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 regardless of where its endpoints lie. Our index is invariant under homotopy with fixed endpoints and is additive for catenations. Duistermaat [4] has proposed a Maslov index for paths which is not additive for catenations but is independent of the choice of the Lagrangian subspace used to define the Maslov cycle. By contrast our Maslov index depends on this choice. We have been motivated by two applications in [10] and [12] as well as the index introduced by Conley and Zehnder in [2] and [3]. In [12] we show how to define a signature for a certain class of one dimensional first order differential operators whose index and coindex are infinite. In [10] we relate the Maslov index to Cauchy Riemann operators such as those that arise in