Complete spacelike hypersurfaces with constant scalar curvature

Shu, Schi Chang

Archivum mathematicum, Tome 44 (2008) no. 2, pp. 105-114.

Résumé

In this paper, we characterize the $n$-dimensional $(n\ge 3)$ complete spacelike hypersurfaces $M^n$ in a de Sitter space $S^{n+1}_1$ with constant scalar curvature and with two distinct principal curvatures one of which is simple.We show that $M^n$ is a locus of moving $(n-1)$-dimensional submanifold $M^{n-1}_1(s)$, along $M^{n-1}_1(s)$ the principal curvature $\lambda $ of multiplicity $n-1$ is constant and $M^{n-1}_1(s)$ is umbilical in $S^{n+1}_1$ and is contained in an $(n-1)$-dimensional sphere $S^{n-1}\big (c(s)\big )=E^n(s)\cap S^{n+1}_1$ and is of constant curvature $\big (\frac{d\lbrace \log |\lambda ^2-(1-R)|^{1/n}\rbrace }{ds}\big )^2-\lambda ^2+1$,where $s$ is the arc length of an orthogonal trajectory of the family $M^{n-1}_1(s)$, $E^n(s)$ is an $n$-dimensional linear subspace in $R^{n+2}_1$ which is parallel to a fixed $E^n(s_0)$ and $u=\big |\lambda ^2-(1-R)\big |^{-\frac{1}{n}}$ satisfies the ordinary differental equation of order 2, $\frac{d^2u}{ds^2}-u\big (\pm \frac{n-2}{2}\frac{1}{u^n}+R-2\big )=0$.

Détail

Classification : 53C20, 53C42
Keywords: de Sitter space; spacelike hypersurface; scalar curvature; principal curvature; umbilical