Periodic Geodesics on Compact Riemannian Manifolds

Abstract

This paper provides a proof of the existence of infinitely many geometrically distinct periodic geodesics on a compact Riemannian manifold $M$, so long as the sequence of Betti numbers on the loop space $\Omega M$ is unbounded. Periodic geodesics in $M$ correspond to critical points of the energy function, $E$. The overall idea is therefore to apply infinite-dimensional Morse theory to the energy function on the Hilbert manifold $H^1(S^1,M)$. In particular, it derives a local homological invariant at the critical points of $E$, which, in combination with iteration formulas for the index and nullity, is used to form a uniform upper bound on the type numbers. This result as well as the Morse inequalities are then used to prove, by contradiction, that there cannot be finitely many periodic geodesics if the Betti numbers are unbounded. It is based on Gromoll and Meyer’s proof from their paper Periodic Geodesics on Compact Riemannian Manifolds (1969).