arXiv subjects

Taechang Byun

Publications and source records attributed to Taechang Byun.

6 recordsLinked to original sources

arXiv2016

Topological property of the holonomy displacement on the principal $U(n)$-bundle over $D_{n,m},$ related to complex surfaces

Consider $D_{n,m} = U(n,m)/\left(U(n) \times U(m)\right)$, the dual of the the Grassmannian manifold and the principal $U(n)$ bundle over $D_{n,m},$ $U(n)\rightarrow U(n,m)/U(m) \stackrelπ \rightarrow D_{n,m}$. Given a nontrivial $X \in M_{m \times n}(\mathbb{C}),$ consider a two dimensional subspace $\mathfrak{m}' \subset \mathfrak{m} \subset \mathfrak{u}(n,m), $ induced by $X, iX \in M_{m \times n}(\mathbb{C}),$ and a complete oriented surface $S,$ related to $(X,g) \in M_{m \times n}(\mathbb{C}) \times U(n,m), $ in the base space $D_{n,m}$ with a complex structure from $\mathfrak{m}'.$ Let $c$ be a smooth, simple, closed, orientation-preserving curve on $S$ parametrized by $0\leq t\leq 1$, and $\hat{c}$ its horizontal lift on the bundle $U(n) \ra U(n,m)/U(m) \stackrelπ\ra D_{n,m} $. Then the holonomy displacement is given by the right action of $e^Ψ$ for some $ Ψ\in \text{Span}_{\bbr}\{i(X^*X)^k\}_{k=1}^{q} \subset \mathfrak{u}(n), \: q=\text{rk}X, $ such that $$ \hat{c}(1) = \hat{c}(0) \cdot e^Ψ\text{24pt and 12pt } \text{Tr}(Ψ)= 2i \, \text{Area}(c), $$ where $\text{Area}(c)$ is the area of the region on the surface $S$ surrounded by $c,$ obtained from a special 2-form $ω_{(X,g)}$ on $S,$ called an area form $ω_{(X,g)}$ related to $(X,g)$ on $S.$

math.DG↗

arXiv2015

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

Consider the principal $U(n)$ bundles over the dual of Grassmann manifolds $U(n)\ra U(n,m)/U(m) \stackrelπ\ra D_{n,m}$. 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 $X^{*}Y = μI_n$ for some $μ\in \bbr$ 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 $γ$ be a piecewise smooth, simple closed curve on $S$ parametrized by $0\leq t\leq 1$, and $\wtγ$ its horizontal lift on the bundle $U(n) \ra π^{-1}(S) \stackrelπ{\rightarrow} S,$ which is immersed in $U(n) \ra U(n,m)/U(m) \stackrelπ\ra D_{n,m} $. Then $$ \wtγ(1)= \wtγ(0) \cdot (e^{i θ} I_n) \text{\hskip24pt or\hskip12pt} \wtγ(1)= \wtγ(0), $$ depending on whether $S$ is a complex submanifold or not, where $A(γ)$ is the area of the region on the surface $S$ surrounded by $γ$ and $θ= 2 \cdot \tfrac{1}{n} A(γ).$

math.DG↗

arXiv2015

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

Consider the principal $U(n)$ bundles over Grassmann manifolds $U(n)\rightarrow U(n+m)/U(m) \stackrelπ\rightarrow G_{n,m}$. 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 $X^{*}Y = μI_n$ for some $μ\in \mathbb{R}$ 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 $γ$ be a piecewise smooth, simple closed curve on $S$ parametrized by $0\leq t\leq 1$, and $\widetildeγ$ its horizontal lift on the bundle $U(n) \rightarrow π^{-1}(S) \stackrelπ{\rightarrow} S,$ which is immersed in $U(n) \rightarrow U(n+m)/U(m) \stackrelπ\rightarrow G_{n,m} $. Then $$ \widetildeγ(1)= \widetildeγ(0) \cdot ( e^{i θ} I_n) \text{\quad or \quad } \widetildeγ(1)= \widetildeγ(0), $$ depending on whether the immersed bundle is flat or not, where $A(γ)$ is the area of the region on the surface $S$ surrounded by $γ$ and $θ= 2 \cdot \tfrac{n+m}{2n} A(γ).$

math.DG↗

arXiv2015

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

Consider the principal $U(n)$ bundles over Grassmann manifolds $U(n)\rightarrow U(n+m)/U(m) \stackrelπ\rightarrow G_{n,m}$. 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 $X^{*}Y = μI_n$ for some $μ\in \mathbb{R}$ 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 $γ$ be a piecewise smooth, simple closed curve on $S$ parametrized by $0\leq t\leq 1$, and $\widetildeγ$ its horizontal lift on the bundle $U(n) \rightarrow π^{-1}(S) \stackrelπ{\rightarrow} S,$ which is immersed in $U(n) \rightarrow U(n+m)/U(m) \stackrelπ\rightarrow G_{n,m} $. Then $$ \widetildeγ(1)= \widetildeγ(0) \cdot ( e^{i θ} I_n) \quad \text{ or } \quad \widetildeγ(1)= \widetildeγ(0), $$ depending on whether the immersed bundle is flat or not, where $A(γ)$ is the area of the region on the surface $S$ surrounded by $γ$ and $θ= 2 \cdot \tfrac{n+m}{2n} A(γ).$

math.DG↗

arXiv2010

Horizontal Displacement Of Curves In Bundle SO(n) -> SO_0(1,N) -> H^n

The Riemannian submersion $ π: \text{SO}_0(1,n) \to \mathbb{H}^n $ is a principal bundle and its fiber at $ π(e) $ is the imbedding of $\text{SO}(n)$ into $ \text{SO}_0(1,n) $, where $e$ is the identity of both $\text{SO}_0(1,n)$ and $\text{SO}(n)$. In this study, we associate a curve, starting from the identity, in $\text{SO}(n)$ to a given surface with boundary, diffeomorphic to the closed disk $D^2$, in $ \mathbb{H}^n $ such that the starting point and the ending point of the curve agree with those of the horizontal lifting of the boundary curve of the given surface with boundary, respectively, and that the length of the curve is as same as the area of the given surface with boundary.

math.DG↗

arXiv2010

SO(n)\SO_0(n,1) has Positive Curvatures

The Lie group SO_0(n, 1) has the left-invariant metric coming from the Killing-Cartan form. The maximal compact subgroup SO(n) of the isometry group acts from the left. The geometry of the quotient space of the homogeneous submersion SO_0(n, 1) -> SO(n)\SO_0(n, 1) is investigated. The space is expressed as a warped product. Its group of isometries and sectional curvatures are calculated.

math.GT↗