Theorem of the three geodesics

[3] In 1905, Henri Poincaré conjectured that every smooth surface topologically equivalent to a sphere likewise contains at least three simple closed geodesics,[4] and in 1929 Lazar Lyusternik and Lev Schnirelmann published a proof of the conjecture; while the general topological argument of the proof was correct, it employed a deformation result that was later found to be flawed.

A universally accepted solution was provided in the 1980s by Grayson, following a suggestion of Karen Uhlenbeck, by means of the curve shortening flow.

[8] On compact hyperbolic Riemann surfaces, there are infinitely many simple closed geodesics, but only finitely many with a given length bound.

The growth rate of the number of simple closed geodesics, as a function of their length, was investigated by Maryam Mirzakhani.

In particular, a simple closed geodesic of a convex polyhedron would necessarily bisect the total angular defect of the vertices, and almost all polyhedra do not have such bisectors.