Every Riemannian manifold with the topology of a sphere has at least three closed 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).[1] 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.[2]
^Poincaré, H. (1905), "Sur les lignes géodésiques des surfaces convexes" [Geodesics lines on convex surfaces], Transactions of the American Mathematical Society (in French), 6 (3): 237–274, doi:10.2307/1986219, JSTOR 1986219.
^Ballmann, W.: On the lengths of closed geodesics on convex surfaces. Invent. Math. 71, 593–597 (1983)
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