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, $\frac{d^2u}{ds^2}-u\big (\pm \frac{n-2}{2}\frac{1}{u^n}+R-2\big )=0$.
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, $\frac{d^2u}{ds^2}-u\big (\pm \frac{n-2}{2}\frac{1}{u^n}+R-2\big )=0$.
@article{ARM_2008_44_2_a2,
author = {Shu, Schi Chang},
title = {Complete spacelike hypersurfaces with constant scalar curvature},
journal = {Archivum mathematicum},
pages = {105--114},
year = {2008},
volume = {44},
number = {2},
mrnumber = {2432847},
zbl = {1212.53084},
language = {en},
url = {http://geodesic.mathdoc.fr/item/ARM_2008_44_2_a2/}
}
TY - JOUR
AU - Shu, Schi Chang
TI - Complete spacelike hypersurfaces with constant scalar curvature
JO - Archivum mathematicum
PY - 2008
SP - 105
EP - 114
VL - 44
IS - 2
UR - http://geodesic.mathdoc.fr/item/ARM_2008_44_2_a2/
LA - en
ID - ARM_2008_44_2_a2
ER -
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