Classification of spacelike conformal Einstein hypersurfaces in Lorentzian space $ \mathbb{R}^{n+1}_1 $

Abstract

Let $ f: M^n\to \mathbb{R}^{n+1}_1 $ be an $ n $-dimensional umbilic-free spacelike hypersurface in the $ (n+1) $-dimensional Lorentzian space $ \mathbb{R}^{n+1}_1 $ with an induced metric $ I $. Let $ II $ be the second fundamental form and $ H $ the mean curvature of $ f $. One can define the conformal metric $ g = \frac{n}{n-1}(\|II\|^2-nH^2)I $ on $ f(M^n) $, which is invariant under the conformal transformation group of $ \mathbb{R}^{n+1}_1 $. If the Ricci curvature of $ g $ is constant, then the spacelike hypersurface $ f $ is called a conformal Einstein hypersurface. In this paper, we completely classify the $ n $-dimensional spacelike conformal Einstein hypersurfaces up to a conformal transformation of $ \mathbb{R}^{n+1}_1 $.

Keywords

• conformal metric
• conformal sectional curvature
• conformal einstein hypersurface
• conformal transformation group