Definition
In a Riemannian manifold (M,g), a closed geodesic is a curve that is a geodesic for the metric g and is periodic.
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.
Read more about this topic: Closed Geodesic
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