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 formula_1 that is a geodesic for the metric "g" and is periodic.
Closed geodesics can be characterized by means of a variational principle. Denoting by formula_2 the space of smooth 1-periodic curves on "M", closed geodesics of period 1 are precisely the critical points of the energy function formula_3, defined by
If formula_5 is a closed geodesic of period "p", the reparametrized curve formula_6 is a closed geodesic of period 1, and therefore it is a critical point of "E". If formula_5 is a critical point of "E", so are the reparametrized curves formula_8, for each formula_9, defined by formula_10. Thus every closed geodesic on "M" gives rise to an infinite sequence of critical points of the energy "E".
Examples.
On the unit sphere formula_11 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.