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 <math>\gamma:\mathbb R\rightarrow M</math> that is a geodesic for the metric ''g'' and is periodic.
Closed geodesics can be characterized by means of a variational principle. Denoting by <math>\Lambda M</math> the space of smooth 1-periodic curves on ''M'', closed geodesics of period 1 are precisely the critical points of the energy function <math>E:\Lambda M\rightarrow\mathbb R</math>, defined by
: <math>E(\gamma)=\int_0^1 g_{\gamma(t)}(\dot\gamma(t),\dot\gamma(t))\,\mathrm{d}t.</math>
If <math>\gamma</math> is a closed geodesic of period ''p'', the reparametrized curve <math>t\mapsto\gamma(pt)</math> is a closed geodesic of period 1, and therefore it is a critical point of ''E''. If <math>\gamma</math> is a critical point of ''E'', so are the reparametrized curves <math>\gamma^m</math>, for each <math>m\in\mathbb N</math>, defined by <math>\gamma^m(t):=\gamma(mt)</math>. Thus every closed geodesic on ''M'' gives rise to an infinite sequence of critical points of the energy ''E''.
==Examples== On the {{tmath|n}}-dimensional unit sphere with the standard metric, every geodesic – a great circle – is 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.
==See also== *Lyusternik–Fet theorem *Theorem of the three geodesics *Curve-shortening flow *Selberg trace formula *Selberg zeta function *Zoll surface
==References== {{Reflist}}
*Besse, A.: "Manifolds all of whose geodesics are closed", ''Ergebisse Grenzgeb. Math.'', no. 93, Springer, Berlin, 1978.
Category:Differential geometry Category:Dynamical systems Category:Geodesic (mathematics)