Transversality Obstructions of Lagrangian Subbundles (Maslov Classes)

Abstract

We already know that a Lagrangian subbundle of a symplectic vector bundle is equivalent to a reduction of the structure group from Sp(n,IR) to O(n), via U(n), and, accordingly, we shall have orthogonal connections. If two Lagrangian subbundles are given, we have two classes of orthogonal connections, whose comparison leads to secondary characteristic classes. The latter turn out to be obstructions to the transversality of the Lagrangian subbundles, and we call them Maslov classes since the simplest of them is precisely the one that yields the Maslov index (See references in Section 1.1.)