Spacelike hypersurfaces with constant conformal sectional curvature in $\mathbb{R}^{n+1}_1$

Xiu Ji, Tongzhu Li and Huafei Sun

Vol. 300 (2019), No. 1, 17–37

DOI: 10.2140/pjm.2019.300.17

Abstract

Let $f:M^n\to {ℝ}_1^{n+1}$ be an $n$-dimensional umbilic-free spacelike hypersurface in the $\left(n+1\right)$-dimensional Lorentzian space ${ℝ}_1^{n+1}$. One can define the conformal metric $g$ on $f$ which is invariant under the conformal transformation group of ${ℝ}_1^{n+1}$. We classify the $n$-dimensional spacelike hypersurfaces with constant sectional curvature with respect to the conformal metric $g$ when $n\ge 3$. Such spacelike hypersurfaces are obtained by the standard construction of cylinders, cones or hypersurfaces of revolution over certain spirals in the $2$-dimensional Lorentzian space forms ${\mathbb{S}}_1^2\left(1\right),{ℝ}_1^2,{ℝ}_{1+}^2$, respectively.

Keywords

conformal metric, conformal sectional curvature, conformal second fundamental form, curvature-spiral

Mathematical Subject Classification 2010

Primary: 53A30, 53B25

Milestones

Received: 11 April 2017

Revised: 16 May 2018

Accepted: 9 September 2018

Published: 20 July 2019