Remarks on η-parallel real hypersurfaces in CP2 and CH2

Let M be a three-dimensional real hypersurface in a nonflat complex space form of complex dimension two. In this paper, we prove that M is eta-parallel with two distinct principal curvatures at each point if and only if it is locally congruent to a geodesic sphere in CP2 or a horosphere, a geodesic sphere or a tube over totally geodesic complex hyperbolic plane in CH2. Moreover, eta-parallel real hypersurfaces in CP2 and CH2 under some other conditions are classified and these results extend Suh's in [Characterizations of real hypersurfaces in complex space forms in terms of Weingarten map, Nihonkai Math. J. 6 (1995) 63-79] and Kon-Loo's in [On characterizations of real hypersurfaces in a complex space form with eta-parallel shape operator, Canad. Math. Bull. 55 (2012) 114-126].