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 \(\mathfrak{D } ^{\bot }\) is invariant by the shape operator.