Abstract
There are two smooth functions sigma and rho 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 sigma is harmonic. In the second part of this paper, we show that presence of a nontrivial concircular vector field v with connecting function rho 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) rho + t = 0, is necessary and sufficient for (M, g) to be isometric to either a sphere S-n(c) or to the Euclidean space E-n, where t is the scalar curvature.