De Sitter space

complete space-like submanifolds in a

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 de Sitter space of constant curvature and of index . E. Calabi, S.Y. Cheng and S.T. Yau proved that a complete maximal space-like hypersurface in a Minkowski space 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 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 are totally umbilical (cf. also Differential geometry) if

1) , when ;

2) , when . The conditions 1) and 2) are best possible. When , Akutagawa and Ramanathan constructed many examples of space-like submanifolds in that are not totally umbilical. When , , where and , is a complete space-like hypersurface in of constant mean curvature that is not totally umbilical and satisfies . Cheng gave a characterization of complete non-compact hypersurfaces in with : a complete non-compact hypersurface in with is either isometric to 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 in with is isometric to if the number of its ends is not less than . S. Montiel [a4] has proved that a compact space-like hypersurface in of constant mean curvature is totally umbilical, and Aiyama has generalized this to compact space-like submanifolds in with parallel mean curvature vector and flat normal bundle. Complete space-like hypersurfaces in 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