Real hypersurfaces in complex two-plane Grassmannians with Reeb parallel Ricci tensor in generalized Tanaka-Webster connection

Abstract

There are several kinds of classification problems for real hypersurfaces in complex two-plane Grassmannians $G_2({\mathbb{C}}^{m+2})$. Among them, Suh classified Hopf hypersurfaces $M$ in $G_2({\mathbb{C}}^{m+2})$ with Reeb parallel Ricci tensor in Levi-Civita connection. In this paper, we introduce a new notion of generalized Tanaka-Webster Reeb parallel Ricci tensor for $M$ in $G_2({\mathbb{C}}^{m+2})$. By using such parallel conditions, we give complete classifications of Hopf hypersurfaces in $G_2({\mathbb{C}}^{m+2})$.