There are two smooth functions σ and ρ associated to a nontrivial concircular vector field v on a connected Riemannian manifold (M,g), called potential function and connecting function. In this paper, we show that presence of a timelike nontrivial concircular vector field influences the geometry of generalized Robertson–Walker space-times. We use a timelike concircular vector field v on an n -dimensional connected conformally flat Lorentzian manifold, n > 2, to find a characterization of generalized Robertson–Walker space-time with fibers Einstein manifolds. It is interesting to note that for n = 4 the concircular vector field annihilates energy-momentum tensor and also that in this case the potential function σ is harmonic. In the second part of this paper, we show that presence of a nontrivial concircular vector field v with connecting function ρ on a complete and connected n -dimensional conformally flat Riemannian manifold (M,g), n > 2, with Ricci curvature Ric(v,v) non-negative, satisfying n(n − 1)ρ + τ = 0, is necessary and sufficient for (M,g) to be isometric to either a sphere Sn(c) or to the Euclidean space En, where τ is the scalar curvature.

中文翻译：

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GRW-时空和共形平面空间上的共圆性

有两个平滑函数σ和ρ关联到一个非平凡的共圆向量场v在连通的黎曼流形上(米,G)，称为势函数和连接函数。在本文中，我们展示了类时非平凡共圆向量场的存在会影响广义 Robertson-Walker 时空的几何形状。我们使用类时同圆矢量场v在一个n-维连接的保形平坦洛伦兹流形，n > 2，以找到具有纤维爱因斯坦流形的广义 Robertson-Walker 时空的表征。有趣的是，对于n = 4同圆矢量场消除能量-动量张量，并且在这种情况下，势函数σ是谐波。在本文的第二部分，我们证明了非平凡的同圆向量场的存在v带连接功能ρ在一个完整的和连接的n维共形平坦黎曼流形(米,G),n > 2, 具有里奇曲率里克(v,v)非负的，令人满意的n(n - 1)ρ + τ = 0, 是必要且充分的(米,G)与任一球体等距小号n(C)或到欧几里得空间乙n， 在哪里τ是标量曲率。