On the semi-Riemannian bumpy metric theorem

TL;DR

This paper proves that on a compact manifold, the set of semi-Riemannian metrics with only nondegenerate closed geodesics is generic, extending a Riemannian genericity result to semi-Riemannian geometry using equivariant variational methods.

Contribution

It extends the higher order genericity result for Riemannian metrics to semi-Riemannian metrics using equivariant variational genericity techniques.

Findings

The set of semi-Riemannian metrics with only nondegenerate closed geodesics is generic.

The proof extends Riemannian genericity results to semi-Riemannian settings.

The theorem applies to metrics of any fixed index on a compact manifold.

Abstract

We prove the semi-Riemannian bumpy metric theorem using equivariant variational genericity. The theorem states that, on a given compact manifold $M$, the set of semi-Riemannian metrics that admit only nondegenerate closed geodesics is generic relatively to the $C^k$-topology, $k=2,...,\infty$, in the set of metrics of a given index on $M$. A higher order genericity Riemannian result of Klingenberg and Takens is extended to semi-Riemannian geometry.