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.