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.
the space of smooth 1-periodic curves on M, closed geodesics of period 1 are precisely the critical points of the energy function
Thus every closed geodesic on M gives rise to an infinite sequence of critical points of the energy E. On the
-dimensional unit sphere with the standard metric, every geodesic – a great circle – is closed.
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.