Ruled real hypersurfaces in the complex hyperbolic quadric

Abstract

In this article, we introduce a new family of real hypersurfaces in the complex hyperbolic quadric Qn∗=SO2,no∕SO2SOn{{Q}^{n}}^{\ast }=S{O}_{2,n}^{o}/S{O}_{2}S{O}_{n}, namely, the ruled real hypersurfaces foliated by complex hypersurfaces. Berndt described an example of such a real hypersurface in Qn∗{{Q}^{n}}^{\ast } as a homogeneous real hypersurface generated by a A{\mathfrak{A}}-principal horocycle in a real form RHn{\mathbb{R}}{H}^{n}. So, in this article, we compute a detailed expression of the shape operator for ruled real hypersurfaces in Qn∗{{Q}^{n}}^{\ast } and investigate their characterizations in terms of the shape operator and the integrable distribution C={X∈TM∣X⊥ξ}{\mathcal{C}}=\left\{X\in TM| X\perp \xi \right\}. Then, by using these observations, we give two kinds of classifications of real hypersurfaces in Qn∗{{Q}^{n}}^{\ast } satisfying η\eta -parallelism under either η\eta -commutativity of the shape operator or integrability of the distribution C{\mathcal{C}}. Moreover, we prove that the unit normal vector field of a real hypersurface with η\eta -parallel shape operator in Qn∗{{Q}^{n}}^{\ast } is A{\mathfrak{A}}-principal. On the other hand, it is known that all contact real hypersurfaces in Qn∗{{Q}^{n}}^{\ast } have a A{\mathfrak{A}}-principal normal vector field. Motivated by these results, we give a characterization of contact real hypersurfaces in Qn∗{{Q}^{n}}^{\ast } in terms of η\eta -parallel shape operator.

Keywords

• ruled real hypersurface
• η-parallel shape operator
• η-commuting shape operator
• singular vector fields
• complex hyperbolic quadric
• 53c40
• 53c55