Classification of real hypersurfaces in complex hyperbolic space in terms of constant \(\phi\)-holomorphic sectional curvatures
If \(M\) is a real hypersurface of the complex hyperbolic space \(\mathbb{C} H^n\) (of constant holomorphic sectional curvature \(-4)\). An almost contact structure \((\varphi,\xi)\) is induced on it by the following conditions: \(J\xi\) is a unit normal of \(M\) and \(\varphi\) is the tensor field with kernel \(\mathbb{R}\xi\) satisfying \(\varphi X=JX\) for \(X\perp\xi\). By definition, \(M\) has constant \(\varphi\)-holomorphical sectional curvature \(H\in\mathbb{R}\), if for every unit vector \(v\in T_xM\) perpendicular to \(\xi_x\) the sectional curvature of \(\text{span} \{v,\varphi v\}\) equals \(H\). The authors derive a classification of these real hypersurfaces of constant \(\varphi\)-holomorphical sectional curvature. But the reviewer cannot see that the classification is complete, because he cannot reduce the problem to the cases (A) and (B) on the basis of the formulas (2.5) and (2.7) (see p. 805; notice that the numbers \(\overline \lambda_j\) are not complex conjugates of the \(\lambda_j)\).
scientific article; zbMATH DE number 1725467