Four-dimensional homogeneous semi-symmetric Lorentzian manifolds
Four-dimensional homogeneous semi-symmetric Lorentzian manifolds
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四维齐次半对称洛伦兹流形
DOI：
10.1016/j.difgeo.2017.08.009
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发表时间：
2016
期刊：
影响因子：
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通讯作者：
A. Ikemakhen
中科院分区：
文献类型：
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作者：
Abderazak Benroumane;M. Boucetta;A. Ikemakhen
伪黎曼流形(M，g)称为半对称的，如果它的曲率张量K满足K，K=0。对于任何向量场X，Y，Z，T，这等价于(1)[K(X，Y)，K(Z，T)]=K(K(X，Y)Z，T)+K(Z，K(X，Y)T)，半对称伪黎曼流形推广了明显的局部对称流形(∇K=0)。它们还推广了二阶局部对称流形(∇2 K=0和∇K≠0)。半对称黎曼流形由E.Cartan[7]首先研究，Takagi[13]给出了半对称非局部对称黎曼流形的第一个例子。最近，Szabo[11]、[12]给出了这些流形的完整描述。在这项研究中，Szabo使用了适用于黎曼坐着的强有力的结果，这表明类似的半对称洛伦兹流形的研究要困难得多。据我们所知，关于三维局部齐次半对称洛伦兹流形的结果很少，D·Alekseevsky和A·Galaev在[1]中对二阶局部对称Lorentz流形进行了分类。在黎曼情形下，每个齐次半对称流形实际上是局部对称的，而在洛伦兹情形下，它们是不局部对称的齐次半对称洛伦兹流形.本文致力于研究洛伦兹向量空间上的半对称曲率代数张量和4维单连通半对称齐次洛伦兹流形的分类.我们的主要成果有：
A pseudo-Riemannian manifold (M, g) is said to be semi-symmetric if its curvature tensor K satisfies K. K= 0. This is equivalent to (1)[K (X, Y), K (Z, T)]= K (K (X, Y) Z, T)+ K (Z, K (X, Y) T), for any vector fields X, Y, Z, T. Semi-symmetric pseudo-Riemannian manifolds generalize obviously locally symmetric manifolds (∇ K= 0). They also generalize second-order locally symmetric manifolds (∇ 2 K= 0 and∇ K≠ 0). Semi-symmetric Riemannian manifolds have been first investigated by E. Cartan [7] and the first example of a semi-symmetric not locally symmetric Riemannian manifold was given by Takagi [13]. More recently, Szabo [11],[12] gave a complete description of these manifolds. In this study, Szabo used strong results proper to the Riemannian sitting which suggests that a similar study of semi-symmetric Lorentzian manifolds is far more difficult. To our knowledge, there are only few results on three dimensional locally homogeneous semi-symmetric Lorentzian manifolds [3],[4] and second-order locally symmetric Lorentzian manifolds have been classified by D. Alekseevsky and A. Galaev in [1]. While in the Riemannian case every homogeneous semi-symmetric manifold is actually locally symmetric, in the Lorentzian case they are homogeneous semi-symmetric Lorentzian manifolds which are not locally symmetric.This paper is devoted to the study of semi-symmetric curvature algebraic tensors on a Lorentzian vector space and to the classification of 4-dimensional simply-connected semi-symmetric homogeneous Lorentzian manifolds. There are our main results: