Holonomy on the principal $U(n)$ bundles over Grassmannian manifolds

Consider the principal $U(n)$ bundles over Grassmann manifolds $U(n)\to U(n+m)\mathrm{/}U(m)G_{n,m}$. Given $X \in U_{m,n}($\mathbb{C}$)$anda2-dimensionalsubspace$\mathfrak{m}’ \subset \mathfrak{m} $ $\subset \mathfrak{u}(m+n),$ assume either ${\mathfrak{m}}^{\mathrm{\prime }}$ is induced by $X,Y\in U_{m,n}(\mathbb{C})$ with $X^{\ast }Y=\mu I_n$ for some $\mu \in $\mathbb{R}$$orby$X,iX \in U_{m,n}($\mathbb{C}$)$\mathrm{.}Then$\mathfrak{m}'$givesrisetoacompletetotallygeodesicsurface$S$ in the base space. Furthermore, let $\gamma$ be a piecewise smooth, simple closed curve on $S$ parametrized by $0\le t\le 1$, and $\tilde{\gamma }$ its horizontal lift on the bundle $U(n)\to {\pi }^{-1}(S)S,$ which is immersed in $U(n) \rightarrow U(n+m)/U(m) \stackrel{\pi}\rightarrow G_{n,m} $\mathrm{.}Then$$ \widetilde{\gamma}(1)= \widetilde{\gamma}(0) \cdot ( e^{i \theta} I_n) $\text{or }$ \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 $\theta =2\cdot \frac{n+m}{2n}A(\gamma )\mathrm{.}$