Geodesic Flow is an Anosov Flow

Question

I am trying to understand why geodesic flow on a compact surface of constant negative curvature is an Anosov flow. Klingenberg's book, Riemannian Geometry, says that in this case, the proof is very short: the Jacobi fields are of the form $\xi_s(t) = e^{-tk}\xi_s(0); \xi_u(t) = e^{tk}\xi_u(0)$ where $-k^2$ is the curvature of the surface. Can someone explain how precisely to identify these Jacobi fields with the expanding and contracting directions of an Anosov flow? Thank you!

Answer 1

At a tangent vector $v \in T_p M$, the tangent space $T_v (TM)$ to the tangent bundle decomposes as a direct sum $T_p M \oplus T_p M$ by sending a vector $X \in T_v TM$ to its horizontal and vertical components. (This decomposition uses the connection on $M$.) I assume you're familiar with this idea. This is an orthogonal decomposition under the Sasaki metric on the tangent bundle (this is in fact one definition of the Sasaki metric), so one can use it efficiently to compute norms.

Similarly, if $SM$ is the unit tangent bundle of $M$, and $v \in SM$, the tangent space $T_v SM$ decomposes into horizontal and vertical components as $T_p M \oplus v^{\perp}$, where $v^{\perp} \subset T_p M$ is the subspace orthogonal to $v$.

Let $\varphi_t : SM \to SM$ denote geodesic flow. The derivative $d\varphi_t : T_v SM \to T_{\varphi_t(v)} SM$ has a nice description in terms of the above decomposition. Namely, suppose $X \in T_v SM$ decomposes as $(X_H, X_V)$, where $X_H$ and $X_V$ are the horizontal and vertical components, respectively. Denote by $\gamma_v$ the geodesic through $v$ (with $\gamma(0) = p$, the footpoint of $v$). Then we can consider the Jacobi field $J$ along $\gamma_v$ defined by the initial conditions $J(0) = X_H, J'(0) = X_V$. Then $d\varphi_t$ takes $X$ to the vector whose decomposition is $J(t), J'(t)$, or, in other words:

Proposition. Let $w = \varphi_t(v)$, and denote by $p, q$ the footpoints of $v, w$, respectively. Then as a map $$ d\varphi_t : S_p M \oplus v^{\perp} \to S_q M \oplus w^{\perp}, $$ $d\varphi_t$ is uniquely defined by the condition that for any Jacobi field $J$ along $\gamma_v$ (with $J'(0) \in v^{\perp}$), we have $$ d\varphi_t\Big(J(0), J'(0)\Big) = \Big(J(t), J'(t)\Big). $$ This theorem is basically obvious when you work through the definitions. Namely, one takes a path in $TM$ representing the tangent vector $\big(J(0), J'(0)\big)$; this gives rise (by geodesic flow) to a variation of geodesics of $\gamma_v$, and one checks that under our identifications $J$ is in fact the variation field of this variation. The claim follows.

With this in mind, it's easy to use the above description to obtain the Anosov decomposition. Namely, the unstable subspace of $T_v SM$ is spanned by pairs $\big(\xi_u(0), \xi_u'(0)\big)$ and the stable by pairs $\big(\xi_s(0), \xi_s'(0)\big)$. That these are in fact stable/unstable subspaces follows by direct computation.