A focal index theorem for null geodesics

A note on the Morse index theorem for geodesics between submanifolds in semi-Riemannian geometry

“…, t N of [a, b] such that t ∈ ]t i , t i+1 [ for some i ≥ 1 (we allow t = t i+1 if t = b and we set i = N − 1). Let's denote by H P J ([a, t]) and H P 0 ([a, t]) the spaces defined in (8), replacing the interval [a, b] by [a, t] (and using the normal partition t 0 , . .…”

“…The case of a Riemannian geodesic with endpoints variable in two submanifolds of M has been treated by several authors, including Ambrose, Bolton and Kalish, (see [1,5,10], see also [17]). Following the approach of Kalish [10], Ehrlich and Kim have then proven in [8] the Morse Index Theorem for lightlike geodesics with endpoints varying on two spacelike submanifolds of a Lorentzian manifold. The case of spacelike geodesics in semi-Riemannian manifolds was treated by Helfer in [9], where an extension of the Index Theorem was proven in terms of the Maslov index of a curve, and by the introduction of a notion of signature for conjugate points.…”

“…We try to keep all the statements and proofs of the paper at the maximum level of generality; in particular, we present an approach that unifies the Riemannian and the causal Lorentzian case, obtaining a proof of all the results for Riemannian and causal Lorentzian geodesics at the same time. In Remark 2.9, among other things we observe that, in the Lorentzian lightlike case, the use of the quotient bundle employed in [2,3,8] is not really essential for the computation of the (non augmented) index, which allows to give an easier statement of the focal index theorem.…”

“…Exactly the same argument works also in the case of timelike geodesics between two submanifolds of a Lorentzian manifold. For the extension to the lightlike Lorentzian case, just minor changes are required and one obtains easily a proof of the focal index theorem presented in [8].…”

Stability of the conjugate index, degenerate conjugate points and the Maslov index in semi-Riemannian geometry

“…Conjugate points along a geodesic correspond to the zeroes of (nontrivial) Jacobi fields along γ, which are vector fields in the kernel of the second variation of the action functional z → b a g(z , z ) dt, called the index form I γ . The celebrated Morse Index Theorem (see for instance [2,3,6,7,9,16,17] for versions of this theorem in different contexts) states that the conjugate index of a Riemannian or nonspacelike Lorentzian geodesic γ is equal to the index of I γ , provided that the final point γ(b) is not considered in the count of conjugate points. It is not too hard to prove that the convergence of a sequence of geodesics to a geodesic implies the norm convergence of the correspondent index forms, seen as bilinear forms on the Hilbert space of H 1 variational vector fields.…”