On a formula for the spectral flow and its applications

TL;DR

This paper derives a formula relating the spectral flow of a path of bilinear forms to that of its restriction to a subspace, with applications to semi-Riemannian geodesics and Maslov index.

Contribution

It introduces a new formula for spectral flow of bilinear form paths and applies it to periodic semi-Riemannian geodesics, linking spectral flow to Maslov index.

Findings

Derived a formula for spectral flow in terms of subspace restrictions.

Applied the formula to semi-Riemannian geodesics.

Connected spectral flow with Maslov index in geometric contexts.

Abstract

We consider a continuous path of bounded symmetric Fredholm bilinear forms with arbitrary endpoints on a real Hilbert space, and we prove a formula that gives the spectral flow of the path in terms of the spectral flow of the restriction to a finite codimensional closed subspace. We also discuss the case of restrictions to a continuous path of finite codimensional closed subspaces. As an application of the formula, we introduce the notion of spectral flow for a periodic semi-Riemannian geodesic, and we compute its value in terms of the Maslov index.