Characterizations of real hypersurfaces of a complex hyperbolic space in terms of Ricci tensor and holomorphic distribution

Abstract

Let CPn and CHn denote the complex projective n-space with constant holomorphic sectional curvature 4, and the complex hyperbolic n-space with constant holomorphic sectional curvature ―4, respectively. Let M be a real hypersurface of CPn or CHn. M has an almost contact metric structure (<fi, ?, fj,g) induced from the complex structure / of CPn or CHn. Real hypersurfaces in CPn and CHn have been studied by many authors (cf. [1], [2], [3], [11], [12], [13], [14], [15] and [17]). For real hypersurfaces in CPn, Takagi ([16]) showed that all homogeneous real hypersurfaces in CPn are realized as the tubes of constant radius over compact Hermitian symmetric spaces of rank 1 or 2 (cf. [2] and [5]). He proved that all homogeneous real hypersurfaces in CPn could be classifiedinto six types which are said to be of type Ax, A2, B, C, D and E. Kimura ([5]) also proved that a real hypersurfaces M in CPn is homogeneous if and only if M has constant principal curvatures and £is principal. Other interesting resultsin real hypersurfaces of CPn are shown by Kimura-Maeda ([8]) and Maeda-Udagawa ([10]):