is the geodesic flow on Hyperbolic Plane completely integrable?

Question

I'm looking for examples of completely integrable systems and specifically geodesic flows. We remember that when we have a symplectic manifold $(M,\omega)$ (with $M$ of dimension $2n$) and $H:M\rightarrow\mathbb{R}$ a smooth function, its symplectic gradient is the unique field $X_H$ over $M$ satisfying

$$\textrm{d}H=\omega(X_H,\cdot)$$

and we say that the system $(M,\omega,H)$ is completely integrable is there exists $f_1,\ldots,f_{n-1}:M\rightarrow\mathbb{R}$ smooth functions Poisson commuting: $\{f_i,f_j\}=\{f_k,H\}=0$, where $\{f,g\}=\omega(X_f,X_g)$, and with $\textrm{d}f_1,\ldots,\textrm{d}f_{n-1},\textrm{d}H$ linearly independent in a dense set of $M$.

In the cotangent bundle $T^*M$ of a manifold $M$, there exists a canonical symplectic form,

$$\omega_\textrm{can}=\sum{\textrm{d}x_i}\wedge\textrm{d}\xi_i$$

$(x_1,\ldots,x_n,\xi_1,\ldots,\xi_n)$ local coordinates of $T^\star M$. Then, if we consider a riemannian manifold $(M,g)$, we can canonically define a symplectic form on $TM$ with the bundle isomorphism $\Phi:TM\rightarrow T^\star M$ given by

$$\Phi(p,v)=(p,v^\star)$$

where $v^*(w)=g(v,w)$ is the Riesz representation of a functional. Hence we can define $\omega=\Phi^{\star}\omega_\textrm{can}$. In this way, the geodesic flow can be viewed as the flow of the symplectic gradient of the metric hamiltonian $H(p,v)=\frac{1}{2}g_p(v,v)$. Then my question is if the geodesic flow on the tangent bundle of the hyperbolic plane is completely integrable and, if yes, what is the another function beside the metric hamiltonian $H$. Any help will be appreciated.

Answer 1

Yes, the geodesic flow on the hyperbolic plane and, in fact, on any Hadamard manifold (${\mathbb R}^n$ provided with a complete Riemannian metric with non-positive curvature) is integrable.

You can easily construct integrals of motion for the geodesic flow in hyperbolic space as follows:

1. Consider the Cayley-Klein model where the hyperbolic space is the interior of the unit ball in ${\mathbb R}^n$ and (oriented) geodesics are (oriented) straight lines.

2. The space of geodesics is then the space of oriented lines passing through the unit ball and this is diffeomorphic to the unit codisc bundle of the standard Riemannian metric on $S^{n-1}$. In particular, the space of geodesics is a smooth manifold.

3. Consider the canonical projection $\pi$ that associates to a unit covector $\xi \in T^*{\mathbb H}^n$ the oriented geodesic that has $\xi$ as initial condition.

4. The pull-back by $\pi$ of any smooth function on the space of geodesics (suitably extended as a homogeneous function) will be an integral of motion.

The key idea in this is that if a Riemannian or Finsler metric is such that its space of geodesics is a manifold, then the geodesic flow is integrable. This has been worked out in detail by Carlos E. Duran in

On the complete integrability of the geodesic flow of manifolds all of whose geodesics are closed Ergodic Theory and Dynamical Systems (1997), 17 : pp 1359-1370 1997.

He considers the compact case, but he also works out the case where the space of geodesics is an orbifold (like lens spaces).

Second answer.

If you are only interested in the hyperbolic plane, then just use its symmetries (i.e., the group $SL(2,{\mathbb R})$) and the classic theorem of Noether that relates one-parameter groups of symmetries with integrals of motion.