# De Sitter space

complete space-like submanifolds in a

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

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

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

Thus, $S _ {p} ^ {n + p } ( c )$ is an $( n + p )$- dimensional indefinite Riemannian manifold of index $p$ and of constant curvature $c$. It is called an $( n + p )$- dimensional de Sitter space of constant curvature $c$ and of index $p$. E. Calabi, S.Y. Cheng and S.T. Yau proved that a complete maximal space-like hypersurface in a Minkowski space $\mathbf R _ {1} ^ {n + 1 }$ 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 $S _ {p} ^ {n + p } ( c )$ 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 $S _ {p} ^ {n + p } ( c )$ are totally umbilical (cf. also Differential geometry) if

1) $H ^ {2} \leq c$, when $n = 2$;

2) $n ^ {2} H ^ {2} < 4 ( n - 1 ) c$, when $n \geq 3$. The conditions 1) and 2) are best possible. When $n = 2$, Akutagawa and Ramanathan constructed many examples of space-like submanifolds in $S _ {1} ^ {3} ( c )$ that are not totally umbilical. When $n \geq 3$, $H ^ {1} ( c _ {1} ) \times S ^ {n - 1 } ( c _ {2} )$, where $c _ {1} = ( 2 - n ) c$ and $c _ {2} = [ { {( n - 2 ) } / {( n - 1 ) } } ] c$, is a complete space-like hypersurface in $S _ {1} ^ {n + 1 } ( c )$ of constant mean curvature $H$ that is not totally umbilical and satisfies $n ^ {2} H ^ {2} = 4 ( n - 1 ) c$. Cheng gave a characterization of complete non-compact hypersurfaces in $S _ {1} ^ {n + 1 } ( c )$ with $n ^ {2} H ^ {2} = 4 ( n - 1 ) c$: a complete non-compact hypersurface in $S _ {1} ^ {n + 1 } ( c )$ with $n ^ {2} H ^ {2} = 4 ( n - 1 ) c$ is either isometric to $H ^ {1} ( c _ {1} ) \times S ^ {n - 1 } ( c _ {2} )$ 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 $M$ in $S _ {1} ^ {n + 1 } ( c )$ with $n ^ {2} H ^ {2} = 4 ( n - 1 ) c$ is isometric to $H ^ {1} ( c _ {1} ) \times S ^ {n - 1 } ( c _ {2} )$ if the number of its ends is not less than $2$. S. Montiel [a4] has proved that a compact space-like hypersurface in $S _ {1} ^ {n + 1 } ( c )$ of constant mean curvature is totally umbilical, and Aiyama has generalized this to compact space-like submanifolds in $S _ {p} ^ {n + p } ( c )$ with parallel mean curvature vector and flat normal bundle. Complete space-like hypersurfaces in $S _ {1} ^ {n + 1 } ( c )$ 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.

#### References

 [a1] K. Akutagawa, "On space-like hypersurfaces with constant mean curvature in the de Sitter space" Math. Z. , 196 (1987) pp. 13–19 [a2] Q. M. Cheng, "Complete space-like submanifolds in a de Sitter space with parallel mean curvature vector" Math. Z. , 206 (1991) pp. 333–339 [a3] Q. M. Cheng, "Hypersurfaces of a Lorentz space form" Arch. Math. , 63 (1994) pp. 271–281 [a4] S. Montiel, "An integral inequality for compact space-like hypersurfaces in a de Sitter space and application to the case of constant mean curvature" Indiana Univ. Math. J. , 37 (1988) pp. 909–917
How to Cite This Entry:
De Sitter space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=De_Sitter_space&oldid=46589
This article was adapted from an original article by Qingming Cheng (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article