Theorem of the three geodesics

In differential geometry the theorem of the three geodesics, also known as Lyusternik–Schnirelmann theorem, states that every Riemannian manifold with the topology of a sphere has at least three simple closed geodesics (i.e. three embedded geodesic circles). The result can also be extended to quasigeodesics on a convex polyhedron, and to closed geodesics of reversible Finsler 2-spheres. The theorem is sharp: although every Riemannian 2-sphere contains infinitely many distinct closed geodesics, only three of them are guaranteed to have no self-intersections. For example, by a result of Morse if the lengths of three principal axes of an ellipsoid are distinct, but sufficiently close to each other, then the ellipsoid has only three simple closed geodesics.