Real hypersurfaces in complex two-plane Grassmannians with certain commuting condition II
Czechoslovak Mathematical Journal, Tome 64 (2014) no. 1, pp. 133-148
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library
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 $G_2({\mathbb C}^{m+2})$.
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 $G_2({\mathbb C}^{m+2})$.
DOI :
10.1007/s10587-014-0089-6
Classification :
32V40, 53C15, 53C40
Keywords: complex two-plane Grassmannians; Hopf hypersurface; $\mathfrak D^{\bot }$-invariant hypersurface; commuting shape operator; Reeb vector field
Keywords: complex two-plane Grassmannians; Hopf hypersurface; $\mathfrak D^{\bot }$-invariant hypersurface; commuting shape operator; Reeb vector field
@article{10_1007_s10587_014_0089_6,
author = {Lee, Hyunjin and Kim, Seonhui and Suh, Young Jin},
title = {Real hypersurfaces in complex two-plane {Grassmannians} with certain commuting condition {II}},
journal = {Czechoslovak Mathematical Journal},
pages = {133--148},
year = {2014},
volume = {64},
number = {1},
doi = {10.1007/s10587-014-0089-6},
mrnumber = {3247450},
zbl = {06391482},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1007/s10587-014-0089-6/}
}
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 SP - 133 EP - 148 VL - 64 IS - 1 UR - http://geodesic.mathdoc.fr/articles/10.1007/s10587-014-0089-6/ DO - 10.1007/s10587-014-0089-6 LA - en ID - 10_1007_s10587_014_0089_6 ER -
%0 Journal Article %A Lee, Hyunjin %A Kim, Seonhui %A Suh, Young Jin %T Real hypersurfaces in complex two-plane Grassmannians with certain commuting condition II %J Czechoslovak Mathematical Journal %D 2014 %P 133-148 %V 64 %N 1 %U http://geodesic.mathdoc.fr/articles/10.1007/s10587-014-0089-6/ %R 10.1007/s10587-014-0089-6 %G en %F 10_1007_s10587_014_0089_6
Lee, Hyunjin; Kim, Seonhui; Suh, Young Jin. Real hypersurfaces in complex two-plane Grassmannians with certain commuting condition II. Czechoslovak Mathematical Journal, Tome 64 (2014) no. 1, pp. 133-148. doi: 10.1007/s10587-014-0089-6
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