Ruled real hypersurfaces in the complex hyperbolic quadric

Abstract

Abstract In this article, we introduce a new family of real hypersurfaces in the complex hyperbolic quadric Q n ∗ = S O 2 , n o ∕ S O 2 S O n {{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 Q n ∗ {{Q}^{n}}^{\ast } as a homogeneous real hypersurface generated by a A {\mathfrak{A}} -principal horocycle in a real form R H n {\mathbb{R}}{H}^{n} . So, in this article, we compute a detailed expression of the shape operator for ruled real hypersurfaces in Q n ∗ {{Q}^{n}}^{\ast } and investigate their characterizations in terms of the shape operator and the integrable distribution C = { X ∈ T M ∣ 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 Q n ∗ {{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 Q n ∗ {{Q}^{n}}^{\ast } is A {\mathfrak{A}} -principal. On the other hand, it is known that all contact real hypersurfaces in Q n ∗ {{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 Q n ∗ {{Q}^{n}}^{\ast } in terms of η \eta -parallel shape operator.