Pseudo focal points along Lorentzian geodesics and Morse index

Details Pseudo focal points along Lorentzian geodesics and Morse index Pseudo focal points along Lorentzian geodesics and Morse index

2007-11-19

arxiv.org/abs/0711.3032v2

Mathematics

Differential Geometry

26 pages, Proposition 3.10, that has become Proposition 3.11 in the current version, has changed introducing a new class of po

Scientific paper

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.

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