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''complete space-like submanifolds in a''
 
''complete space-like submanifolds in a''
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110040/d1100401.png" /> be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110040/d1100402.png" />-dimensional [[Minkowski space|Minkowski space]] of index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110040/d1100403.png" />, i.e., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110040/d1100404.png" /> and is equipped with the Lorentz metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110040/d1100405.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110040/d1100406.png" />, let
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110040/d1100407.png" /></td> </tr></table>
+
$$
 +
S _ {p} ^ {n + p } ( c ) = \{ {x \in \mathbf R _ {p} ^ {n + p + 1 } } :  
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110040/d1100408.png" /></td> </tr></table>
+
$$
 +
\
 +
{} {x _ {1}  ^ {2} + \dots + x _ {n + 1 }  ^ {2} - x _ {n + 2 }  ^ {2} - \dots - x _ {n + p + 1 }  ^ {2} = {1 / c } } \} .
 +
$$
  
Thus, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110040/d1100409.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110040/d11004010.png" />-dimensional indefinite [[Riemannian manifold|Riemannian manifold]] of index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110040/d11004011.png" /> and of constant [[Curvature|curvature]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110040/d11004012.png" />. It is called an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110040/d11004013.png" />-dimensional de Sitter space of constant curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110040/d11004014.png" /> and of index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110040/d11004015.png" />. E. Calabi, S.Y. Cheng and S.T. Yau proved that a complete maximal space-like hypersurface in a Minkowski space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110040/d11004016.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110040/d11004017.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110040/d11004018.png" /> are totally umbilical (cf. also [[Differential geometry|Differential geometry]]) if
+
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) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110040/d11004019.png" />, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110040/d11004020.png" />;
+
1) $  H  ^ {2} \leq  c $,  
 +
when $  n = 2 $;
  
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110040/d11004021.png" />, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110040/d11004022.png" />. The conditions 1) and 2) are best possible. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110040/d11004023.png" />, Akutagawa and Ramanathan constructed many examples of space-like submanifolds in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110040/d11004024.png" /> that are not totally umbilical. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110040/d11004025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110040/d11004026.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110040/d11004027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110040/d11004028.png" />, is a complete space-like hypersurface in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110040/d11004029.png" /> of constant mean curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110040/d11004030.png" /> that is not totally umbilical and satisfies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110040/d11004031.png" />. Cheng gave a characterization of complete non-compact hypersurfaces in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110040/d11004032.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110040/d11004033.png" />: a complete non-compact hypersurface in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110040/d11004034.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110040/d11004035.png" /> is either isometric to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110040/d11004036.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110040/d11004037.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110040/d11004038.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110040/d11004039.png" /> is isometric to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110040/d11004040.png" /> if the number of its ends is not less than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110040/d11004041.png" />. S. Montiel [[#References|[a4]]] has proved that a compact space-like hypersurface in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110040/d11004042.png" /> of constant mean curvature is totally umbilical, and Aiyama has generalized this to compact space-like submanifolds in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110040/d11004043.png" /> with parallel mean curvature vector and flat normal bundle. Complete space-like hypersurfaces in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110040/d11004044.png" /> with constant mean curvature have also been characterized under conditions on the squared norm of the [[Second fundamental form|second fundamental form]].
+
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]].
 
Cf. also [[Anti-de Sitter space|Anti-de Sitter space]].

Revision as of 17:32, 5 June 2020


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=13196
This article was adapted from an original article by Qingming Cheng (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article