Chern-Simons-Maslov classes of some symplectic vector bundles

Abstract

Let E 0 , J 0 {E_0},\;{J_0} , and L 0 {L_0} be the symplectic 2 n 2n -vector bundle, the compatible complex operator, and the Lagrangian subbundle that are determined by the U ( n ) U(n) -extension of the principal O ( n ) O(n) -bundle U ( n ) → U ( n ) / O ( n ) U(n) \to U(n)/O(n) . We compute the Chern-Simons-Maslov class μ 1 ( E 0 , J 0 , L 0 ) {\mu ^1}({E_0},{J_0},{L_0}) . Then for a trivial symplectic 2 n 2n -bundle E E , a compatible complex operator J J , and a Lagrangian subbundle L L , we compute Chern-Simons-Maslov classes μ h ( E , J , L ) {\mu ^h}(E,J,L) under some condition on the base space of E E .