Stability of the Focal and Geometric Index in semi-Riemannian Geometry via the Maslov Index

Francesco Mercuri, Paolo Piccione, Daniel V. Tausk

Published 1999-05-17, updated 2000-04-24Version 3

We investigate the problem of the stability of the number of conjugate or focal points (counted with multiplicity) along a semi-Riemannian geodesic $\gamma$. For a Riemannian or a non spacelike Lorentzian geodesic, such number is equal to the intersection number (Maslov index) of a continuous curve with a subvariety of codimension one of the Lagrangian Grassmannian of a symplectic space. Such intersection number is proven to be stable in a large variety of circumstances. In the general semi-Riemannian case, under suitable hypotheses this number is equal to an algebraic count of the multiplicities of the conjugate points, and it is related to the spectral properties of a non self-adjoint differential operator. This last relation gives a weak extension of the classical Morse Index Theorem in Riemannian and Lorentzian geometry. In this paper we reprove some results that were incorrectly stated by Helfer in a previous reference; in particular, a counterexample to one of Helfer's results, which is essential for the theory, is given. In the last part of the paper we discuss a general technique for the construction of examples and counterexamples in the index theory for semi-Riemannian metrics, in which some new phenomena appear.