Pseudo focal points along Lorentzian geodesics and Morse index

Abstract

Given a Lorentzian manifold $(M,g)$, a geodesic $\gamma$ in $M$ and a timelike Jacobi field $\mathcal{Y}$ along $\gamma$, we introduce a special class of instants along $\gamma$ that we call $\mathcal{Y}$-pseudo conjugate (or focal relatively to some initial orthogonal submanifold). We prove that the $\mathcal{Y}$-pseudo conjugate instants form a finite set, and their number equals the Morse index of (a suitable restriction of) the index form. This gives a Riemannian-like Morse index theorem. As special cases of the theory, we will consider geodesics in stationary and static Lorentzian manifolds, where the Jacobi field $\mathcal{Y}$ is obtained as the restriction of a globally defined timelike Killing vector field.

26 pages, Proposition 3.10, that has become Proposition 3.11 in the current version, has changed introducing a new class of points that we call pseudo-focal points. The new Section 5 is devoted to describe some properties to these points

References in corpus (1)

• On a formula for the spectral flow and its applications