On the Maslov index of symplectic paths that are not transversal to the Maslov cycle. Semi-Riemannian index theorems in the degenerate case

Abstract

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. We discuss some applications of the theory, with special emphasis on the study of the Jacobi equation along a semi-Riemannian geodesic. The research work exposed in this paper originated from a suggestion given by the third author relating the spectral flow with the partial signatures. He pointed out several references and formulated the statement that in the invertible endpoints case, the spectral flow of a real analytic path of self-adjoint Fredholm operators is given by the sum of the odd partial signatures at each degeneracy instant. In the present version of the article, this statement is part of Proposition 2.9; totally, the contribution of the third author to the theory consists in Definition 2.3, formulas (2.1) and (2.2), the first statement of Proposition 2.4, parts of Remark 2.5, Definition 2.6, and parts of the statement and parts of the proof of Proposition 2.9. Apart from this original contribution, the material contained in this paper was entirely developed and written by the first two authors at the Universita' di Camerino (Italy), Universidade de Sao Paulo (Brazil).