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. ,


Introduction
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 = −dt 2 + ϕ 2 g, 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 n-dimensional generalized Robertson-Walker spacetime (GRW-spacetime) is I × ϕ N, with metric g = −dt 2 + ϕ 2 g, where (N, g) is an (n − 1)-dimensional Riemannian manifold (cf. [1,2]). An interesting characterization of GRW-spacetime was obtained by Chen (cf. [3]), 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. Additionally, for interesting characterizations of GRW-spacetimes using torse-forming vector fields and Weyl tensors, we refer to (cf. [4,5]).
Yano generalized concircular vector fields by introducing a torse-forming vector field on semi-Riemannian manifold (M, g) (cf. [10]), defined by: where α is a 1-form called the torsed 1-form. Naturally, if α = 0, then a torse-forming vector field is a concircular vector field. These vector fields are also used in characterizing a GRW-spacetime (cf. [2,4]). In [11], Chen considered an interesting special class of torseforming vector field, requiring ξ to be nowhere zero and satisfying α(ξ) = 0, that is the torse-forming vector field is perpendicular to the dual-vector field to torsed form α, called torqued vector fields.
In the present paper, we introduce on a Lorentzian manifold a special type of torseforming vector field. A unit time-like torse-forming vector field ξ on a Lorentzian manifold (M, g) is said to be a special torse-forming vector field if it satisfies: where ρ is a non-zero function and η is 1-form dual to ξ. We call ρ the potential function of the special torse-forming vector field ξ. Note that for a special torse-forming vector field, using Equation (1), we have α(U) = −ρη(U), that is ξ is a torse-forming vector field, which is parallel to the vector field dual α as opposed to the torqued vector field where ξ is orthogonal to the vector field dual α. Moreover, from the definition of special torse-forming vector field ξ on a Lorentzian manifold, it follows that under no situation, it reduces to a concircular vector field. 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). We also show that if the function ξ(ρ) is nowhere zero on M, then the fibers of GRW-spacetime I × ϕ N are compact (see Theorem 2).
If ξ is a special torse-forming vector field on a simply connected Lorentzian manifold (M, g), then the dual-1-form η is closed (see Equation (15)), and, therefore, there is a function f such that η = d f . Thus, the special torse-forming vector field ξ on a simply connected Lorentzian manifold (M, g) satisfies ξ = ∇ f , call this function f the associated function of ξ. Recall that a Lorentzian manifold (M, g) is said to be a quasi-Einstein manifold (cf. [13]) if its Ricci tensor has the following expression: where f 1 , f 2 are scalars and β is a 1-form on M. Exact solutions of the Einstein field equations can provide very important information about quasi-Einstein manifolds. For example, the Robertson-Walker spacetimes are quasi-Einstein manifolds. For this reason, the study of quasi-Einstein manifolds is important. It is shown that if the associated function f of the special torse-forming vector field ξ on a simply connected Lorentzian manifold (M, g), dimM > 4, satisfies (i) f is nowhere zero and (ii) f is a solution of the Fischer-Marsden equation, then (M, g) is a quasi-Einstein manifold (see Theorem 3). Additionally, it is shown that if the scalar curvature τ of a simply 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, then the potential function ρ is an eigenfunction of the Laplace operator ∆ (see Corollary 1).

Preliminaries
Let ϕ be a smooth function on an n-dimensional connected Lorentzian (M, g). The Hessian operator H ϕ is defined by: where ∇ϕ is the gradient of ϕ and Hessian Hess(ϕ) is defined by (cf. [14]): The Laplacian ∆ϕ of the function ϕ is given by ∆ϕ = div(∇ϕ), and it satisfies: Let ξ be a time-like special torse-forming vector field on a Lorentzian (M, g). Then, using the expression for the curvature tensor field and Equation (2), we compute: Above equation gives expression for the Ricci tensor Ric of the Lorentzian manifold (M, g): Note that the Ricci operator Q of the Lorentzian manifold (M, g) is given by Ric(U, V) = g(QU, V), U ∈ X(M), and, therefore, Equation (7) implies: and: The Laplace operator ∆ acting on vector fields on the Lorentzian manifold (M, g) is defined by: where {v 1 , . . . , v n } is a local orthonormal frame on M. The de Rham-Laplace operator on the Lorentzian manifold (M, g) is : X(M) → X(M) given by (cf. [12]): Lemma 1. Let ξ be a time-like special torse-forming vector on an n-dimensional Lorentzian manifold (M, g) with potential function ρ. Then: Proof. Using Equation (2), for U ∈ X(M), we have: Since ξ is a time-like unit vector field, choosing a local frame {v 1 , . . . , v n−1 , ξ} on M, where v i , i = 1, . . . , n − 1 are spacelike unit vector fields in the above equation, to conclude: Thus, using Equations (8) and (11) with the above equation, we conclude:
Proof. Let (M, g) be a connected Lorentzian manifold, n > 5, ξ be a time-like special torse-forming vector field on (M, g) with ξ = λξ, λ being a scalar. We denote by ∇ the Levi-Civita connection on (M, g); using Equation (2), we have: Define a smooth distribution D on M by: Note that Equation (2) gives: that is the dual-1-form η to ξ is closed. Thus, for E, proving that the distribution D is integrable. Let N be a leaf of D. Then, N is a hypersurface of M with unit normal ξ. Using Equation (2), we observe that for E ∈ X(N), that is the shape operator S of N is given by: Now, as ξ = λξ, where λ is a scalar on M, using Lemma 1, we get: On taking the inner product in above equation with ξ yields λ = (n − 5)ξ(ρ) + 2(n − 2)ρ 2 and substituting this value of λ in Equation (18), we have: Above equation on taking the inner product with E ∈ X(N), gives (n − 5)E(ρ) = 0, and the assumption n > 5 implies E(ρ) = 0, that is ρ is a constant on the hypersurface N. Therefore, Equation (17) implies that N is a totally umbilical hypersurface of M. Moreover, the orthogonal complementary distribution D ⊥ to D is one-dimensional spanned by ξ, and by Equation (13), the integral curves of the distribution D ⊥ are geodesics on M. Thus, (M, g) is the warped product I × ϕ N (cf. [15]), that is (M, g) is a GRW-spacetime.
Conversely, we have already seen that a GRW-spacetime I × ϕ N admits a special torse-forming vector field ξ, which is an eigenvector of .
In the above result we have seen that the presence of a time-like special torse-forming vector field ξ on a Lorentzian manifold (M, g) satisfying ξ = λξ for scalar λ is a GRWspacetime I × ϕ N. It is interesting to observe if in addition ξ(ρ) is nowhere zero, then this condition has effect on the topology of N.

Theorem 2.
Let ξ be a time-like special torse-forming vector field with potential function ρ on an n-dimensional complete and connected Lorentzian manifold (M, g), n > 5. If ξ is an eigenvector of the de Rham-Laplace operator on (M, g) and the function ξ(ρ) is nowhere zero, then (M, g) is GRW-spacetime I × ϕ N, with N compact.
Proof. Let ξ be a time-like special torse-forming vector field on a Lorentzian manifold (M, g), n > 5, with ξ being an eigenvector of the de Rham Laplace operator on (M, g) and the function ξ(ρ) = 0 everywhere on M. Since n > 5, Equation (19) implies: As ξ is a time-like unit vector field and ξ(ρ) is nowhere zero, the above equation implies that ∇ρ is nowhere zero on M. Therefore, the potential function ρ : M → E is a submersion, and each fiber F x = ρ −1 {ρ(x)}, x ∈ M, is an (n − 1)-dimensional smooth manifold; as {ρ(x)} is compact in E, we obtain that F x is compact. Consider a smooth vector field: that has no zeros on M. Then, it follows that u(ρ) = −1 and u has a local flow {φ s } that satisfies: Recall the escape Lemma (cf. [16]), which states that if γ is a integral curve of u whose maximal domain is not all of E, then the image of γ cannot lie in any compact subset of M. Using the escape lemma and Equation (21) on a complete and connected M, we obtain that u is complete and has global flow {φ s }. Now, define f : Then, f is smooth, and for each u ∈ M, we find s ∈ E such that φ s (u) = y ∈ F x , satisfying u = φ −s (y). Thus, f (−s, y) = u, that is f is an on-to map. We observe that, on taking (s 1 , , and using Equation (21), we obtain ρ(u 1 ) − s 1 = ρ(u 2 ) − s 2 . As u 1 , u 2 ∈ F x , ρ(u 1 ) = ρ(u 2 ), and we obtain s 1 = s 2 . Thus, using φ s 1 (u 1 ) = φ s 2 (u 2 ), we arrive at u 1 = u 2 , that is f is one-to-one. Furthermore, we have: where F x is a compact subset of M. Using Theorem 3.1, we see that I × N is diffeomorphic to E × F x , and as the open interval I is diffeomorphic to E, the fiber N must be diffeomorphic to F x . As F x is compact, we obtain that N is compact.

Lorentzian Manifolds as Quasi-Einstein Manifolds
Fischer-Marsden considered the following differential equation on a semi-Riemannian manifold (M, g) (cf. [17]): where f is a smooth function on M. We call the above differential equation the Fischer-Marsden equation. This differential equation is closely associated with Einstein spaces. A generalization of Einstein manifolds was considered in [13], where the authors defined quasi-Einstein manifolds. A semi-Riemannian manifold (M, g) is said to be a quasi-Einstein manifold if its Ricci tensor satisfies Equation (3). In this section, we use a unit time-like special torse-forming vector field ξ on a Lorentzian manifold (M, g) to find conditions under which (M, g) is a quasi-Einstein manifold.
Let ξ be a time-like special torse-forming vector field on a simply connected Lorentzian manifold (M, g). On using Equations (2) and (15), we have dη = 0, that is η is a closed 1-form and M is simply connected η = d f (exact) for a smooth function f on M. Thus, for a time-like special torse-forming ξ on a simply connected Lorentzian manifold (M, g), we have: