Holonomy on the principal $U(n)$ bundles over Grassmannian manifolds
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 $\mathfrak{m}'$...
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Main Authors | , |
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Format | Journal Article |
Language | English |
Published |
16.06.2012
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Subjects | |
Online Access | Get full text |
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Summary: | 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 $\theta= 2 \cdot \tfrac{n+m}{2n} A(\gamma).$ |
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DOI: | 10.48550/arxiv.1206.3652 |