A generalized index theorem for Morse–Sturm systems and applications to semi-Riemannian geometry

Abstract

ABSTRACT. We prove an extension of the Index Theorem for Morse-Sturm systems of the form −V ′′ + RV = 0, where R is symmetric with respect to a (non positive) symmetric bilinear form, and thus the corresponding differential operator is not self-adjoint. The result is then applied to the case of a Jacobi equation along a geodesic in a Lorentzian manifold, obtaining an extension of the Morse Index Theorem for Lorentzian geodesics with variable initial endpoints. Given a Lorentzian manifold (M, g), we consider a geodesic γ in M starting orthogonally to a smooth submanifold P of M. Under suitable hypotheses, satisfied, for instance, if (M, g) is stationary, the theorem gives an equality between the index of the second variation of the action functional f at γ and the sum of the Maslov index of γ with the index of the metric g on P. Under generic circumstances, the Maslov index of γ is given by an algebraic count of the P-focal points along γ. Using the Maslov index, we obtain the global Morse relations for geodesics between two fixed points in a stationary Lorentzian manifold.

Pseudo Focal Points Along Lorentzian Geodesics and Morse Index

“…(f) If an instant t 0 ∈ ]0, 1[ is a jump instant for µ, then t 0 is a focal instant along γ. This follows from the main result of [10], since the P-Maslov of γ has jumps only at the focal instants. Thus, µ is constant on every interval that does not contain focal instants.…”

“…Using the theory developed in this paper and some results in the recent literature, we can now summarize a few facts about the distribution of focal and pseudo focal instants along a geodesic. (d) µ(t) is equal to the P-Maslov index of γ| [0,t] , as proved in [10].…”

“…Assume now that Y is singular and Y (0) is orthogonal to P . We will see that in this case H P (σ) = H * P (σ) , so that we can apply [10,Corollary 4.5] …”

“…Such restriction has the same critical points as the original geodesic action functional, but its second variation is essentially positive at each critical point. Thus, one has finite Morse index, and in [10] it is proven that this index is equal to the Maslov index.…”

On a formula for the spectral flow and its applications

“…This is a symplectic invariant, which is computed as an intersection number in the Lagrangian Grasmannian of a symplectic vector space. Details on the definition and the computation of the Maslov index for a given geodesic γ, that will be denoted by i Maslov (γ) can be found in [15,16,22].…”