REAL HYPERSURFACES IN COMPLEX SPACE FORMS WITH ε-PARALLEL RICCI TENSOR AND STRUCTURE JACOBI OPERATOR

Abstract

We know that there are no real hypersurfaces with parallel Ricci tensor or parallel structure Jacobi operator in a nonflat complex space form (See [4], [6], [10] and [11]). In this paper we investigate real hypersurfaces M in a nonflat complex space form Mn(c) under the condition that ∇ξS = 0 and ∇ξRξ = 0, where S and Rξ respectively denote the Ricci tensor and the structure Jacobi operator of M in Mn(c). 0. Introduction A Kaehler manifold of constant holomorphic sectional curvature c is called a complex space form, which is denoted by Mn(c). It is well known that complete and simply connected complex space forms are isometric to a complex projective space PnC, a complex Euclidean space C or a complex hyperbolic space HnC according as c > 0, c = 0 and c < 0. In this paper we consider a real hypersurface M in a complex space form Mn(c), c6=0. Then M has an almost contact metric structure (φ, ξ, η, g) induced from the complex structure J and the Kaehler metric of Mn(c). The structure vector field ξ is said to be principal if Aξ = αξ is satisfied, where A denotes the shape operator of M and α = η(Aξ). A real hypersurface is said to be a Hopf hypersurface if the structure vector field ξ of M is principal. In the study of real hypersurfaces in PnC, Takagi [12] classified all homogeneous real hypersurfaces and Cecil and Ryan [2] showed that they can be regarded as the tubes of constant radius over Kaehler submanifolds when the structure vector field ξ is principal. Such tubes can be divided into six kinds of type A1, A2, B, C, D and E. Received August 8, 2005; Revised May 30, 2006. 2000 Mathematics Subject Classification. Primary 53C40; Secondary 53C15.