Ruled real hypersurfaces in the complex hyperbolic quadric

초록

In this article, we introduce a new family of real hypersurfaces in the complex hyperbolic quadric Q(n & lowast;)=SO2,no/SO2SOn , namely, the ruled real hypersurfaces foliated by complex hypersurfaces. Berndt described an example of such a real hypersurface in Q(n & lowast;) as a homogeneous real hypersurface generated by a (sic) -principal horocycle in a real form RHn . So, in this article, we compute a detailed expression of the shape operator for ruled real hypersurfaces in Q(n & lowast;) and investigate their characterizations in terms of the shape operator and the integrable distribution C={X is an element of TM divided by X perpendicular to xi} . Then, by using these observations, we give two kinds of classifications of real hypersurfaces in Q(n & lowast;) satisfying eta -parallelism under either eta -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 eta -parallel shape operator in Q(n & lowast;) is (sic) -principal. On the other hand, it is known that all contact real hypersurfaces in Q(n & lowast;) have a (sic) -principal normal vector field. Motivated by these results, we give a characterization of contact real hypersurfaces in Q(n & lowast;) in terms of eta -parallel shape operator.

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