On the Maslov Index of Lagrangian Paths That Are Not Transversal to the Maslov Cycle. Semi-riemannian Index Theorems in the Degenerate Case

The Maslov index of a Lagrangian path, under a certain transversality assumption, is given by an algebraic count of the intersections of the path with a subvariety of the Lagrangian Grassmannian called the Maslov cycle. In these notes we use the notion of generalized signatures at a singularity of a smooth curve of symmetric bilinear forms to determine a formula for the computation of the Maslov index in the case of a real-analytic path having possibly non transversal intersections. Using this formula we give a general definition of Maslov index for continuous curves in the Lagrangian Grassmannian, both in the finite and in the infinite dimensional (Fredholm) case, and having arbitrary endpoints. Other notions of Maslov index are also considered, like the index for pairs of Lagrangian paths, the Kashiwara’s triple Maslov index, and Hörmander’s four-fold index. We discuss some applications of the theory, with special emphasis on the study of the Jacobi equation along a semi-Riemannian geodesic. In this context, we prove an extension of several versions of the Morse index theorems for geodesics having possibly conjugate endpoints.