Complete spacelike hypersurfaces with constant scalar curvature

Shu, Schi Chang. "Complete spacelike hypersurfaces with constant scalar curvature." Archivum Mathematicum 044.2 (2008): 105-114. <http://eudml.org/doc/252263>.

@article{Shu2008,
abstract = {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, $\frac\{d^2u\}\{ds^2\}-u\big (\pm \frac\{n-2\}\{2\}\frac\{1\}\{u^n\}+R-2\big )=0$.},
author = {Shu, Schi Chang},
journal = {Archivum Mathematicum},
keywords = {de Sitter space; spacelike hypersurface; scalar curvature; principal curvature; umbilical; de Sitter space; space-like hypersurface; scalar curvature; principal curvature; umbilical},
language = {eng},
number = {2},
pages = {105-114},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Complete spacelike hypersurfaces with constant scalar curvature},
url = {http://eudml.org/doc/252263},
volume = {044},
year = {2008},
}

TY - JOUR
AU - Shu, Schi Chang
TI - Complete spacelike hypersurfaces with constant scalar curvature
JO - Archivum Mathematicum
PY - 2008
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 044
IS - 2
SP - 105
EP - 114
AB - 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, $\frac{d^2u}{ds^2}-u\big (\pm \frac{n-2}{2}\frac{1}{u^n}+R-2\big )=0$.
LA - eng
KW - de Sitter space; spacelike hypersurface; scalar curvature; principal curvature; umbilical; de Sitter space; space-like hypersurface; scalar curvature; principal curvature; umbilical
UR - http://eudml.org/doc/252263
ER -