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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 $X^{*}Y = \mu I_n$ forsome $\mu \in \bbr$ or by $X,iX \in U_{m,n}(\bbc)$. Then $\frakm'$ gives riseto 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) \raU(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 $\theta= 2 \cdot \tfrac{1}{n} A(\gamma).$

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