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 γ : R → M {\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 Λ M {\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 E : Λ M → R {\displaystyle E:\Lambda M\rightarrow \mathbb {R} } , defined by
E ( γ ) = ∫ 0 1 g γ ( t ) ( γ ˙ ( t ) , γ ˙ ( t ) ) d t . {\displaystyle E(\gamma )=\int _{0}^{1}g_{\gamma (t)}({\dot {\gamma }}(t),{\dot {\gamma }}(t))\,\mathrm {d} t.}
If γ {\displaystyle \gamma } is a closed geodesic of period p, the reparametrized curve t ↦ γ ( p t ) {\displaystyle t\mapsto \gamma (pt)} is a closed geodesic of period 1, and therefore it is a critical point of E. If γ {\displaystyle \gamma } is a critical point of E, so are the reparametrized curves γ m {\displaystyle \gamma ^{m}} , for each m ∈ N {\displaystyle m\in \mathbb {N} } , defined by γ m ( t ) := γ ( m t ) {\displaystyle \gamma ^{m}(t):=\gamma (mt)} . Thus every closed geodesic on M gives rise to an infinite sequence of critical points of the energy E.
Examples On the n {\displaystyle 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
Besse, A.: "Manifolds all of whose geodesics are closed", Ergebisse Grenzgeb. Math., no. 93, Springer, Berlin, 1978.