Closed geodesic

In differential geometry and dynamical systems, a closed geodesic on a Riemannian manifold is a geodesic that forms a simple closed curve. It may be formalized as the projection of a closed orbit of the geodesic flow on the tangent space of the manifold.

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

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