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

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

Czechoslovak Mathematical Journal, Tome 62 (2012) no. 3, pp. 849-861.

Résumé

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 $.

Détail

DOI : 10.1007/s10587-012-0049-y

Classification : 11R52, 53C40, 53C50, 53C55
Mots-clés : real hypersurface; complex two-plane Grassmannians; Hopf hypersurface; commuting shape operator