Spectral flow, Maslov index and bifurcation of semi-Riemannian geodesics

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 imply that each nondegenerate and nonnull conjugate (or P-focal) point along a semi-Riemannian geodesic is a bifurcation point. In particular, the semi-Riemannian exponential map is not injective in any neighborhood of a nondegenerate conjugate point, extending a classical Riemannian result originally due to Morse and Littauer.