A Characterization of GRW Spacetimes

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 ξ with potential function ρ on a Lorentzian manifold ( M , g ) , dim M > 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 ξ ( ρ ) 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 ξ , there is a function f called the associated function of ξ . It is shown that if a connected Lorentzian manifold ( M , g ) , dim M > 4 , admits a time-like special torse-forming vector field ξ with associated function f nowhere zero and satisfies the Fischer–Marsden equation, then ( M , g ) is a quasi-Einstein manifold.