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 $TM$), we can consider the geodesic $\gamma_v(t)$ starting at $p$ at time $t=0$ with initial velocity $v$. This defines a map $\phi_t:TM \to TM$ by $\phi_t(v) = \dot{\gamma}_v(t)$, which is the derivative of the geodesic at time $t$.
This map $\phi_t$ is the geodesic flow. It's a flow on the tangent bundle $TM$ (considered as a differentiable manifold in its own right) which captures how geodesics on $M$ evolve over time. For each fixed $t$, the map $\phi_t:TM \to TM$ is a diffeomorphism.
Intuitively, if we think of each point in $TM$ 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 $TM$, 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.
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Author of the notes: Antonio J. Pan-Collantes
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