Abstract
We show presence a special torse-forming vector field (a particular form of torse-forming of a vector field) on generalized Robertson-Walker (GRW) spacetime, which is an eigenvector of the de Rham-Laplace operator. This paves the way to showing that the presence of a time-like special torse-forming vector field xi with potential function rho on a Lorentzian manifold (M, g), dimM > 5, which is an eigenvector of the de Rham Laplace operator, gives a characterization of a GRW-spacetime. We show that if, in addition, the function xi(rho) is nowhere zero, then the fibers of the GRW-spacetime are compact. Finally, we show that on a simply connected Lorentzian manifold (M, g) that admits a time-like special torse-forming vector field xi, there is a function f called the associated function of xi. It is shown that if a connected Lorentzian manifold (M, g), dimM > 4, admits a time-like special torse-forming vector field xi. with associated function f nowhere zero and satisfies the Fischer-Marsden equation, then (M, g) is a quasi-Einstein manifold.