Spectral Flow, Maslov Index and Bifurcation of semi-Riemannian Geodesics

Abstract

We give a functional analytical proof of the equality between the Maslov index of a semi-Riemannian geodesic and the spectral flow of the path of self-adjoint Fredholm operators obtained from the index form. This fact, together with recent results on the bifurcation for critical points of strongly indefinite functionals (see P. M. Fitzpatrick, J. Pejsachowicz, L. Recht, Spectral Flow and Bifurcation of Strongly Indefinite Functionals Part I. General Theory, J. Funct. Anal. 162 (1) (1999), 52-95.) imply that each non degenerate and non null conjugate (or $P$-focal) point along a semi-Riemannian geodesic is a bifurcation point.