Chern-Simons-Maslov classes of some symplectic vector bundles

Abstract:

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