Geodesic flow - Knowlpedia

Definition

Let $(M,g)$ be a complete Riemannian manifold and $SM$ its unit tangent bundle. The geodesic flow is the one-parameter family

\varphi_t:SM\to SM,\qquad \varphi_t(x,v)=(\gamma_{x,v}(t),\dot\gamma_{x,v}(t)),

where $\gamma_{x,v}$ is the unique geodesic with initial data $(x,v)$ .

Flow law

Uniqueness for the geodesic equation gives $\varphi_0=\operatorname{id}$ and $\varphi_{t+s}=\varphi_t\circ\varphi_s$ . Completeness ensures that $\varphi_t$ is defined for all $t\in\mathbb R$ .

Negative curvature

On a compact manifold of strictly negative sectional curvature, the geodesic flow is an Anosov flow. Exponential stable and unstable behavior is the dynamical source of the fractal sets used in quantum-chaos uncertainty arguments.

References

Gabriel P. Paternain, Geodesic Flows, Birkhäuser, 1999. DOI record.