Spectral flow and iteration of closed semi-Riemannian geodesics

TL;DR

This paper introduces spectral flow as an analogue of Morse index for semi-Riemannian geodesics and analyzes its growth under iteration, providing insights into the stability and multiplicity of closed geodesics.

Contribution

It defines spectral flow for semi-Riemannian geodesics and investigates its asymptotic behavior under iteration, extending Morse theory concepts to indefinite metrics.

Findings

Spectral flow serves as a Morse index analogue in semi-Riemannian geometry.

The asymptotic growth rate of spectral flow under iteration is characterized.

Results have implications for the stability and multiplicity of closed geodesics.

Abstract

We introduce the notion of spectral flow along a periodic semi-Riemannian geodesic, as a suitable substitute of the Morse index in the Riemannian case. We study the growth of the spectral flow along a closed geodesic under iteration, determining its asymptotic behavior.