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

Lee, Hyunjin ; Kim, Seonhui ; Suh, Young Jin

Czechoslovak Mathematical Journal, Tome 64 (2014) no. 1, pp. 133-148.

Résumé

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 $G_2({\mathbb C}^{m+2})$.

Détail

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