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
anti-de Sitter space of constant curvature
and of
index
.
A hypersurface
of
is said to be
space-like
if the metric on
induced by that of ambient space
is positive definite. The
mean curvature
of
is defined as in the case of Riemannian manifolds. By definition,
is a maximal hypersurface if the mean curvature
of
is identically zero.
S. Ishihara
proved that a complete maximal space-like hypersurface
in
satisfies
,
and
if and only if
is isometric to the hyperbolic cylinder
,
where
is the squared norm of the
second fundamental form
of
and
,
,
is a
-dimensional
hyperbolic space of constant curvature
.
The rigidity of the hyperbolic cylinder
in
was
proved
by
U.-H. Ki,
H.S. Kim
and
H. Nakagawa
[a3]:
for a given
integer
and constant
,
there exists a constant
,
depending on
and
,
such that the hyperbolic cylinder
is the only complete maximal space-like hypersurface in
of constant scalar curvature and such that
.
In particular, for
,
Q.M. Cheng
[a1]
has characterized the complete maximal space-like hypersurfaces in
under the condition of constant Gauss–Kronecker curvature (cf.
Gaussian curvature):
Let
be a
-dimensional
complete maximal space-like hypersurface of
.
Now:
1)
if the Gauss–Kronecker curvature of
is a non-zero constant, then
is the hyperbolic cylinder
;
2)
if the
scalar curvature
is constant and
,
then
is the hyperbolic cylinder
.
There are no complete maximal space-like hypersurfaces in
with constant scalar curvature and
.
On the other hand, complete space-like submanifolds in anti-de Sitter
spaces with parallel mean curvature have been investigated by many authors.
Cf. also
De Sitter space.