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

Yayun Chen; Tongzhu Li; Yayun Chen; Tongzhu Li

doi:10.3934/math.20231182

AIMS Mathematics

2023, Volume 8, Issue 10: 23247-23271. doi: 10.3934/math.20231182

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Research article

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

Yayun Chen ¹,
Tongzhu Li ^{2
,
,}

1.
Department of Mathematics, Beijing Institute of Technology, Beijing 100081, China
2.
Beijing Key Laboratory on MCAACI, Beijing 100081, China

Received: 29 March 2023 Revised: 26 June 2023 Accepted: 28 June 2023 Published: 21 July 2023
MSC : 53A30, 53B30

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 $.
- conformal metric,
- conformal sectional curvature,
- conformal Einstein hypersurface,
- conformal transformation group
Citation: Yayun Chen, Tongzhu Li. Classification of spacelike conformal Einstein hypersurfaces in Lorentzian space $ \mathbb{R}^{n+1}_1 $[J]. AIMS Mathematics, 2023, 8(10): 23247-23271. doi: 10.3934/math.20231182

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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 $.

References

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