On ruled real hypersurfaces in complex space forms
An immersed real hypersurface of a complex space form \(M_\kappa^{m+1}\) of constant holomorphic sectional curvature \(\kappa\) is said to be ruled if it is foliated by totally geodesic complex hypersurfaces of \(M_\kappa^{m+1}\). Such hypersurfaces can be parametrized by maps of the form \(f:\mathbb{R}\times M_\kappa^m\to M^{m+1}_\kappa\) for which the maps \(f_t:M^m_\kappa\to M_\kappa^{m+1}\), \(q\mapsto f(t,q)\) are totally geodesic, holomorphic, isometric immersions, and there exists a point \(q_0\in M^m_\kappa\) such that the curve \(\alpha:t\mapsto f(t,q_0)\) intersects \(f_t(M^m_\kappa)\) perpendicularly at each point and for each \(v\in T_{q_0}M^m_\kappa\) the vector field \(t\mapsto F_{t*}v\) along \(\alpha\) is parallel in the tangent bundle of \(M^m_\kappa\). The authors study the geometry of ruled real hypersurfaces in complex space forms via such parametrizations. The main results are as follows. The set of points where \(f\) is not immersive is thin and its complement connected. The pullback via \(f\) of the metric on \(M^{m+1}_\kappa\) turns \(\mathbb{R}\times M^m_\kappa\) into a twisted product with singularities. The authors also classify all minimal ruled real hypersurfaces in non-flat complex space forms. Finally, they show that the ruled real hypersurface in complex hyperbolic space \(\mathbb{C} H^{m+1}\), which is induced by a horocycle in some totally geodesic totally real \(\mathbb{R} H^2\subset\mathbb{C} H^{m+1}\), is homogeneous. This provides the first example of a homogeneous real hypersurface in complex hyperbolic space for which the structure vector field is not a principal curvature vector everywhere. In contrast, it is known that the structure vector field on any homogeneous real hypersurface in complex projective space has this property.
scientific article; zbMATH DE number 1270439