Let
be an
-dimensional
Minkowski space
of index
,
i.e.,
and is equipped with the
Lorentz metric
.
For
,
let
Thus,
is an
-dimensional
indefinite
Riemannian manifold
of index
and of constant
curvature
.
It is called an
-dimensional
de Sitter space of constant curvature
and of index
.
E. Calabi,
S.Y. Cheng
and
S.T. Yau
proved that a complete maximal space-like hypersurface in a Minkowski space
possesses a remarkable Bernstein property. As a generalization of the Bernstein-type
problem,
S. Ishihara
proved that a complete maximal space-like submanifold in a de Sitter space
is totally geodesic (cf.
Totally-geodesic manifold).
It was proved by
K. Akutagawa
[a1],
Q.M. Cheng
[a2]
and
K.G. Ramanathan
that complete space-like submanifolds with parallel
mean curvature
vector in a de Sitter space
are totally umbilical (cf. also
Differential geometry)
if
1)
,
when
;
2)
,
when
.
The conditions 1) and 2) are best possible. When
,
Akutagawa and Ramanathan constructed many examples of space-like submanifolds
in
that are not totally umbilical. When
,
,
where
and
,
is a complete space-like hypersurface in
of constant mean curvature
that is not totally umbilical and satisfies
.
Cheng
gave a characterization of complete non-compact hypersurfaces in
with
:
a complete non-compact hypersurface in
with
is either isometric to
or its
Ricci curvature
is positive and the squared norm of the
second fundamental form
is a subharmonic function. Therefore, the
Cheeger–Gromoll splitting theorem
implies that a complete non-compact hypersurface
in
with
is isometric to
if the number of its ends is not less than
.
S. Montiel
[a4]
has proved that a compact space-like hypersurface in
of constant mean curvature is totally umbilical, and
Aiyama
has generalized this to compact space-like submanifolds in
with parallel mean curvature vector and flat normal bundle. Complete space-like
hypersurfaces in
with constant mean curvature have also been characterized under conditions on the
squared norm of the
second fundamental form.
Cf. also
Anti-de Sitter space.