A study of real hypersurfaces with Ricci operators in 2-dimensional complex space forms

Abstract

We prove that a real hypersurface M in complex projective space P2.C/ or complex hyperbolic space H2.C/, whose Ricci operator is -parallel and commutes with the structure tensor on the holomorphic distribution, is a Hopf hypersurface. We also give a characterization of this hypersurface. In this paper we consider a real hypersurface M in a complex space form M2.c/, c6D 0. Then M has an almost contact metric structure.; g;;/ induced from the Kahler metric and complex structure J on Mn.c/. The structure vector field is said to be principal if A D is satisfied, where A is the shape operator of M and D. A/ . In this case, it is known that is locally constant (Ki and Suh 1990) and that M is called a Hopf hypersurface. Takagi (1973) classified homogeneous real hypersurfaces in Pn.C/ into six model spaces A1, A2, B, C, D and E of Hopf hypersurfaces with constant principal curvatures. Berndt (1989) classified all homogeneous Hopf hypersurfaces in Hn.C/ as four model spaces, which are said to be A0, A1, A2 and B. A real hypersurface M of type A1 or A2 in Pn.C/ or type A0, A1 or A2 in Hn.C/ is said to be of type A for simplicity. As a typical characterization of real hypersurfaces of type A, the following is due to Okumura (1975) for c> 0, and Montiel and Romero (1986) for c< 0.