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 $U(n)\ra U(n,m)/U(m) \stackrelπ\ra D_{n,m}$. 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 $X^{\ast }Y=\mu I_n$ for some $μ\in \bbr$ 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 $0\le t\le 1$, and $\wtγ$ its horizontal lift on the bundle $U(n) \ra π^{-1}(S) \stackrelπ{\rightarrow} S,$ which is immersed in $U(n) \ra U(n,m)/U(m) \stackrelπ\ra D_{n,m} $. Then $$ \wtγ(1)= \wtγ(0) \cdot (e^{i θ} I_n) \text{\hskip24pt or\hskip12pt} \wtγ(1)= \wtγ(0), $$ 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 $\theta =2\cdot \frac{1}{n}A(\gamma ).$

arXiv commentSubstantial text overlap deals with the dual one of the bundle which was studied in the arXiv:1503.03177. It plays a key role how the structure of the hopf bundle can be immersed in both of them