Real Hypersurfaces in Complex Two-Plane Grassmannians with Recurrent Structure Jacobi Operator

Abstract

In this paper, we introduce a new notion of recurrent structure Jacobi operator, that is, \((\nabla _{X}R_{\xi })Y =\omega (X)R_{\xi }Y\) for any tangent vector fields X and Y on a real hypersurface M in a complex two-plane Grassmannian, where R ξ denotes the structure Jacobi operator and ω a certain 1-form on M. Next, we prove that there does not exist any Hopf hypersurface M in a complex two-plane Grassmannian with recurrent structure Jacobi operator.