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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. We study a real hypersurface M in a complex space form M n (c), c = 0, whose shape operator and structure tensor commute each other on the holomorphic distribution of M.

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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