Closed geodesic

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. 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. In a Riemannian manifold (M,g), a closed geodesic is a curve γ : R → M {displaystyle gamma :mathbb {R} ightarrow 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 ightarrow mathbb {R} } , defined by If γ {displaystyle gamma } is a closed geodesic of period p, the reparametrized curve t ↦ γ ( p t ) {displaystyle tmapsto 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 min 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. On the unit sphere S n ⊂ R n + 1 {displaystyle S^{n}subset mathbb {R} ^{n+1}} with the standard round Riemannian metric, every great circle is an example of a closed geodesic. Thus, on the sphere, all geodesics are 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.