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

TL;DR

This paper introduces a new generalized Tanaka-Webster Reeb parallel Ricci tensor condition for real hypersurfaces in complex two-plane Grassmannians and provides a complete classification of Hopf hypersurfaces under this condition.

Contribution

It proposes a novel notion of Reeb parallel Ricci tensor in the generalized Tanaka-Webster connection and classifies Hopf hypersurfaces satisfying this condition.

Findings

Complete classification of Hopf hypersurfaces with generalized Tanaka-Webster Reeb parallel Ricci tensor.

Extension of previous classifications to a broader parallel condition.

New geometric insights into the structure of hypersurfaces in complex Grassmannians.

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