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Ruled real hypersurfaces in the complex hyperbolic quadric

Lee Hyunjin, Suh Young Jin, Woo Changhwa

Demonstratio Mathematica

2023

Vol. 56, nr 1

art. no. 20230258

In this article, we introduce a new family of real hypersurfaces in the complex hyperbolic quadric [formula] , namely, the ruled real hypersurfaces foliated by complex hypersurfaces. Berndt described an example of such a real hypersurface in [formula] as a homogeneous real hypersurface generated by a [symbol] -principal horocycle in a real form [formula]. So, in this article, we compute a detailed expression of the shape operator for ruled real hypersurfaces in [formula] and investigate their characterizations in terms of the shape operator and the integrable distribution [formula]. Then, by using these observations, we give two kinds of classifications of real hypersurfaces in [formula] satisfying η -parallelism under either η -commutativity of the shape operator or integrability of the distribution C. Moreover, we prove that the unit normal vector field of a real hypersurface with η -parallel shape operator in [formula] is [symbol] -principal. On the other hand, it is known that all contact real hypersurfaces in [formula] have a [symbol] -principal normal vector field. Motivated by these results, we give a characterization of contact real hypersurfaces in [formula] in terms of η -parallel shape operator.