The topological aspect of the holonomy displacement on the principal U(n) bundles over Grassmannian manifolds

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• 作者：Taechang Byun^a ; ^{tcbyun@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; Younggi Choi^b ; ¹ ; ^{yochoi@snu.ac.kr" class="auth_mail" title="E-mail the corresponding author}
• 关键词：53C29 ; 57N35 ; 57R22 ; 57R55
• 刊名：Topology and its Applications
• 出版年：2015
• 出版时间：December 2015
• 年：2015
• 卷：196
• 期：part_PA
• 页码：8-21
• 全文大小：340 K

文摘

Consider the principal U(n)$U(n)$ bundles over Grassmann manifolds $U(n)\to U(n+m)/U(m)\stackrel{\pi }{\to }{G}_{n,m}$. Given X∈U_m,n(C)$X\in U_{m,n}(\mathbb{C})$ and a 2-dimensional subspace m^′⊂m${\mathfrak{m}}^{\prime }\subset \mathfrak{m}$⊂u(m+n)$\subset \mathfrak{u}(m+n)$, assume either m^′${\mathfrak{m}}^{\prime }$ is induced by X,Y∈U_m,n(C)$X,Y\in U_{m,n}(\mathbb{C})$ with X^⁎Y=μI_n$X^{⁎}Y=\mu I_n$ for some μ∈R$\mu \in \mathbb{R}$ or by a72b96ae6c2" title="Click to view the MathML source">X,iX∈U_m,n(C)$X,iX\in U_{m,n}(\mathbb{C})$. Then m^′${\mathfrak{m}}^{\prime }$ gives rise to a complete totally geodesic surface S in the base space. Furthermore, let γ be a piecewise smooth, simple closed curve on S   parametrized by a75b6f26a" title="Click to view the MathML source">0≤t≤1$0\le t\le 1$, and $\tilde{\gamma }$ be 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 depending on whether the immersed bundle is flat or not, where A(γ)$A(\gamma )$ is the area of the region on the surface S surrounded by γ   and $\theta =2\cdot \frac{n+m}{2n}A(\gamma )$.