Real Hypersurfaces of Type A in Complex Two-Plane Grassmannians Related to The Reeb Vector Field

TL;DR

This paper characterizes real hypersurfaces of type A in complex two-plane Grassmannians using invariance properties related to the Reeb vector field and specific geometric structures.

Contribution

It introduces a new characterization of type A hypersurfaces based on the invariance of a vector bundle under the shape operator involving the Reeb vector field.

Findings

Characterization of type A hypersurfaces using invariance of JTM^ot

Relation between Reeb vector field and geometric structures

Extension of previous characterizations to type A hypersurfaces

Abstract

Y. J. Suh and H. Lee (Bull. Korean. Math. Soc. 47, 551-561 (2010)) characterized real hypersurfaces $M$ of type $B$ by the invariance of vector bundle $JTM^\perp$ under the shape operator and the orthogonality of $JTM^\perp$ and $\mathcal {J}TM^\perp$, where $TM^\perp$, $J$ and $\mathcal J$ are the normal bundle of $M$, K\"ahler structure and Quaternionic K\"ahler structure of $G_2({\mathbb{C}}^{m+2})$ respectively. In this paper, we characterize real hypersurfaces $M$ of type A by the invariance of the vector bundle $JTM^\perp$ under the shape operator with the Reeb vector field in $\mathcal {J}TM^\perp$.