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

TL;DR

This paper studies the holonomy of principal $U(n)$ bundles over Grassmannian manifolds, characterizing when the bundle is flat or not based on the geometry of certain totally geodesic surfaces and the area enclosed by loops.

Contribution

It provides a geometric criterion for the holonomy of principal $U(n)$ bundles over Grassmannians using totally geodesic surfaces and area calculations.

Findings

Holonomy depends on the area enclosed by loops on specific geodesic surfaces.

The bundle is flat if the holonomy is trivial, otherwise it involves a phase factor.

Explicit relation between holonomy angle and surface area is established.

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 $$ \widetilde{\gamma}(1)= \widetilde{\gamma}(0) \cdot ( e^{i \theta} I_n) \text{\quad or \quad } \widetilde{\gamma}(1)= \widetilde{\gamma}(0), $$ 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 $\theta= 2 \cdot \tfrac{n+m}{2n} A(\gamma).$