Closed geodesic

In differential geometry and dynamical systems, a closed geodesic on a Riemannian manifold is a geodesic that returns to its starting point with the same tangent direction. It may be formalized as the projection of a closed orbit of the geodesic flow on the tangent space of the manifold.

Definition

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

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

${\displaystyle E(\gamma )=\int _{0}^{1}g_{\gamma (t)}({\dot {\gamma }}(t),{\dot {\gamma }}(t))\,\mathrm {d} t.}$

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

Examples

On the unit sphere ${\displaystyle S^{n}\subset \mathbb {R} ^{n+1}}$ with the standard round Riemannian metric, every great circle is an example of a closed geodesic. Thus, on the sphere, all geodesics are closed. On a smooth surface topologically equivalent to the sphere, this may not be true, but there are always at least three simple closed geodesics; this is the theorem of the three geodesics.[1] 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.