The topological aspect of the holonomy displacement on the principal $U(n)$ bundles over Grassmanian manifolds

Abstract: Consider the principal $U(n)$ bundles over Grassmann manifolds $U(n)\rightarrow U(n+m)/U(m) \stackrel{\pi}\rightarrow G_{n,m}$. Given $X \in U_{m,n}(\mathbb{C})$ and a 2-dimensional subspace $\mathfrak{m}' \subset \mathfrak{m} $ $ \subset \mathfrak{u}(m+n), $ assume either $\mathfrak{m}'$ is induced by $X,Y \in U_{m,n}(\mathbb{C})$ with $X^{*}Y = \mu I_n$ for some $\mu \in \mathbb{R}$ or by $X,iX \in U_{m,n}(\mathbb{C})$. Then $\mathfrak{m}'$ 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 $\widetilde{\gamma}$ its horizontal lift on the bundle $U(n) \rightarrow \pi^{-1}(S) \stackrel{\pi}{\rightarrow} S,$ which is immersed in $U(n) \rightarrow U(n+m)/U(m) \stackrel{\pi}\rightarrow G_{n,m} $. Then $$ \widetilde{\gamma}(1)= \widetilde{\gamma}(0) \cdot ( e^{i \theta} I_n) \quad \text{ or } \quad \widetilde{\gamma}(1)= \widetilde{\gamma}(0), $$ depending on whether the immersed bundle is flat or not, where $A(\gamma)$ is the area of the region on the surface $S$ surrounded by $\gamma$ and $\theta= 2 \cdot \tfrac{n+m}{2n} A(\gamma).$