Perturbed closed geodesics are periodic orbits: Index and Transversality

Abstract

We study the classical action functional $\SMC_V$ on the free loop space of a closed, finite dimensional Riemannian manifold $M$ and the symplectic action $\AMC_V$ on the free loop space of its cotangent bundle. The critical points of both functionals can be identified with the set of perturbed closed geodesics in $M$. The potential $V\in C^\infty(M\times S^1,\R)$ serves as perturbation and we show that both functionals are Morse for generic $V$. In this case we prove that the Morse index of a critical point $x$ of $\SMC_V$ equals minus its Conley-Zehnder index when viewed as a critical point of $\AMC_V$ and if $x^*TM \to S^1$ is trivial. Otherwise a correction term +1 appears.