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2. Hopf hypersurfaces in complex two-plane Grassmannians with generalized Tanaka-Webster parallel normal Jacobi operator
- Creator:
- Pak, Eunmi, De Dios Pérez, Juan, Machado, Carlos J. G., and Woo, Changhwa
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- real hypersurface, complex two-plane Grassmannian, Hopf hypersurface, normal Jacobi operator, and generalized Tanaka-Webster parallel normal Jacobi operator
- Language:
- English
- Description:
- We study the classifying problem of immersed submanifolds in Hermitian symmetric spaces. Typically in this paper, we deal with real hypersurfaces in a complex two-plane Grassmannian $G_2({\mathbb C}^{m+2})$ which has a remarkable geometric structure as a Hermitian symmetric space of rank 2. In relation to the generalized Tanaka-Webster connection, we consider a new concept of the parallel normal Jacobi operator for real hypersurfaces in $G_2({\mathbb C}^{m+2})$ and prove non-existence of real hypersurfaces in $G_2({\mathbb C}^{m+2})$ with generalized Tanaka-Webster parallel normal Jacobi operator.
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public
3. Real hypersurfaces in complex hyperbolic two-plane Grassmannians with commuting restricted normal Jacobi operators
- Creator:
- Hwang, Doo Hyun, Pak, Eunmi, and Woo, Changhwa
- Format:
- print, bez média, and svazek
- Type:
- model:article and TEXT
- Subject:
- mathematics, real hypersurface, complex hyperbolic two-plane Grassmannians, Hopf hypersurface, shape operator, Ricci tensor, normal Jacobi operator, commuting condition, 13, and 51
- Language:
- English
- Description:
- We give a classification of Hopf real hypersurfaces in complex hyperbolic two-plane Grassmannians ${\rm SU}_{2,m}/S(U_2{\cdot}U_m)$ with commuting conditions between the restricted normal Jacobi operator $\overline{R}_N\phi$ and the shape operator $A$ (or the Ricci tensor $S$)., Doo Hyun Hwang, Eunmi Pak, Changhwa Woo., and Obsahuje bibliografii
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public
4. Real hypersurfaces in complex two-plane Grassmannians with certain commuting condition
- Creator:
- Lee, Hyunjin, Kim, Seonhui, and Suh, Young Jin
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- real hypersurface, complex two-plane Grassmannians, Hopf hypersurface, and commuting shape operator
- Language:
- English
- Description:
- In this paper, first we introduce a new notion of commuting condition that $\phi \phi _{1} A = A \phi _{1} \phi $ between the shape operator $A$ and the structure tensors $\phi $ and $\phi _{1}$ for real hypersurfaces in $G_2({\mathbb C}^{m+2})$. Suprisingly, real hypersurfaces of type $(A)$, that is, a tube over a totally geodesic $G_{2}(\mathbb C^{m+1})$ in complex two plane Grassmannians $G_2({\mathbb C}^{m+2})$ satisfy this commuting condition. Next we consider a complete classification of Hopf hypersurfaces in $G_2({\mathbb C}^{m+2})$ satisfying the commuting condition. Finally we get a characterization of Type $(A)$ in terms of such commuting condition $\phi \phi _{1} A = A \phi _{1} \phi $.
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public
5. Real hypersurfaces in complex two-plane Grassmannians with certain commuting condition. II
- Creator:
- Lee, Hyunjin, Kim, Seonhui, and Suh, Young Jin
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- complex two-plane Grassmannians, Hopf hypersurface, $\mathfrak D^{\bot }$-invariant hypersurface, commuting shape operator, and Reeb vector field
- Language:
- English
- Description:
- Lee, Kim and Suh (2012) gave a characterization for real hypersurfaces $M$ of Type ${\rm (A)}$ in complex two plane Grassmannians $G_2({\mathbb C}^{m+2})$ with a commuting condition between the shape operator $A$ and the structure tensors $\phi $ and $\phi _{1}$ for $M$ in $G_2({\mathbb C}^{m+2})$. Motivated by this geometrical notion, in this paper we consider a new commuting condition in relation to the shape operator $A$ and a new operator $\phi \phi _{1}$ induced by two structure tensors $\phi $ and $\phi _{1}$. That is, this commuting shape operator is given by $\phi \phi _{1} A = A \phi \phi _{1}$. Using this condition, we prove that $M$ is locally congruent to a tube of radius $r$ over a totally geodesic $G_2({\mathbb C}^{m+1})$ in $G_2({\mathbb C}^{m+2})$.
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public