A family of multiply warped product semi-Riemannian Einstein metrics

In this paper, we characterize multiply warped product semi -Riemannian manifolds when the base is conformal to an \begin{document}$ n $\end{document} -dimensional pseudo-Euclidean space. We prove some conditions on warped product semi- Riemannian manifolds to be an Einstein manifold which is invariant under the action of an \begin{document}$ (n-1) $\end{document} -dimensional translation group. After that we apply this result for the case of Ricci-flat multiply warped product space when the fibers are Ricci-flat. We also discuss the existence of infinitely many Ricci-flat multiply warped product spaces under the same action with null like vector.

where for each i ∈ {1, 2, ...m}, smooth functions h i : B → (0, ∞) are called warping functions, π and σ i are the natural projections on base B and fibers F i respectively.
R. L. Bishop and B. O'Neill in [4], first studied about warped product space for construct the examples of Riemannian manifolds with negative curvature. In ( [1], [2]), authors expressed the exact solutions of Einstein's field equation in term of warped product. In ( [7], [8]), they have also constructed new examples of complete locally conformally flat manifolds of negative curvature on warped product and multiply warped product spaces. In 2005, authors [6] obtained necessary and sufficient conditions for a semi-Riemannian warped product to be locally conformally flat. In 2012, authors [14] described warped product Einstein metrics when the base is locally conformally flat. In ( [16], [17]), authors studied Ricci flat Einstein warped product space and Einstein warped product space with quarter symmetric connection. In 2000, B. Unal [19] derived covariant derivative formulas for multiply warped products and also studied the geodesic equations for such type of spaces. In 2000, J. Choi [10] investigated the curvature of a multiply warped product with C 0warping functions and represented the interior Schwarzschild space-time as a multiply warped product space-time with warping functions. In 2005, F. Dobarro and B. Unal [11], studied Ricci-flat and Einstein-Lorentzian multiply warped products and provided some results on generalized Kanser space-times. In 2016, D. Dumitru [12], calculated warping functions for multiply generalized Robertson-Walker spacetime to be an Einstein manifold when all fibers are Ricci flat. In 2017, F. Gholami, F. Darabi and A. Haji-Badali [13] studied multiply warped product metrics and reduced Einstein equation for generalized Friedmann-Robrtson-Walker spacetime.
In 2017, Sousa and Pina [18] studied warped product semi-Riemannian Einstein manifolds under consideration that base is conformal to an n-dimensional pseudo-Euclidean space and invariant under the action of an (n − 1)-dimensional group. More recently, in [5] authors generalized the work of Sousa and Pina (see, [18]) for Quasi-Einstein manifolds.
The purpose of this article is to extend the work of Sousa and Pina (see [18]) for multiply warped space .. ⊕ h 2 l g F l , whereḡ = 1 ϕ 2 g, g is pseudo-Euclidean metric on R n with coordinates x = (x 1 , ..., x n ), g ij = δ ij ε i and for each s ∈ {1, 2, ...l}, h s , ϕ : R n → R are smooth functions, where h s are positive smooth functions, F ms s is a semi-Riemannian manifolds with constant Ricci curvatures λ Fs , m s ≥ 1. We will classify multiply warped product Einstein manifold when the base is locally conformally flat.
We organize the paper as follows: In Theorem 2.1, we compute the necessary and sufficient conditions for the multiply warped product metricg =ḡ ⊕ h 2 1 g F1 ⊕ h 2 2 g F2 ....⊕h 2 l g F l to be an Einstein. In Theorem 2.2, we apply the concept of Theorem prove necessary and sufficient conditions for the multiply warped product (M,g) to be a Ricci-flat manifold when fibers F s are Ricci-flat manifolds and the direction of α is timelike or spacelike. In the last Theorem, we see that if the direction of α is null, then there are infinitely many solutions. More explicitly, for any given positive differentiable functions h s , the function ϕ(ξ) satisfy a linear ordinary differential equation of second order. We clarify this fact by giving an example. We give explicite solutions for Einstein field equation, in the vacume case, that are not locally conformally flat.
2. Main results. In the following theorems, ϕ ,xixj and h s,xixj denote the second order derivatives of ϕ and h s with respect to x i x j , for all s ∈ {1, 2, ...l}.
Theorem 2.1. Let (R n , g) be a semi-Euclidean space, n ≥ 3, with coordinates x = (x 1 , ..., x n ) and g ij = δ ij ε i . Consider a multiply warped product Proof. For each s ∈ {1, 2, ...l}, assume that m s > 1. Let X 1 , X 2 , ..., X n ∈ L(R n ) and Y s p ∈ L(F ms s ), for each s ∈ {1, 2, ..., l} and p ∈ {1, 2, ..., m s } (L(R n ) and L(F ms s ) are respectively the lift of vector field on R n and F ms s to M respectively, for each s ∈ {1, 2, ..., l}). Then from proposition 2.5 of [11], it follows that Note that ifḡ = 1 ϕ 2 g then from (ex. [3]) we get As g(X i , X j ) = δ ij ε i , thus the above equation implies that Recall that whereΓ r ij are the Christoffel symbols of the metricḡ. For i, j, r different, we havē Hence ∀ s ∈ {1, 2, ...l}, After using (5) and (6) in the first equation of system (4), we obtain and Next, Substituting (9) in the fourth equation of system (4), we have where In the next theorem, we will convert the equations (1) Theorem 2.2. Let (R n , g) be a semi-Euclidean space, n ≥ 3, with coordinates x = (x 1 , ..., x n ) and g ij = δ ij ε i . Consider a multiply warped product M = (R n ,ḡ) × h1 F m1 Proof. We are assuming that for all s ∈ {1, 2, ...l}, ϕ(ξ) and h s (ξ), are functions of After using the above expressions in (1), we have If there exist i = j such that α i α j = 0 then this equation implies that Similarly equation (2), implies that Substituting equation (13), the above equation reduces to Similarly, the equation (3) reduces to Thus if we consider the case n i=1 ε i α 2 i = ε i0 then equations (13), (14) and (15), provided the system of equations (11). On the other hand using n i=1 ε i α 2 i = 0, in the equations (13), (14) and (15), we get (12). Now, if α i α j = 0 for all i = j, then ξ = x i0 the equation (1) trivially satisfied ∀ i = j. Now for the other equations we will consider two cases: and equation (3) will convert into * i = i 0 In this case, since α i = 0 ∀ i = i 0 the equation (2) provides and equation (3) again will convert into equation (17). Hence from (16) and (18) we get the first equation of (11). This conclude the proof of the theorem.
Remark 1. If we take M = (R n ,ḡ) × f R and M = (R n ,ḡ) × f F m then the Theorem 2.3 will convert into Theorem 1.3 and Theorem 1.4 respectively of [18]. Both Theorems provide the explicite solutions of the functions.
The next theorem shows that there are infinitely many multiply warped product M = (R n ,ḡ) × h1 F m1 Then there exists a function ϕ(ξ) satisfy (12) and M = (R n ,ḡ) × h1 F m1 is a Ricci-flat manifold.
First, we present one example illustrating Theorem 2.4, after that we will prove the Theorem 2.