Instability of semi-Riemannian closed geodesics

TL;DR

This paper establishes a new instability criterion for semi-Riemannian closed geodesics using spectral flow and introduces a generalized Bott iteration formula, advancing understanding of geodesic stability in semi-Riemannian geometry.

Contribution

It introduces a spectral index based on spectral flow, generalizes Bott's iteration formula to semi-Riemannian cases, and provides new instability criteria for closed geodesics.

Findings

Spectral index as a topological invariant for geodesic stability

Generalized Bott iteration formula for semi-Riemannian geodesics

Instability criteria based on Morse index control

Abstract

A celebrated result due to Poincar\'e affirms that a closed non-degenerate minimizing geodesic $\gamma$ on an oriented Riemannian surface is hyperbolic. Starting from this classical theorem, our first main result is a general instability criterion for timelike and spacelike closed semi-Riemannian geodesics on a (non)oriented manifold. A key role is played by the spectral index, a new topological invariant that we define through the spectral flow (being the Morse index truly infinite) of a path of selfadjoint Fredholm operators. A major step in the proof of this result is a em new spectral flow formula. Bott's iteration formula, introduced by author in 1956, relates in a clear way the Morse index of an iterated closed Riemannian geodesic and the so-called $\omega$-Morse indices. Our second result is a semi-Riemannian generalization of the famous Bott-type iteration formula in the case of closed (resp. timelike closed) Riemannian (resp. Lorentzian) geodesics. Our last result is a strong instability result obtained by controlling the Morse index of the geodesic and of all of its iterations.