On some real hypersurfaces of a complex hyperbolic space

Abstract

Given a real hypersurface of a complex hyperbolic space CH', we construct a principal circle bundle over it which is a Lorentzian hypersurface of the anti-De Sitter space H~ "+ 1. Relations between the respective second fundamental forms are obtained permitting us to classify a remarkable family of real hypersurfaces of CH ".Geometriae Dedicata 20 (1986) 245-261.

Commuting Conditions of the k-th Cho operator with the structure Jacobi operator of real hypersurfaces in complex space forms

“…Finally, the following Theorem plays an important role in the study of real hypersurfaces in M n .c/, which is due to Okumura in case of CP n (see [10]) and to Montiel and Romero in case of CH n (see [8]). It provides the classification of real hypersurfaces in M n .c/, n 2, whose shape operator commutes with the structure tensor field '.…”

Real Hypersurfaces With Cyclic-Parallel Structure Jacobi Operators in a Nonflat Complex Space Form

“…By the classification theorems of real hypersurfaces in M n (c), c = 0, due to Okumura [11] and Montiel and Romero [9], M is locally congruent to one of the real hypersurfaces of type A 1 or A 2 in P n C or of type A 0 , A 1 or A 2 in H n C. Suppose that α = 0. We first prove that all principal curvatures are constant.…”

“…In the case of complex hyperbolic space H n C, Montiel and Romero started the study of real hypersurfaces in [9] and constructed some homogeneous real hypersurfaces in H n C which are said to be of type A 0 , A 1 and A 2 . Those hypersurfaces have a lot of nice geometric properties (see Berndt [1] and Montiel and Romero [9]).…”

“…Those hypersurfaces have a lot of nice geometric properties (see Berndt [1] and Montiel and Romero [9]). In 2007 Berndt and Tamaru [2] classified all homogeneous real hypersurfaces in H n C.…”

Real Hypersurfaces in <i>CP<sup>2</sup></i> and <i>CH<sup>2</sup></i> Equipped With Structure Jacobi Operator Satisfying L<sub>ξ</sub>l =▽<sub>ξ</sub>l

“…Okumura (see [4]) in and Montiel and Romero (see [5]) in H  gave the classification of real hypersurfaces satisfying relation = 0 A A    . Ki and Liu (see [6]) have given the above classification as follows:…”