Semi-parallelism of normal Jacobi operator for Hopf hypersurfaces in complex two-plane Grassmannians

TL;DR

This paper proves the non-existence of certain Hopf hypersurfaces in complex two-plane Grassmannians with semi-parallel normal Jacobi operators under specific curvature and invariance conditions.

Contribution

It establishes a non-existence result for Hopf hypersurfaces with semi-parallel normal Jacobi operators in complex two-plane Grassmannians under particular geometric constraints.

Findings

No Hopf hypersurfaces with semi-parallel normal Jacobi operator exist under the given conditions.

The result applies when the principal curvature of the Reeb vector field is non-zero.

Invariant conditions on the Reeb vector field components are crucial for the proof.

Abstract

It is proved the non-existence of Hopf hypersurfaces in $G_{2}({\Bbb C}^{m+2})$, $m \geq 3$, whose normal Jacobi operator is semi-parallel, if the principal curvature of the Reeb vector field is non-vanishing and the component of the Reeb vector field in the maximal quaternionic subbundle ${\frak D}$ or its orthogonal complement ${\frak D}^{\bot}$ is invariant by the shape operator.