Theorem of the three geodesics

Remarks

The idea goes back to Lyusternik and Schnirelmann. Consider the space of unoriented simple closed curves on $S^2$, together with the point curves. Its topology is rich enough to force three minimax levels, in the spirit of Lusternik–Schnirelmann theory. The difficulty is to deform curves by a length-decreasing process that keeps them simple; this caused gaps in the early arguments. One complete proof uses the curve-shortening flow, which preserves simplicity and, by a theorem of Grayson, either shrinks a curve to a point or converges to a closed geodesic.

Example: on a round sphere every great circle is a simple closed geodesic, so there are infinitely many. On an ellipsoid in $\R^3$ with three distinct axes, the three principal sections in the coordinate planes are simple closed geodesics. The bound three cannot be improved in general.

Going deeper

For a student

The theorem is a sharp lower bound for simple closed geodesics. When self-intersections are allowed, the theorem of Bangert and Franks gives infinitely many closed geodesics on any Riemannian $S^2$. Its proof is a model of Lusternik–Schnirelmann theory on a space of curves: three independent families of curves sweep out the sphere, and each family yields a critical level. The first step, a single closed geodesic on $S^2$, is the Birkhoff min-max method.