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It has been proved there are no semi-parallel real hypersurfaces in the complex projective space CP n , n 3, and in any non-at complex space form of complex dimension 2. Also, characterizations of geodesic hyperspheres and ruled real hypersurfaces in CP n , n 3, have been obtained by considering some other curvature conditions. We generalize these results by studying two new conditions for real hypersurfaces in non-at complex space forms. As a corollary, we extend the known characterizations to real hypersurfaces of type A0 and A1 and ruled real hypersurfaces in non-at complex space forms. In particular, we prove that there are no semi-parallel real hypersurfaces in non-at complex space forms of complex dimension at least 2.

Classifications of real hypersurfaces in complex space forms by means of curvature conditions