Theorem of the three geodesics

In differential geometry the theorem of the three geodesics states that every Riemannian manifold with the topology of a sphere has three closed geodesics that form simple closed curves without self-intersections. The result can also be extended to quasigeodesics on a convex polyhedron.

This result stems from the mathematics of ocean navigation, where the surface of the earth can be modeled accurately by an ellipsoid, and from the study of the geodesics on an ellipsoid, the shortest paths for ships to travel. In particular, a nearly-spherical triaxial ellipsoid has only three simple closed geodesics, its equators. In 1905, Henri Poincaré conjectured that every smooth surface topologically equivalent to a sphere likewise contains at least three simple closed geodesics, and in 1929 Lazar Lyusternik and Lev Schnirelmann published a proof of the conjecture, which was later found to be flawed. The proof was repaired by Ballmann in 1978.

One proof of this conjecture examines the homology of the space of smooth curves on the sphere, and uses the curve-shortening flow to find a simple closed geodesic that represents each of the three nontrivial homology classes of this space.

More strongly, there necessarily exist three simple closed geodesics whose length is at most proportional to the diameter of the surface.

The number of closed geodesics of length at most L on a smooth topological sphere grows in proportion to L/log L, but not all such geodesics can be guaranteed to be simple.

On compact hyperbolic Riemann surfaces, there are infinitely many simple closed geodesics, but only finitely many with a given length bound. They are encoded analytically by the Selberg zeta function. The growth rate of the number of simple closed geodesics, as a function of their length, was investigated by Maryam Mirzakhani.

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