Pseudo focal points along Lorentzian geodesics and Morse index
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Given a Lorentzian manifold $(M,g)$, a geodesic $γ$ in $M$ and a timelike Jacobi field $\mathcal Y$ along $γ$, we introduce a special class of instants along $γ$ 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
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