Pseudo Focal Points Along Lorentzian Geodesics and Morse Index

Abstract

Abstract Given a Lorentzian manifold (M, g), a geodesic γ in M and a timelike Jacobi field γ along γ, we introduce a special class of instants along γ that we call γ- pseudo conjugate (or focal relatively to some initial orthogonal submanifold). We prove that the γ-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 γ is obtained as the restriction of a globally defined timelike Killing vector field.