Prescribed Schouten Tensor in Locally Conformally Flat Manifolds

We consider the pseudo-Euclidean space (Rn,g)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({\mathbb {R}}^n,g)$$\end{document}, with n≥3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \ge 3$$\end{document} and gij=δijεi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_{ij} = \delta _{ij} \varepsilon _{i}$$\end{document}, where εi=±1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon _{i} = \pm 1$$\end{document}, with at least one positive εi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon _{i}$$\end{document} and non-diagonal symmetric tensors T=∑i,jfij(x)dxi⊗dxj\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T = \sum \nolimits _{i,j}f_{ij}(x) dx_i \otimes dx_{j} $$\end{document}. Assuming that the solutions are invariant by the action of a translation (n-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(n-1)$$\end{document}- dimensional group, we find the necessary and sufficient conditions for the existence of a metric g¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar{g}$$\end{document} conformal to g, such that the Schouten tensor g¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar{g}$$\end{document}, is equal to T. From the obtained results, we show that for certain functions h, defined in Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^{n}$$\end{document}, there exist complete metrics g¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar{g}$$\end{document}, conformal to the Euclidean metric g, whose curvature σ2(g¯)=h\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _{2}(\bar{g}) = h$$\end{document}.


Introduction
In recent years, problems involving the Ricci curvature have aroused great interest. Among the problems studied, we highlight the Einstein manifolds, Ricci solitons gradient, prescribed Ricci and Schouten tensor, prescribed curvature tensor, and Einstein field equation. For more details see [1,3,6,10,14,19]. In [5], Deturck and Yang considered the following problem: Given a Riemannian manifold (M n , g 0 ), with n ≥ 3, and a symmetric tensor of order 2, T , defined in M n , find a Riemannian metric g such that Ric g + λK g g = T (1.1) where λ ∈ R is a constant and Ric g and K g are the Ricci tensor and scalar curvature of g, respectively. They have shown that when T is non-singular, Problem (1.1) has also been studied locally by Robert Brayant for any value of λ, proving that the problem always has local solutions when the components of the tensor are analytic functions. Motivated by the work of Deturck and Yang [5], our goal is to find global solutions to the following problem: Given a (0, 2)symmetrical tensor T defined in a manifold (M n , g 0 ), with n ≥ 3, is there a metric g such that This problem corresponds to studying a system of nonlinear second order partial differential equations. The importance of Schouten tensors in conformal geometry can be seen in the following decomposition of the Riemann curvature tensor where R g is the Riemann curvature tensor, is the Kulkarni-Nomizu product, and W g is the Weyl tensor of g (see [2]). Because the Weyl tensor is conformally invariant, i. e., g −1 W g is invariant in a given conformal class, in a conformal class the Schouten tensor is important, especially when g is locally conformally flat (W g = 0). Therefore, if g is locally conformally flat, the Riemann curvature tensor is determined by the Schouten tensor. From the Schouten tensor, curvatures that extend the concept of the scalar curvature can be defined. This study was first conducted by Jeff Viaclovsky in [22]. For an integer 1 ≤ k ≤ n and σ k -or k-scalar curvature, the Schouten curvature is defined by where (g −1 · A g ) is defined locally by (g −1 · A g ) ij = k g ik (A g ) kj and σ k and the k-th symmetrical elementary function. Thus, we define σ k (g) as being the k-th elementary symmetric function of the auto-values of the operator g −1 A g , to 1 ≤ k ≤ n, where σ 0 (g) = 1. Considering the eigenvalues of the Schouten tensor A g (λ 1 , λ 2 , . . . , λ n ) with respect to the metric g, to 1 ≤ k ≤ n, the kth polymorphic elementary symmetric functions σ k are given by σ k (A g ) = σ k (λ) = i1<···<i k λ i1 · · · λ i k . When k = 1, a σ 1 (g)-scalar curvature is exactly the scalar curvature (less than one constant). Thus, σ 1 (g) is constant if and only if (M n , g) has constant scalar curvature. In [11], the authors considered the problem of classifying compact Riemannian manifolds locally conformally flat with σ k (g) constant for some k ≥ 2.
In [22], Viaclovsky also noted that σ 2 (g) has still a variational structure. For k > 2, σ k (g) has a variational structure, if and only if the manifold considered is locally conformally flat. From this work of Viaclovisky and the work of Chang, Gursky, and Yang in [4],an intensive investigation started for the variational problem related to the Schouten curvature σ k (g), seeking to find a metric g, in the class of [g 0 ], satisfying and Γ + k is a convex open cone (the Garding cone) defined by k represents the Schouten tensor A g (x) ∈ Γ + k , for any x ∈ M ; an important fact is that g ∈ Γ + k guarantees that equation (1.3) is elliptic. Several authors have recently studied subjects related to the Schouten curvature, see for example [7,8], and [11]. In [7], the authors consider the problem σ2 σ1 = f , where f is a given differentiable function. In [12], Simon et al. showed that, if (M n ; g) is a compact locally conformally flat manifold with nonzero curvature σ k (A g ) for some 2 ≤ k ≤ n and A g defined as semi-positive, then (M n , g) is a space form of positive sectional curvature. In [13], Simon et al. studied the extreme properties of the Schouten function defined in the quotient of the Riemannian metric space by the group of diffeomorphisms.
In [15], the authors considered the pseudo-Euclidean space (R n , g), with n ≥ 3 and g ij = δ ij ε i , ε i = ±1, and tensors of the form T = i ε i f i (x)dx 2 i , and found necessary and sufficient conditions for the existence of a metricḡ, conformal to g, such that Aḡ = T . The solution to this problem was explicitly given for special cases of the tensor T , including a case where the metricḡ is complete in R n . Similar problems were considered for locally conformally flat manifolds. As an application of these results, the authors considered the problem of finding metricsḡ, conformal to g, such that σ 2 (ḡ) or σ2(ḡ) σ1(ḡ) are equal to a certain function.
In this work we will consider the pseudo-Euclidean space (R n , g), with n ≥ 3, coordinates x = (x 1 , .., x n ), and metric g, where g ij = δ ij ε i , with ε i = ±1, with at least one positive ε i , and a non-diagonal tensor of order 2 of the form functions. We want to find metricsḡ = 1 ϕ 2 g, such that the Schouten tensor of the metricḡ is T , that is, we want to solve the following problem: To obtain solutions to the problem (1.4), let us assume that the metricḡ is invariant by the action of a (n − 1) − dimensional translation group. In this case, we find necessary conditions on the tensor T , so that the problem admits solution (Lemma 2.2). For this special class of metrics, we find necessary and sufficient conditions for the problem to have solutions (Theorem 2.3). As a consequence of the Theorems (2.3) and (2.5) we obtain complete metrics in Euclidean space R n , with prescribed Schouten tensors. The results obtained were extended to locally conformally flat manifolds (Theorem 2.6).
As applications of these results, we show explicit solutions for a second order nonlinear partial differential equation in R n . The geometric interpretation of this result is equivalent to finding conformal metrics in R n with σ 2 (ḡ) prescribed. In particular, by considering f : R n −→ R, we find examples of complete metricsḡ, conformal to the Euclidean metric, such that σ 2 (ḡ) = f .

Main Results
Let ϕ ,xixj and f ij,x k denote the second order derivatives of ϕ with respect to x i x j and the derivative of f ij with respect to x k , respectively.
Then, there exists a positive function ϕ such that the metricḡ = 1 ϕ 2 g satisfies Aḡ = T if and only if the functions f ij and ϕ satisfy the following set of equations: In an attempt to find solutions to the system (2.1) we will consider the so- Initially, we will determine the necessary conditions on the tensor T .
From Lemma (2.2), we can state one of the main theorems of this section. We consider n k=1 k a 2 k = 0 and, without loss of generality, we assume that n k=1 k a 2 k = = ±1, and the case where n k=1 k a 2 k = 0 will be dealt with later.
Vol. 74 (2019) Prescribed Schouten Tensor in Locally Conformally Flat Manifolds Page 5 of 12 168 Theorem 2.3. Let (R n , g), with n ≥ 3, be the pseudo Euclidean space, with coordinates x = (x 1 , . . . , x n ) and metric g ij = δ ij ε i . Consider the non-diagonal tensor of Then, there is a metricḡ = 1 ϕ 2 g such that Aḡ = T , with ϕ = ϕ(ξ), if and only if the components of the tensor satisfy the following equations

2)
and As a direct consequence of the Theorem (2.3), we obtain the following example in the Riemannian case.
Example. Consider the Euclidean space (R n , g) n ≥ 3, with coordinates The case where ξ = a k x k for some fixed k, will be treated in the next Theorem. Note that, in this case, in (2.1), ϕ xi = ϕ a i , ϕ xixi = ϕ a 2 i ∀i = 1, . . . n, and ϕ xixj = 0, ∀ i = j. Then, for the second equation, we have f ij ϕ = 0, which is equivalent to the given tensor T being diagonal, because, in this case, f ij = 0 ∀ i = j. Without loss of generality, let us consider ξ = a 1 x 1 and 1 a 2 1 = . Theorem 2.4. Let (R n , g), with n ≥ 3, be the pseudo Euclidean space, with coordinates x = (x 1 , . . . , x n ), and metric Then, there is a metricḡ = 1 ϕ 2 g, with ϕ = ϕ(ξ), such that Aḡ = T , if and only if the components of the tensor satisfy the equations: where K is a positive constant and 0 = ±1.
As a consequence of Theorem (2.4), we will show an example in the Riemannian case.
Example. Consider the Euclidean space (R n , g) n ≥ 3, with coordinates x = (x 1 , . . . , x n ). Given the tensor T = f 11 √ −2g , = 1, 0 = −1, and f ii = f jj = g ∀i = j = 2, . . . , n, where g is a differentiable function smaller than zero. Then, there is a metricḡ = 1 Note that in the above theorem, conditions (2.4) are checked. Because the g < 0 function is arbitrary, it can be chosen so that √ −2gdξ is limited, and, in this case, the metricḡ will be complete in R n .
Considering now the case where ξ = a i x i and a 2 i i = 0, we obtain the following result. Example. Let (R n , g), with n ≥ 3, be a pseudo Euclidean space, with coordinates x = (x 1 , . . . , x n ) and metric . . , n, and f ij = ka i a j , ∀ i = j = 1, . . . , n, and n i=1 a 2 i i = 0. Then, there exists aḡ = 1 ϕ 2 g such that Aḡ = T , if and only if the function ϕ is satisfies: Example. Let (R n , g), with n ≥ 3, be a pseudo Euclidean space, with coordinates x = (x 1 , . . . , x n ) and metric . , n and f ij = (h 2 +h )a i a j ∀ i = j = 1, . . . , n, and n i=1 a 2 i i = 0, and a differentiable function h. Then, there exists ā g = 1 ϕ 2 g such that Aḡ = T , if and only if the function ϕ is given by where C 1 and C 2 are arbitrary constants and ϕ 0 = e hdξ , with h = h(ξ).
Next, we present a generalization of the Theorem (2.3), for locally conformally flat manifolds.
Let us now consider a locally conformally flat Riemannian manifold (M n , g). We can consider the problem (1.4) for a neighborhood V ⊂ M with coordinates x = (x 1 , . . . , x n ) such that g ij = δij F 2 , where F is a non-null, differentiable function in V . Given a tensor T = i,j f ij (x)dx i ⊗ dx j defined in V , we want to find a metricḡ = 1 φ 2 g such that Aḡ = T . Considering that g andḡ are translation invariant, where ξ = a i x i is the basic invariant of the action, we have, in a way analogous to Lemma (2.2), that the components of the given tensor T , are necessarily given by f ij = a i a j f (ξ) and f ii = f ii (ξ), where f is a differentiable function.
, such that Aḡ = T if and only if the functions f ij and ϕ are given in Theorem (2.3) and φ = ϕ F . Analogously, the Theorems (2.4) and (2.5) can be extended to locally conformally flat manifolds.
In [22], Vioclovisk extended the concept of scalar curvature using the Schouten tensor, by introducing the σ k curvatures, which are obtained from the eigenvalues of the Schouten tensor. Recall that σ 0 (g) has been defined as 1 and σ 1 (g) is the scalar curvature less than constant. Recently, many works have considered the σ 2 (g) prescribed problem. For more details see [7,11], and [22].
We know that, given a function h : M n −→ R, finding a metricḡ = 1 ϕ 2 g such that σ 2 (ḡ) = h is equivalent to studying the following partial differential equation (ver [9]).
(Δϕ) 2 − |Hess g ϕ| 2 ϕ 2 − (n − 1)Δϕ|∇ϕ| 2 ϕ + n(n − 1) 4 |∇ϕ| 4 = 2h. (2.5) Let us show that, for certain functions h, the above equation admits infinite solutions. Corollary 2.7. Let (R n , g), with n ≥ 3, be the Euclidean space, with coordinates x = (x 1 , . . . , x n ) and metric g ij = δ ij . Given the function g, which can be any positive function, K is a positive constant and a ∈ R Then, the partial differential equation, has infinite solutions, globally defined in R n , given by The geometric interpretation of this result is presented below. We show an example of a metricḡ, conformal to the Euclidean metric, complete in R n , with curvature σ 2 (ḡ) prescribed for h. Corollary 2.8. Let (R n , g), with n ≥ 3, be the Euclidean space, with coordinates x = (x 1 , . . . , x n ) and metric g ij = δ ij ε i . Given a function h, in (2.6) there exists a metricḡ = 1 ϕ 2 g with ϕ given by (2.3) such that σ 2 (ḡ) = h.
By choosing the function g so that ϕ is limited, the metrics obtained in the corollary (2.8) will be complete in R n .

Proof of the Main Results
Before proving our results, it follows from [15], that if (R n , g) is a pseudo-Euclidean space andḡ = g/ϕ 2 Is a conformal metric, then the Ricci tensorḡ is given by Ricḡ = 1 ϕ 2 (n − 2)ϕHess g ϕ + ϕΔ g ϕ − (n − 1)|∇ g ϕ| 2 g (3.1) and the scalar curvature ofḡ is given bȳ In the rest of this section, we demonstrate the main results of this article.
Proof [Theorem 2.1]. Using the expressions (3.1) and (3.2), we can write the Schouten tensor of the metricḡ as: where ∇ g denotes the gradient of the pseudo-Euclidean metric g.
In this case, studying the problem (1.4) with T = n ij f ij (x)dx i ⊗ dx j , is equivalent to studying the following system of equations: