On the Equality between the Maslov Index and the Spectral Index for the Semi-Riemannian Jacobi Operator1

We consider a Morse–Sturm system in n whose coefficient matrix is symmetric with respect to a (not necessarily positive definite) nondegenerate symmetric bilinear form on n. The main motivation for studying such systems comes from semi-Riemannian geometry, where the Morse–Sturm system is obtained from the Jacobi equation along a geodesic by writing the equation in terms of a parallelly transported basis of the tangent bundle along the geodesic. Two integer numbers are naturally associated to such systems: the Maslov index, which gives a sort of algebraic count of the conjugate instants, and the spectral index, which gives an algebraic count of the negative eigenvalues of the corresponding second-order differential operator. In this paper we prove that these two integer numbers are equal; in the case of Riemannian geometry, this equality is precisely the Morse Index Theorem. Such equality is already known to hold under a suitable nondegeneracy assumption on the eigenvalues of the Jacobi operator; we give a proof of the equality in the degenerate case using a perturbation argument