The Maslov index and the spectral flow—revisited

The Spectral flow and the Maslov index

exist and have no zero eigenvalue. A typical example for A(t) is the div-grad-curl operator on a 3-manifold twisted by a connection which depends on t. Atiyah et al proved that the Fredholm index of such an operator DA is equal to minus the “spectral flow” of the family {A(t)}t∈R. This spectral flow represents the net change in the number of negative eigenvalues of A(t) as t runs from −∞ to ∞. ...

The Maslov Index , the Spectral Flow , and Decompositions of Manifolds

The Maslov Index Revisited

Abs t r ac t . Let 7) be a Hermitian symmetric space of tube type, S its Shilov boundary and G the neutral component of the group of bi-holomorphic diffeomorphisms of 7). In the model situation 7) is the Siegel disc, S is the manifold of Lagrangian subspaces and G is the symplectic group. We introduce a notion of transversality for pairs of elements in S, and then study the action of G on the s...

The Maslov index for paths

Maslov’s famous index for a loop of Lagrangian subspaces was interpreted by Arnold [1] as an intersection number with an algebraic variety known as the Maslov cycle. Arnold’s general position arguments apply equally well to the case of a path of Lagrangian subspaces whose endpoints lie in the complement of the Maslov cycle. Our aim in this paper is to define a Maslov index for any path regardle...

Computation of the Maslov Index and the Spectral Flow via Partial Signatures

Given a smooth Lagrangian path, both in the finite and in the infinite dimensional (Fredholm) case, we introduce the notion of partial signatures at each isolated intersection of the path with the Maslov cycle. For real-analytic paths, we give a formula for the computation of the Maslov index using the partial signatures; a similar formula holds for the spectral flow of real-analytic paths of F...