# How to show that geodesics exist for all of time in a compact manifold?

Let $M$ be a compact manifold and the tangent bundle $TM$ have a Riemannian metric $g$ so that it is isomorphic to the cotangent bundle $T^*M$. Consider the pull-back of the canonical symplectic form $\omega_{std}$ to define a symplectic form $\omega_g$ on $TM.$ Assuming the following theorem:

The geodesics in $M$ are the images by the standard projection $$\pi: TM \to M$$ of the integral curves of the vector field $X_H$ where $H$ is the norm function squared in TM defined by the metric $g$

I want to show that the geodesics exist for all of time.

This implies that an ODE on a compact manifold (not quite your situation) admits a solution for all time: Cover by finitely many half-charts, get the bounds in terms of the coefficients and the sizes of the charts, and get a global $\epsilon > 0$ such that the ODE can always be solved for time $\epsilon$. Then it can be solved for all time just by "moving forward in the interval" and repeatedly solving for $\epsilon$ more time starting from the end of your previous solution.
Your situation is slightly different, because your manifold, $TM$, is not compact. But you can prove that your integral curves always stay the same length because they remain on a level set for $H$. So in fact if you start in the interior of a compact $B_R(TM)$ (consisting of the closed $R$ ball around $0$ in each tangent space) then you will remain there (and also remain at a fixed distance from the boundary of this). So the same proof as in the compact case works.
• The point in the compact case is just that there's a global $\epsilon$ such that you can solve for time $\epsilon$ from any starting point. Then if you solve for time $[0,\epsilon]$, start again at $t = \epsilon$ as your initial value and solve for $\epsilon$ more time. Now you have a solution on $[0,2\epsilon]$. And so on. The theorem sounds fancy but it's all just based on the ODE case in coordinates. I may have time to add more exposition later. I believe Hirsch Differential Topology discusses this sort of stuff. – aes Dec 14 '14 at 20:22