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

Abstract

Consider the principal $U(n)$ bundles over Grassmann manifolds $U(n)\rightarrow U(n+m)/U(m) \stackrelπ\rightarrow G_{n,m}$. 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 $X^{*}Y = μI_n$ for some $μ\in \mathbb{R}$ 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 $γ$ be a piecewise smooth, simple closed curve on $S$ parametrized by $0\leq t\leq 1$, and $\widetildeγ$ its horizontal lift on the bundle $U(n) \rightarrow π^{-1}(S) \stackrelπ{\rightarrow} S,$ which is immersed in $U(n) \rightarrow U(n+m)/U(m) \stackrelπ\rightarrow G_{n,m} $. Then $$ \widetildeγ(1)= \widetildeγ(0) \cdot ( e^{i θ} I_n) \text{\quad or \quad } \widetildeγ(1)= \widetildeγ(0), $$ depending on whether the immersed bundle is flat or not, where $A(γ)$ is the area of the region on the surface $S$ surrounded by $γ$ and $θ= 2 \cdot \tfrac{n+m}{2n} A(γ).$