# Anti-de Sitter space

complete maximal space-like hypersurfaces in an

Let $\mathbf R _ {p + 1 } ^ {n + p + 1 }$ be an $( n + p + 1 )$- dimensional Minkowski space of index $p + 1$, i.e., $\mathbf R _ {p + 1 } ^ {n + p + 1 } = \{ ( x _ {1} \dots x _ {n + p + 1 } ) \in \mathbf R ^ {n + p + 1 } \}$ and is equipped with the Lorentz metric $\sum _ {i = 1 } ^ {n} ( dx _ {i} ) ^ {2} - \sum _ {j = 1 } ^ {p + 1 } ( dx _ {n + j } ) ^ {2}$. For $c > 0$, let

$$H _ {p} ^ {n + p } ( c ) = \{ {x \in \mathbf R _ {p + 1 } ^ {n + p + 1 } } :$$

$$\ {} {x _ {1} ^ {2} + \dots + x _ {n} ^ {2} - x _ {n + 1 } ^ {2} - \dots - x _ {n + p + 1 } ^ {2} = - {1 / c } } \} .$$

Thus, $H _ {p} ^ {n + p } ( c )$ is an $( n + p )$- dimensional indefinite Riemannian manifold of index $p$ and of constant curvature $- c$. It is called an $( n + p )$- dimensional anti-de Sitter space of constant curvature $- c$ and of index $p$. A hypersurface $M$ of $H _ {1} ^ {n + 1 } ( c )$ is said to be space-like if the metric on $M$ induced by that of ambient space $H _ {1} ^ {n + 1 } ( c )$ is positive definite. The mean curvature $H$ of $M$ is defined as in the case of Riemannian manifolds. By definition, $M$ is a maximal hypersurface if the mean curvature $H$ of $M$ is identically zero. S. Ishihara proved that a complete maximal space-like hypersurface $M$ in $H _ {1} ^ {n + 1 } ( c )$ satisfies $S \leq nc$, and $S = nc$ if and only if $M$ is isometric to the hyperbolic cylinder $H ^ {k} ( c _ {1} ) \times H ^ {n - k } ( c _ {2} )$, where $S$ is the squared norm of the second fundamental form of $M$ and $H ^ {k} ( c _ {i} )$, $i = 1, 2$, is a $k$- dimensional hyperbolic space of constant curvature $c _ {i}$. The rigidity of the hyperbolic cylinder $H ^ {k} ( c _ {1} ) \times H ^ {n - k } ( c _ {2} )$ in $H _ {1} ^ {n + 1 } ( c )$ was proved by U.-H. Ki, H.S. Kim and H. Nakagawa [a3]: for a given integer $n$ and constant $c > 0$, there exists a constant $C < nc$, depending on $n$ and $c$, such that the hyperbolic cylinder $H ^ {k} ( c _ {1} ) \times H ^ {n - k } ( c _ {2} )$ is the only complete maximal space-like hypersurface in $H _ {1} ^ {n + 1 } ( c )$ of constant scalar curvature and such that $S > C$. In particular, for $n = 3$, Q.M. Cheng [a1] has characterized the complete maximal space-like hypersurfaces in $H _ {1} ^ {4} ( c )$ under the condition of constant Gauss–Kronecker curvature (cf. Gaussian curvature): Let $M$ be a $3$- dimensional complete maximal space-like hypersurface of $H _ {1} ^ {4} ( c )$. Now:

1) if the Gauss–Kronecker curvature of $M$ is a non-zero constant, then $M$ is the hyperbolic cylinder $H ^ {1} ( c _ {1} ) \times H ^ {2} ( c _ {2} )$;

2) if the scalar curvature $K$ is constant and $\inf K ^ {2} > 0$, then $M$ is the hyperbolic cylinder $H ^ {1} ( c _ {1} ) \times H ^ {2} ( c _ {2} )$. There are no complete maximal space-like hypersurfaces in $H _ {1} ^ {4} ( c )$ with constant scalar curvature and $\sup K ^ {2} < { {S ^ {3} } / {54 } }$.

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 $H_1^4(c)$" 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
How to Cite This Entry:
Anti-de Sitter space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Anti-de_Sitter_space&oldid=53263
This article was adapted from an original article by Qingming Cheng (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article