Area and holonomy of the principal $U(n)$ bundles over the dual of grassmannian manifolds

Abstract

Consider the principal $U(n)$ bundles over the dual of Grassmann manifolds $U(n)\ra U(n,m)/U(m) \stackrel{\pi}\ra D_{n,m}$. Given a 2-dimensional subspace $\frakm' \subset \frakm $ $ \subset \mathfrak{u}(n,m), $ assume either $\frakm'$ is induced by $X,Y \in U_{m,n}(\bbc)$ with $X^{*}Y = \mu I_n$ for some $\mu \in \bbr$ or by $X,iX \in U_{m,n}(\bbc)$. Then $\frakm'$ gives rise to a complete totally geodesic surface $S$ in the base space. Furthermore, let $\gamma$ be a piecewise smooth, simple closed curve on $S$ parametrized by $0\leq t\leq 1$, and $\wt\gamma$ its horizontal lift on the bundle $U(n) \ra \pi^{-1}(S) \stackrel{\pi}{\rightarrow} S,$ which is immersed in $U(n) \ra U(n,m)/U(m) \stackrel{\pi}\ra D_{n,m} $. Then $$ \wt\gamma(1)= \wt\gamma(0) \cdot (e^{i \theta} I_n) \text{\hskip24pt or\hskip12pt} \wt\gamma(1)= \wt\gamma(0), $$ depending on whether $S$ is a complex submanifold or not, where $A(\gamma)$ is the area of the region on the surface $S$ surrounded by $\gamma$ and $\theta= 2 \cdot \tfrac{1}{n} A(\gamma).$