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

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 geodesies with variable initial endpoints. Given a Lorentzian manifold (M.,g), we consider a geodesic 7 in A4 starting orthogonally to a smooth submanifold V 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 / at 7 and the sum of the Maslov index of 7 with the index of the metric g on V. Under generic circumstances, the Maslov index of 7 is given by an algebraic count of the 'P-focal points along 7. Using the Maslov index, we obtain the global Morse relations for geodesies between two fixed points in a stationary Lorentzian manifold.