Real hypersurfaces in complex two-plane Grassmannians with certain commuting condition II

Hyunjin Lee; Seonhui Kim; Young Jin Suh

Czechoslovak Mathematical Journal (2014)

  • Volume: 64, Issue: 1, page 133-148
  • ISSN: 0011-4642

Abstract

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Lee, Kim and Suh (2012) gave a characterization for real hypersurfaces M of Type ( A ) in complex two plane Grassmannians G 2 ( ℂ m + 2 ) with a commuting condition between the shape operator A and the structure tensors φ and φ 1 for M in G 2 ( 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 φ φ 1 induced by two structure tensors φ and φ 1 . That is, this commuting shape operator is given by φ φ 1 A = A φ φ 1 . Using this condition, we prove that M is locally congruent to a tube of radius r over a totally geodesic G 2 ( ℂ m + 1 ) in G 2 ( ℂ m + 2 ) .

How to cite

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Lee, Hyunjin, Kim, Seonhui, and Suh, Young Jin. "Real hypersurfaces in complex two-plane Grassmannians with certain commuting condition II." Czechoslovak Mathematical Journal 64.1 (2014): 133-148. <http://eudml.org/doc/262050>.

@article{Lee2014,
abstract = {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 $G_2(\{\mathbb \{C\}\}^\{m+2\})$.},
author = {Lee, Hyunjin, Kim, Seonhui, Suh, Young Jin},
journal = {Czechoslovak Mathematical Journal},
keywords = {complex two-plane Grassmannians; Hopf hypersurface; $\mathfrak \{D\}^\{\bot \}$-invariant hypersurface; commuting shape operator; Reeb vector field; complex two-plane Grassmannian; Hopf hypersurface; Reeb vector field; shape operator; $\mathfrak \{D\}^\{\bot \}$-invariant hypersurface},
language = {eng},
number = {1},
pages = {133-148},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Real hypersurfaces in complex two-plane Grassmannians with certain commuting condition II},
url = {http://eudml.org/doc/262050},
volume = {64},
year = {2014},
}

TY - JOUR
AU - Lee, Hyunjin
AU - Kim, Seonhui
AU - Suh, Young Jin
TI - Real hypersurfaces in complex two-plane Grassmannians with certain commuting condition II
JO - Czechoslovak Mathematical Journal
PY - 2014
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 64
IS - 1
SP - 133
EP - 148
AB - 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 $G_2({\mathbb {C}}^{m+2})$.
LA - eng
KW - complex two-plane Grassmannians; Hopf hypersurface; $\mathfrak {D}^{\bot }$-invariant hypersurface; commuting shape operator; Reeb vector field; complex two-plane Grassmannian; Hopf hypersurface; Reeb vector field; shape operator; $\mathfrak {D}^{\bot }$-invariant hypersurface
UR - http://eudml.org/doc/262050
ER -

References

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