Geodesic flow

Example

Let $M$ be a complete Riemannian manifold with tangent bundle $TM$. Let $SM$ be its unit tangent bundle, the set of tangent vectors of length $1$. For $v \in SM$ let $\gamma_v$ be the geodesic with $\gamma_v(0)$ the base point of $v$ and $\gamma_v'(0) = v$. The geodesic flow is $\varphi_t(v) = \gamma_v'(t)$ for $t \in \R$. It is defined for all $t$ because $M$ is complete, and $\varphi_{s+t} = \varphi_s \circ \varphi_t$ because $t \mapsto \gamma_v(s + t)$ is the geodesic with initial vector $\gamma_v'(s)$. So it is a smooth dynamical system in continuous time on $SM$.

Check on the flat torus $M = \R^2/\Z^2$: geodesics are straight lines, so $SM = M \times S^1$ and $\varphi_t(x, v) = (x + t v, v)$. For a direction $v$ of irrational slope, the orbit of $(x, v)$ is dense in the torus $M \times \set{v}$. For a direction of rational slope, every orbit is periodic.