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 ξ with potential function ρ 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 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), dimM>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.
Highlights
It is well known that through cosmological considerations the space being homogeneous and isotropic in the large scale, picks the Robertson–Walker metrics
It amounts to the fact that an n-dimensional spacetime, n > 3, acquires the form I ×φ N, with metric g = −dt2 + φ2g, where I is an open interval, φ is a smooth positive function defined on I, and (N, g) is an (n − 1)-dimensional Riemannian manifold of constant curvature
An interesting characterization of generalized Robertson–Walker (GRW)-spacetime was obtained by Chen, by proving that a Lorentzian manifold (M, g) admits a non-trivial time-like concircular vector field, if, and only if, it is a GRW-spacetime
Summary
It is well known that through cosmological considerations the space being homogeneous and isotropic in the large scale, picks the Robertson–Walker metrics. We study the role of a time-like special torse-forming vector field ξ on a Lorentzian manifold (M, g) in characterizing GRW-spacetimes It is achieved by using the de Rham– Laplace operator (cf [12]) and a time-like special torse-forming vector field ξ with potential function ρ on a connected Lorentzian manifold (M, g), dimM > 5, through showing that ξ = σξ holds for a smooth function σ, if, and only if, (M, g) is a GRWspacetime (see Theorem 1). It is shown that if the associated function f of the special torse-forming vector field ξ on a connected Lorentzian manifold (M, g), dimM > 4, satisfies (i) f is nowhere zero and (ii) f is a solution of the Fischer–Marsden equation, (M, g) is a quasi-Einstein manifold (see Theorem 3). It is shown that if the scalar curvature τ of a connected Lorentzian manifold (M, g), dimM ≥ 4, is a constant and possesses a special torse-forming vector field ξ with potential function ρ and associated function f satisfying the above two conditions, the potential function ρ is an eigenfunction of the Laplace operator ∆ (see Corollary 1)
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