Complete space-like hypersurfaces with constant scalar curvature. (English) Zbl 1212.53084

Shu, Shichang

Arch. Math., Brno 44, No. 2, 105-114 (2008).

Summary: We characterize the \(n\)-dimensional \((n\geq 3)\) complete space-like 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 \(\big (\frac {d\{\log | \lambda ^2-(1-R)| ^{1/n}\}}{ds}\big )^2 -\lambda ^2+1\), 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 of \(\mathbb R^{n+2}_1\) which is parallel to a fixed \(E^n(s_0)\) and \(u=| \lambda ^2-(1-R)| ^{-1/n}\) satisfies the ordinary differential equation of order \(2\), \(\frac {d^2u}{ds^2}-u\big (\pm \frac {n-2}{2} \frac {1}{u^n}+R-2\big )=0\).

MSC:

53C42	Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53C20	Global Riemannian geometry, including pinching

Keywords:

de Sitter space; space-like hypersurface; scalar curvature; principal curvature; umbilical