Dave Richeson on Nostr: Given a Riemannian manifold that is topologically a sphere, we can talk about ...

Given a Riemannian manifold that is topologically a sphere, we can talk about "geodesics"—paths as close to straight as possible on the surface. On a true sphere, geodesics are great circles, but on topological spheres, the geodesics may roam around the sphere forever without closing up.

In 1905, Poincaré conjectured that any such sphere must have at least three simple closed geodesics. That is, they are closed curves without self-intersections. For example, for an ellipsoid x²/a²+y²/b²+z²/c²=1, think of the ellipses where the ellipsoid intersects the coordinate planes.

In 1929, Lyusternik and Schnirelmann proved the conjecture. Although the idea of the proof was correct, it contained a flaw that was fixed in the 1980s by Grayson.

Their result was about simple closed geodesics. What about closed geodesics in general? You head off in some direction, you may cross your path multiple times, but you end up back where you started from, heading in the same direction. What can we say about those?

In the early 1990s, Franks and Bangert each published an article covering a special case: the sphere has positive Gaussian curvature (Franks) and all other spheres (Bangert). Together, their work proved the remarkable theorem:

Every Riemannian manifold that is topologically a sphere has infinitely many closed geodesics!

I was telling my topology class about this theorem because we were discussing the annulus, and the key to Franks's result was to show that a certain map on the annulus had infinitely many periodic points.