A characterization of the homogeneous ruled real hypersurface in a complex hyperbolic space in terms of the first curvature of some integral curves

Abstract

Due to the works of Berndt and others (J Reine Angew Math 395:132–141, 1989; Geom Dedicata 138:129–150, 2009; Trans Am Math Soc 359:3425–3438, 2007), we find that the class of all homogeneous real hypersurfaces M in \({\mathbb{C}H^n(c)}\) has just one example which is minimal in this space. Furthermore, in this ambient space, a homogeneous real hypersurface M is minimal if and only if it is ruled. This fact implies that it is interesting to study this minimal homogeneous real hypersurface from the viewpoint of the geometry of ruled real hypersurfaces. It is known that the shape operator A of every ruled real hypersurface M is given by the characteristic field \({\xi}\) and the vector field U. The purpose of this paper is to characterize this minimal homogeneous real hypersurface in \({\mathbb{C}H^n(c)}\) in the class of all ruled real hypersurfaces M by investigating the first curvature of all integral curves of the vector fields \({\xi}\) and U. Note that there exist minimal non-homogeneous ruled real hypersurfaces in \({\mathbb{C}H^n(c)}\) (see Geom Dedicata 79:267–286, 1999; Hokkaido Math J 43:1–14, 2014).