# Closed geodesic

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

## Definition

In a Riemannian manifold (M,g), a closed geodesic is a curve $\gamma:\mathbb R\rightarrow M$ that is a geodesic for the metric g and is periodic.

Closed geodesics can be characterized by means of a variational principle. Denoting by $\Lambda M$ the space of smooth 1-periodic curves on M, closed geodesics of period 1 are precisely the critical points of the energy function $E:\Lambda M\rightarrow\mathbb R$, defined by

$E(\gamma)=\int_0^1 g_{\gamma(t)}(\dot\gamma(t),\dot\gamma(t))\,\mathrm{d}t.$

If $\gamma$ is a closed geodesic of period p, the reparametrized curve $t\mapsto\gamma(pt)$ is a closed geodesic of period 1, and therefore it is a critical point of E. If $\gamma$ is a critical point of E, so are the reparametrized curves $\gamma^m$, for each $m\in\mathbb N$, defined by $\gamma^m(t):=\gamma(mt)$. Thus every closed geodesic on M gives rise to an infinite sequence of critical points of the energy E.

## Examples

On the unit sphere $S^n\subset\mathbb R^{n+1}$ with the standard round Riemannian metric, every great circle is an example of a closed geodesic. Manifolds all of whose geodesics are closed have been thoroughly investigated in the mathematical literature. On a compact hyperbolic surface, whose fundamental group has no torsion, closed geodesics are in one-to-one correspondence with non-trivial conjugacy classes of elements in the Fuchsian group of the surface.