Perturbed closed geodesics are periodic orbits: Index and transversality

Abstract.

We study the classical action functional ${\cal S}_V$ on the free loop space of a closed, finite dimensional Riemannian manifold M and the symplectic action ${\cal A}_V$ 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 $V\in C^\infty(M\times S^1,\mathbb{R})$ 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 ${\cal S}_V$ equals minus its Conley-Zehnder index when viewed as a critical point of ${\cal A}_V$ and if $x^*TM\to S^1$ is trivial. Otherwise a correction term +1 appears.