The study of closed geodesics on a Riemannian manifold is a classical and important part of differential geometry. In 1969 Gromoll and Meyer used Morse - Bott theory to give a topological condition on the loop space of compact manifold M which ensures that any Riemannian metric on M has an infinite number of closed geodesics. This makes a very close connection between closed geodesics and the topology of loop spaces.
Nowadays it is known that there is a rich algebraic structure associated to the topology of loop spaces — this is the theory of string homology initiated by Chas and Sullivan in 1999. In recent work, in collaboration with John McCleary, we have used the ideas of string homology to give new results on the existence of an infinite number of closed geodesics. I will explain some of the key ideas in our approach to what has come to be known as the closed geodesics problem.
- Topology Seminar