As part of a seminar course on Riemannian geometry, i wrote a small paper on the basics of Lyusternik-Schnirelmann theory, with a goal of proving the Lyusternik-Fet Theorem. The project can be found here.

The theorem concerns closed geodeics, which are defined as follows.

Definition. A closed geodesic in a Riemannian manifold, \(M\), is a non-constant geodesic segment \(c: [0,1] \to M\) such that \(c(0)=c(1)\) and \(c'(0) = c'(1)\).

Intuitively, a closed geodesic is thus a ‘straight’ line, which wraps around the manifold and ends up where it started. I.e. periodic!

The easiest example of closed geodesics are those of the spheres \(S^n\), where any great circle (i.e. any geodesic) is closed! (Think of the equator of the earth and how you end up at the same place you started, if you walk around it).

But they exist in much more generality, as proven by Lazar Lyusternik and Abram Ilyich Fet!

Theorem (Lyusternik-Fet Theorem) On every compact Riemannian manifold without boundary, there exists a closed geodesic.

The proof is quite techincal and is based on the fact that closed geodesics are exactly the ctritical points of the energy functional. See the paper for more.

There is a lot more interesting theory regarding closed geodeics, and this theorem is only the beginning. This is the basis of Lyusternik-Schnirelmann theory. Going furhter, one may find the ‘Theorem of Three Geodesics’.

Theorem (Theorem of Three Geodesics) On the 2-dimensional sphere with an arbitrary Riemannian metric, there exists three closed geodesics without self-intersections.

There are still many questions left unanswered regarding closed geodesics. Does every complete Riemannian manifold with finite volume have at least one closed geodesic? For more questions see Burns and Matveev’s survey of open problems related to geodesics.