Real hypersurfaces in complex hyperbolic two-plane Grassmannians with commuting structure Jacobi operators

TL;DR

This paper classifies real hypersurfaces in complex hyperbolic two-plane Grassmannians based on a new commuting condition involving the structure Jacobi operator and symmetric tensor fields, advancing understanding of their geometric structure.

Contribution

It introduces a novel commuting condition between the structure Jacobi operator and symmetric (1,1)-tensor fields, leading to a complete classification of such hypersurfaces.

Findings

Complete classification of hypersurfaces with the commuting condition

Identification of conditions for simultaneous diagonalization

New insights into geometric structures of hypersurfaces

Abstract

In this paper, we introduce a new commuting condition between the structure Jacobi operator and symmetric (1,1)-type tensor field $T$, that is, $R_{\xi}\phi T=TR_{\xi}\phi$, where $T=A$ or $T=S$ for Hopf hypersurfaces in complex hyperbolic two-plane Grassmannians. By using simultaneous diagonalzation for commuting symmetric operators, we give a complete classification of real hypersurfaces in complex hyperbolic two-plane Grassmannians with commuting condition respectively.