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

TL;DR

This paper explores the geometric properties of principal $U(n)$ bundles over dual Grassmann manifolds, focusing on area, holonomy, and geodesic surfaces, revealing how these relate to the bundle's structure and curvature.

Contribution

It characterizes the holonomy of $U(n)$ bundles over dual Grassmannians via totally geodesic surfaces and area relations, extending understanding of bundle geometry in complex symmetric spaces.

Findings

Holonomy depends on the area enclosed by curves on geodesic surfaces.

Complete totally geodesic surfaces are induced by specific subspaces in the Lie algebra.

Holonomy is trivial or a phase factor depending on the submanifold's complex structure.

Abstract

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 $X^{*}Y = \mu I_n$ for some $\mu \in \bbr$ 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 $0\leq t\leq 1$, 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) \ra U(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 $\theta= 2 \cdot \tfrac{1}{n} A(\gamma).$