TITLE:
Commuting Structure Jacobi Operator for Real Hypersurfaces in Complex Space Forms
AUTHORS:
U-Hang Ki, Hiroyuki Kurihara
KEYWORDS:
Complex Space Form; Hopf Hypersurface; Structure Jacobi Operator; Shape Operator; Ricci Tensor
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.3 No.2,
March
14,
2013
ABSTRACT: Let M be a real hypersurface of a complex space form with almost contact metric structure (φ,ξ,η,g). In this paper, we prove that if the structure Jacobi operator Rξ=(·,ξ) ξ is φ▽ξξ-parallel and Rξ commute with the shape operator, then M is a Hopf hypersurface. Further, if Rξ is φ▽ξξ-parallel and Rξ commute with the Ricci tensor, then M is also a Hopf hypersurface provided that TrRξ is constant.