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

TL;DR

This paper classifies Hopf hypersurfaces in complex two-plane Grassmannians based on a new commuting condition involving restricted Jacobi operators and the Ricci tensor.

Contribution

Introduces a new commuting condition between restricted Jacobi operators and the Ricci tensor, leading to a complete classification of Hopf hypersurfaces in complex two-plane Grassmannians.

Findings

Complete classification of Hopf hypersurfaces under the new condition

Identification of geometric structures satisfying the commuting condition

Extension of understanding of hypersurface geometry in complex Grassmannians

Abstract

In this paper, we have considered a new commuting condition, that is, $(R_\xi\phi) S = S (R_\xi\phi)$ \big(resp. $(\Bar{R}_N\phi) S = S (\Bar{R}_N\phi$)\big) between the restricted Jacobi operator~$R_\xi\phi$ (resp. $\Bar{R}_N\phi$), and the Ricci tensor $S$ for real hypersurfaces $M$ in $G_2({\mathbb C}^{m+2})$. In terms of this condition we give a complete classification for Hopf hypersurfaces $M$ in $G_2({\mathbb C}^{m+2})$.