#### Vol. 300, No. 1, 2019

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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
##### 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 ${\mathbb{S}}_{1}^{2}\left(1\right),\phantom{\rule{1em}{0ex}}{ℝ}_{1}^{2},\phantom{\rule{1em}{0ex}}{ℝ}_{1+}^{2}$, respectively.

##### Keywords
conformal metric, conformal sectional curvature, conformal second fundamental form, curvature-spiral
##### Mathematical Subject Classification 2010
Primary: 53A30, 53B25