same exact problem. Could you please post what you figured out, or tell me where to look? – Krishnan Mody Jun 12 '15 at 15:12
Hi Krishnan, sory for the late response. It was clear at the time, but now I am not in this field anymore and I can't recall the argument unfortunately. I can look at the work if i find it and let you know. – Marlo Jun 20 '15 at 11:37

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