Closed geodesic

In differential geometry and dynamical systems, a closed geodesic on a Riemannian manifold M is the projection of a closed orbit of the geodesic flow on M.

Examples

On the unit sphere, every great circle is an example of a closed geodesic. On a compact hyperbolic surface, closed geodesics are in one-to-one correspondence with non-trivial conjugacy classes of elements in the Fuchsian group of the surface. A prime geodesic is an example of a closed geodesic.

Definition

Geodesic flow is an $\mathbb R$-action on tangent bundle T(M) of a manifold M defined in the following way

$G^t(V)=\dot\gamma_V(t)$

where $t\in \mathbb R$, $V\in T(M)$ and γ_V denotes the geodesic with initial data $\dot\gamma_V(0)=V$.

It defines a Hamiltonian flow on (co)tangent bundle with the (pseudo-)Riemannian metric as the Hamiltonian. In particular it preserves the (pseudo-)Riemannian metric g, i.e.

$g(G^t(V),G^t(V))=g(V,V).\,$

That makes possible to define geodesic flow on unit tangent bundle UT(M) of the Riemannian manifold M when the geodesic γ_V is of unit speed.

See also

• Selberg trace formula
• Zoll surface
• geodesic

References

• Besse, A.: "Manifolds all of whose geodesics are closed", Ergebisse Grenzgeb. Math., no. 93, Springer, Berlin, 1978.