We prove that a real hypersurface M in complex projective space P 2 ‫)ރ(‬ or complex hyperbolic space H 2 ‫,)ރ(‬ whose Ricci operator is η-parallel and commutes with the structure tensor on the holomorphic distribution, is a Hopf hypersurface. We also give a characterization of this hypersurface.

Abstract. Let M be a real hypersurface with almost contact metric structure (φ, g, ξ, η) in a complex space form Mn(c), c = 0. In this paper we prove that if R ξ L ξ g = 0 holds on M , then M is a Hopf hypersurface in Mn(c), where R ξ and L ξ denote the structure Jacobi operator and the operator of the Lie derivative with respect to the structure vector field ξ respectively. We characterize such Hopf hypersurfaces of Mn(c).

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In this paper, we study a real hypersurface M in a non-at 2-dimensional complex space form M 2 (c) with η-parallel Ricci and shape operators. The characterizations of these real hypersurfaces are obtained.

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