Real hypersurfaces in complex two-plane Grassmannians with GTW Reeb Lie derivative structure Jacobi operator

TL;DR

This paper classifies certain real hypersurfaces in complex two-plane Grassmannians based on conditions involving the GTW Reeb Lie derivative of the structure Jacobi operator, identifying them as tubes around totally geodesic submanifolds.

Contribution

It provides a complete classification of real hypersurfaces satisfying a specific GTW Reeb Lie derivative condition in complex two-plane Grassmannians, identifying them as tubes around totally geodesic submanifolds.

Findings

Hypersurfaces satisfying the GTW Reeb Lie derivative condition are classified.

Such hypersurfaces are shown to be tubes around totally geodesic submanifolds.

The classification is achieved using simultaneous diagonalization methods.

Abstract

Using generalized Tanaka-Webster connection, we considered a real hypersurface $M$ in a complex two-plane Grassmannian $G_2({\mathbb C}^{m+2})$ when the GTW Reeb Lie derivative of the structure Jacobi operator coincides with the Reeb Lie derivative. Next using the method of simultaneous diagonalization, we prove a complete classification for a real hypersurface in $G_2({\mathbb C}^{m+2})$ satisfying such a condition. In this case, we have proved that $M$ is an open part of a tube around a totally geodesic $G_2({\mathbb C}^{m+1})$ in $G_2({\mathbb C}^{m+2})$.