Geodesic flow

The geodesic flow on a Riemannian manifold is indeed a flow on the tangent bundle of the manifold, defined by the geodesics of the manifold.

More formally, given a Riemannian manifold $(M, g)$ , for each tangent vector $v$ at a point $p$ in $M$ (that is, for each point in the tangent bundle $T M$ ), we can consider the geodesic $γ_{v} (t)$ starting at $p$ at time $t = 0$ with initial velocity $v$ . This defines a map $ϕ_{t} : T M \to T M$ by $ϕ_{t} (v) = {\dot{γ}}_{v} (t)$ , which is the derivative of the geodesic at time $t$ .

This map $ϕ_{t}$ is the geodesic flow. It's a flow on the tangent bundle $T M$ (considered as a differentiable manifold in its own right) which captures how geodesics on $M$ evolve over time. For each fixed $t$ , the map $ϕ_{t} : T M \to T M$ is a diffeomorphism.

Intuitively, if we think of each point in $T M$ as a "state" which consists of a position on $M$ and a velocity vector, then the geodesic flow describes the "dynamics" on $M$ where each state moves along the geodesic determined by its position and velocity.

Note that this flow is generated by a certain vector field on $T M$ , called the geodesic spray. This vector field encodes the second order differential equation that geodesics satisfy (the geodesic equation).

An important thing to remember is that the geodesic flow is a global concept that requires the entire manifold structure to define, while the geodesic equation is a local differential equation defined using the metric in a neighborhood of each point.