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

Abstract

It is proved the non-existence of Hopf hypersurfaces in $G_2({\mathbb{C}}^{m+2})$, $m\ge 3$, 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 $\mathfrak{D}$ or its orthogonal complement ${\mathfrak{D}}^{\perp }$ is invariant by the shape operator.

11 pages