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''complete space-like submanifolds in a''

Let  $  \mathbf R _ {p} ^ {n + p + 1 } $
be an  $  ( n + p + 1 ) $-
dimensional [[Minkowski space|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|Riemannian manifold]] of index  $  p $
and of constant [[Curvature|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|Totally-geodesic manifold]]). It was proved by K. Akutagawa [[#References|[a1]]], Q.M. Cheng [[#References|[a2]]] and K.G. Ramanathan that complete space-like submanifolds with parallel [[Mean curvature|mean curvature]] vector in a de Sitter space  $  S _ {p} ^ {n + p } ( c ) $
are totally umbilical (cf. also [[Differential geometry|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|Ricci curvature]] is positive and the squared norm of the [[Second fundamental form|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 [[#References|[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|second fundamental form]].

Cf. also [[Anti-de Sitter space|Anti-de Sitter space]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K. Akutagawa,   "On space-like hypersurfaces with constant mean curvature in the de Sitter space"  ''Math. Z.'' , '''196'''  (1987)  pp. 13–19</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  Q. M. Cheng,   "Complete space-like submanifolds in a de Sitter space with parallel mean curvature vector"  ''Math. Z.'' , '''206'''  (1991)  pp. 333–339</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  Q. M. Cheng,   "Hypersurfaces of a Lorentz space form"  ''Arch. Math.'' , '''63'''  (1994)  pp. 271–281</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  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</TD></TR></table>