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.
Closed geodesics can be characterized by means of a variational principle. Denoting by the space of smooth 1-periodic curves on M, closed geodesics of period 1 are precisely the critical points of the energy function , defined by
If is a closed geodesic of period p, the reparametrized curve is a closed geodesic of period 1, and therefore it is a critical point of E. If is a critical point of E, so are the reparametrized curves , for each , defined by . Thus every closed geodesic on M gives rise to an infinite sequence of critical points of the energy E.
On the unit sphere 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. 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.