A characterization of a real hypersurface of type \(B\)
Let \(N\) be the \(n\)-dimensional complex projective space \(P_n\mathbb{C}\) or the \(n\)-dimensional complex hyperbolic space \(H_ n\mathbb{C}\) of constant holomorphic curvature \(4\) resp. \(-4\). Distinguished examples of real hypersurfaces of \(N\) are the six classes \(A_1\), \(A_2\), \(B\)-\(E\) of homogeneous hypersurfaces of \(P_n\mathbb{C}\) [see \textit{R. Takagi}, Osaka J. Math. 10, 495--506 (1973; Zbl 0274.53062)] and the four classes \(A_0\)-\(A_1\) and \(B\) of hypersurfaces with constant principal curvatures for which the normal bundle is mapped into a principal direction by the complex structure of \(N\) [see \textit{J. Berndt}, J. Reine Angew. Math. 395, 132--141 (1989; Zbl 0655.53046)]. The hypersurfaces of type \(A_1\), \(A_2\) and \(B\) are the tubes around the totally geodesic complete submanifolds of \(N\); the horospheres of \(H_n\mathbb{C}\) form the class \(A_0\). The authors derive characterizations for the classes \(A_0\)-\(A_2\) and \(B\) by special properties of the shape operator (pseudo-parallelity, cyclic parallelity, and special formulas for the norm of the covariant derivative of the shape operator).
scientific article; zbMATH DE number 4198946