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Commuting Structure Jacobi Operator for Real Hypersurfaces in Complex Space Forms

Author(s): U-Hang Ki | Hiroyuki Kurihara

Journal: Advances in Pure Mathematics
ISSN 2160-0368

Volume: 03;
Issue: 02;
Start page: 264;
Date: 2013;
Original page

Keywords: Complex Space Form | Hopf Hypersurface | Structure Jacobi Operator | Shape Operator | Ricci Tensor

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 φξξ-parallel and Rξ commute with the Ricci tensor, then M is also a Hopf  hypersurface provided that TrRξ is constant.
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