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

Mathematical Subject Classification 2010

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