J. Geom. (2018) 109:16
2018 Springer International Publishing AG,
part of Springer Nature
published online February 21, 2018
Journal of Geometry
The homogeneous ruled real hypersurface
in a complex hyperbolic space
Makoto Kimura, Sadahiro Maeda, and Hiromasa Tanabe
Abstract. We characterize the homogeneous ruled real hyperurface of a
complex hyperbolic space in the class of ruled real hypersurfaces having
constant mean curvature.
Mathematics Subject Classiﬁcation. Primary 53B25; Secondary 53C40.
Keywords. Complex hyperbolic space, Homogeneous ruled real hypersur-
face, Constant mean curvature, Integral curves of the vector ﬁeld U,The
ﬁrst curvature function.
For a nonzero constant c, we denote by
(c)ann-dimensional nonﬂat com-
plex space form of constant holomorphic sectional curvature c, namely a com-
plex n-dimensional complete and simply connected K¨ahler manifold of con-
stant holomorphic sectional curvature c. This space is holomorphically isomet-
ric to a complex projective space CP
(c) when c>0 or a complex hyperbolic
(c) when c<0.
Among typical examples of real hypersurfaces isometrically immersed into
(c), we pay particular attention to the class of ruled real hypersurfaces.
A ruled real hypersurface in
(c) is, roughly speaking, a real hypersurface
having a one-codimensional foliation whose leaves are totally geodesic complex
(c) (for details, see Sect. 3). In particular, studying ruled real
hypersurfaces in a complex hyperbolic space CH
(c) gives fruitful results. For
example, a homogeneous real hypersurface M
(c), i.e., M is an
orbit of some subgroup of the full isometry group I(CH
(c))(= U(1,n)), is
minimal if and only if M is ruled (cf. ). On the other hand, there exist two
classes of non-homogeneous minimal ruled real hypersurfaces in CH
respect to I(CH
(c)) ([1,4]). Motivated by these facts, in this paper we give a
Makoto Kimura is supported by JSPS KAKENHI Grant Number JP16K05119.