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

Abstract

Consider the principal $U(n)$ bundles over Grassmann manifolds $U(n)\to U(n+m)/U(m)\stackrel{\pi }{\to }{G}_{n,m}$. Given $X\in U_{m,n}(\mathbb{C})$ and a 2-dimensional subspace ${\mathfrak{m}}^{\prime }\subset \mathfrak{m}$ $\subset \mathfrak{u}(m+n),$ assume either ${\mathfrak{m}}^{\prime }$ is induced by $X,Y\in U_{m,n}(\mathbb{C})$ with $X^{\ast }Y=\mu I_n$ for some $\mu \in \mathbb{R}$ or by $X,iX\in U_{m,n}(\mathbb{C})$. Then ${\mathfrak{m}}^{\prime }$ 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 $\tilde{\gamma }$ its horizontal lift on the bundle $U(n)\to {\pi }^{-1}(S)\stackrel{\pi }{\to }S,$ which is immersed in $U(n)\to U(n+m)/U(m)\stackrel{\pi }{\to }{G}_{n,m}$. Then $$\tilde{\gamma }(1)=\tilde{\gamma }(0)\cdot (e^{i\theta }{I}_n)\text{or }\tilde{\gamma }(1)=\tilde{\gamma }(0),$$ 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 $\theta =2\cdot \frac{n+m}{2n}A(\gamma ).$

arXiv commentThis paper has been withdrawn by the author due to a crucial error in Theorem 2.6 under the metric of Grassmannian manifolds induced from the riemannian submersion