Anti-de Sitter space
complete maximal space-like hypersurfaces in an
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.
References
| [a1] | Q.M. Cheng, "Complete maximal space-like hypersurfaces of  " Manuscr. Math. , 82 (1994) pp. 149–160 |
| [a2] | T. Ishikawa, "Maximal space-like submanifolds of a pseudo–Riemannian space of constant curvature" Michigan Math. J. , 35 (1988) pp. 345–352 |
| [a3] | U-H. Ki, H.S. Kim, H. Nakagawa, "Complete maximal space-like hypersurfaces of an anti-de Sitter space" Kyungpook Math. J. , 31 (1991) pp. 131–141 |
Anti-de Sitter space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Anti-de_Sitter_space&oldid=45191