On Warped product gradient Ricci Soliton

Transcription

1 On Warped product gradient Ricci Soliton Márcio Lemes de Sousa arxiv: v2 [math.dg] 15 Apr 2016 ICET - CUA, Universidade Federal de Mato Grosso, Av. Universitária nº 3.500, Pontal do Araguaia, MT, Brazil marciolemesew@yahoo.com.br Romildo Pina IME, Universidade Federal de Goiás, Caixa Postal 131, , Goiânia, GO, Brazil romildo@ug.br October 15, 2018 Abstract In this paper we consider M = B F warped product gradient Ricci solitons. We show that when the Hessian matrix o is not zero then the potential unction depends only on the base and the iber F is necessarily Einstein maniold. Moreover, we provide all such solutions in the case o steady gradient Ricci solitons when the base is conormal to an n-dimensional pseudo-euclidean space, invariant under the action o an (n 1)-dimensional translation group, and the iber F is Ricci-lat Mathematics Subject Classiication: 53C21, 53C50, 53C25 Key words: semi-riemannian metric, gradient Ricci solitons, warped product 1 Introduction and main statements Let (M, g) be a semi-riemannian maniold o dimension n 3. We say that (M, g) is a gradient Ricci soliton i there exists a dierentiable unction h : M R (called the Partially supported by CAPES-PROCAD.

2 potential unction) such that Ric g + Hess g (h) = ρg, ρ R, where Ric g is the Ricci tensor, Hess g (h) is the Hessian o h with respect to the metric g, and ρ is a real number. A gradient Ricci soliton is said to be shrinking, steady or expanding i ρ > 0, ρ = 0 or ρ < 0, respectively. When M is a Riemannian maniold usually one requires the maniold to be complete. In the case o semi-riemannian maniold, one does not require (M, g) to be complete. Ricci solitons are natural extensions o Einstein maniolds and they appear as selsimilar solutions o the Ricci low g(t)/ t = 2Ricg(t). Moreover, Ricci solitons are important in understanding singularities o the Ricci low. Simple examples o gradient Ricci solitons are obtained by considering R n with the canonical metric g. Then (R n, g) is a gradient Ricci soliton, where h(x) = A x 2 /2 + g(x, B) + C, A, C R and B R n are all the potential unctions. In this case, (R n, g) is a shrinking, steady or expanding soliton according to the sign o the constant A. Bryant [9] proved that there exists a complete, steady, gradient Ricci soliton spherically symmetric or any n 3, which is known as Bryant s soliton. Recently, Cao and Chen [10] showed that any complete, steady, gradient Ricci soliton, locally conormally lat, up to homothety, is either isometric to the Bryant s soliton or is lat. Complete, shrinking gradient solitons, conormally lat, have been characterized as being quotients o R n, S n or R S n 1 (see [13]). Although the Ricci soliton equation was introduced and studied initially in the Riemannian context, Lorentzian Ricci solitons have been recently investigated in [2], [4] and [16], where the authors show that there are important dierences with the Riemannian case. The existence o Lorentzian, steady, gradient Ricci solitons which are locally conormally lat and distinct rom Bryant s solitons, was proved in [2]. In [5], the authors gave a local characterization o the Lorentzian gradient Ricci solitons which are locally conormally lat. In [1], Barbosa-Pina-Tenenblat, considered gradient Ricci solitons, conormal to an n-dimensional pseudo-euclidean space, which are invariant under the action o an (n 1)- dimensional translation group. The one provided all such solutions in the case o steady gradient Ricci solitons. Our purpose is to generalize the results in [1]. To get these generalizations, we have to use warped product maniolds to study gradient Ricci solitons that are non locally 2

3 conormally lat. Then considering (B, g B ) and (F, g F ) semi-riemannian maniolds, and let > 0 be a smooth unction on B, the warped product M = B F is the product maniold B F urnished with metric tensor g = g B + 2 g F, B is called the base o M = B F, F the iber and is the warping unction. For example, polar coordinates determine a warped product in the case o constant curvature spaces, the case corresponds to R + r S n 1. There are several studies correlating warped product maniolds and locally conormally lat maniolds, see [6], [8] and their reerences. Many o the original examples o gradient Ricci solitons arise as warped products over a one dimensional base (c. [11], [14]). In this paper, initially we prove that i a warped product M = B F is a gradient Ricci soliton with the hypothesis that there is at least one pair o vector (X i, X k ) o the base, such that Hess gb ()(X i, X k ) 0 then the potential unction depends only on the base and the iber is necessarily an Einstein maniold (see Theorem 1.1 and Corollary 1.1). The Theorem 1.1 and Corollary 1.1 generalize the results obtained in [7] where the authors studied warped product gradient Ricci solitons with one-dimensional base. In what ollows, we consider warped product M = B F gradient Ricci solitons, where the base is conormal to a pseudo-euclidean space which are invariant under the action o an (n 1)-dimensional translation group and the iber is an Einstein maniold. More precisely, let (R n, g) be the pseudo-euclidean space, n 3, with coordinates x = (x 1,, x n ) and g ij = δ ij ε i and let M = (R n, g) F m be a warped product where g = 1 g, F a semi Riemannian Einstein maniold with constant Ricci curvature λ 2 F, m 1,,, h : R n R, smooth unctions, and is a positive unction. In Theorem 1.2 we ind necessary and suicient conditions or the warped product metric g = g + 2 g F be a gradient Ricci soliton, namely Ric g + Hess g (h) = ρ g, ρ R. (1) In Theorem 1.3, we consider, and h invariant under the action o an (n-1) dimensional translation group and let ξ = n i=1 α i x i, α i R, be a basic invariant or an (n 1)-dimensional translation group. We want to obtain dierentiable unctions (ξ), (ξ) and h(ξ), such that the metric g is a gradient Ricci soliton. We irst obtain necessary and suicient conditions on (ξ), (ξ) and h(ξ) or the existence o g. We show 3

4 that these conditions are dierent depending on the direction α = n i=1 α i / x i being null (lightlike) or not. We observe that in the null case the metrics g and g F are necessarily gradient Ricci solitons steady and Ricci-lat, respectively. Considering M = (R n, g) F m and F Ricci-lat maniold, we obtain all the metrics g = g + 2 g F, which are gradient Ricci solitons steady and are invariant under the action o an (n 1)-dimensional translation group. We prove that i the direction α is timelike or spacelike, the unctions, and h depend on the dimensions n, m and also on a inite number o parameters. In act, the solutions are explicitly given in Theorems 1.4, 1.5 and 1.6. I the direction α is null, then there are ininitely many solutions. In act, in this case, or any given positive dierentiable unctions (ξ) and (ξ), the unction h(ξ) satisies a linear ordinary dierential equation o second order (see Theorem 1.7). We illustrate this act with some explicit examples. When the dimension o the iber F is m = 1 we consider M = (R n, g) R and in this case λ F = 0. In what ollows, we state our main results. We denote by,xi x j,,xi x j and h,xi x j the second order derivative o, and h, with respect to x i x j. Theorem 1.1 Let M = B n F m be a warped product semi-riemannian maniold with metric g. I the warped product g = g B + 2 g F is a gradient Ricci soliton with h : M R as potential unction, and there is at least one pair o vector (X i, X k ) o the base, such that Hess gb ()(X i, X k ) 0, then h depends only on the base. Corollary 1.1 Let M = B F be a warped product semi-riemannian maniold with metric g. I the warped product metric g = g B + 2 g F is a gradient Ricci soliton with h : B R as potential unction, and is non-constant, then the iber is an Einstein maniold. The results obtained in Theorem 1.1 and Corollary 1.1 are valid in the Riemannian case. Motivated by the previous results we study the problem considering warped product with h depending only on the base and the iber an Einstein maniold. Theorem 1.2 Let (R n, g) be a pseudo-euclidean space, n 3 with coordinates x = (x 1,, x n ) and g ij = δ ij ε i. Consider M = (R n, g) F m a warped product, where g = 1 g, F a semi Riemannian Einstein maniold with constant Ricci curvature λ 2 F, m 1,,, h : R n R, smooth unctions and is a positive unction. Then the warped 4

5 product metric g = g + 2 g F is a gradient Ricci soliton with h as potential unction i, and only i, the unctions, and h satisy: (n 2),xi x j +h,xi x j m,xi x j m,xi,xj m,xj,xi +,xi h,xj +,xj h,xi = 0, (2) where 1 i j n, and [ (n 2),xi x i + h,xi x i m,xi x i 2m,xi,xi + 2,xi h,xi ] n ε i ε k [,xk x k (n 1) 2,x k + m,xk,xk,xk h,xk ] + = ε i ρ, 1 i n ε k [ 2,xk x k + (n 2),xk,xk (m 1) 2 2,x k + 2,xk h,xk ] = ρ 2 λ F. (4) We want to ind solutions o the system (2), (3) and (4) o the orm (ξ), (ξ) and h(ξ), where ξ = n i=1 α i x i, α i R. Whenever n i=1 ε i α 2 i (3) 0, without loss o generality, we may consider n i=1 ε i α 2 i = ±1. The ollowing theorem provides the system o ordinary dierential equations that must be satisied by such solutions. Theorem 1.3. Let (R n, g) be a pseudo-euclidean space, n 3, with coordinates x = (x 1,, x n ) and g ij = δ ij ε i. Consider M = (R n, g) F m, where g = 1 g, F m a semi 2 Riemannian Einstein maniold with constant Ricci curvature λ F and smooth unctions (ξ), (ξ) and h(ξ), where ξ = n i=1 α i x i, α i R, and n i=1 ε i αi 2 = ε i 0 or n i=1 ε i αi 2 = 0. Then the warped product metric g = g+ 2 g F is a gradient Ricci soliton with h as potential unction i, and only i, the unctions, and h satisy: (i) (n 2) + h m 2m + 2 h = 0, ε k αk[ 2 (n 1) 2 + m h ] = ρ ε k αk[ (n 2) (m 1) h ] = ρ 2 λ F, whenever (ii) ε i αi 2 = ε i 0, and i=1 (5) (n 2) + h m 2m + 2 h = 0, and ρ = λ F = 0, (6) 5

6 whenever ε i αi 2 = 0. i=1 In the ollowing two results we provide all the solutions o (5) when g is steady gradient Ricci solitons, i.e., ρ = 0, and F m is a Ricci-lat maniold, i.e., λ F = 0. Theorem 1.4. Let (R n, g) be a pseudo-euclidean space, n 3, with coordinates x = (x 1,, x n ) and g ij = δ ij ε i. Consider smooth unctions (ξ), (ξ) and h(ξ), where ξ = n i=1 α i x i, α i R, n i=1 ε i αi 2 = ε i0, given by ± (ξ) = c 2 (N ± ξ + b) k N ±, ± (ξ) = c 1 (N ± ξ + b) 1 N ±, (7) h ± (ξ) = m (n 2)k + N ± N ± ln (N ± ξ + b) where k, c 1, c 2, N ± and b are constant with k, c 1, c 2 > 0 and N ± = k ± m + k 2 (n 1). Then the warped product M = (R n, g) F m, with g = 1 2 g and F Ricci-lat, is a steady gradient Ricci soliton with h as potential unction. These solutions are deined on the hal space determined by n i=1 α i x i < b k+ m+k or n b 2 (n 1) i=1 α i x i > k m+k (n 1) 2 Theorem 1.5 Let (R n, g) be a pseudo-euclidean space, n 3, with coordinates x = (x 1,, x n ) and g ij = δ ij ε i. Consider smooth unctions x(ξ), and z(ξ), where ξ = ni=1 α i x i, α i R, n i=1 ε i α 2 i = ±1, given by x = c 3 (z + k z = c 3 (z + k ) a 1 ( ) m + k 2 2 a+1 (n 1) z + k + m + k 2 2 (n 1) ) a+1 ( ) m + k 2 2 a 1 (n 1) z + k + m + k 2 2 (n 1) (8) (9) where a, k, c 3 are constants with, k, c 3 > 0 and a = k m + k 2 (n 1). Let (ξ), (ξ) and h(ξ) be unctions obtained by integrating, (ξ) = k (ξ), (ξ) = x(ξ), h (ξ) = [z(ξ) + m k(n 2)]x(ξ) (10) Then the warped product M = (R n, g) F m, with g = 1 2 g and F Ricci-lat, is a steady gradient Ricci soliton with h as potential unction. We remark that the obtained metric, in the Theorems 1.4 and 1.5, are non locally conormally lat. 6

7 Theorem 1.6 Let (R n, g) be a pseudo-euclidean space, n 3, with coordinates x = (x 1,, x n ) and g ij = δ ij ε i. Consider M = (R n, g) F m a warped product where g = 1 2g. Let g = g + 2 g F, be a steady gradient Ricci soliton with h as potential unction and F Ricci-lat. Then, and h are invariant under an (n-1)-dimensional translation group whose basic invariant is ξ = n i=1 α i x i, where α = n i=1 α i / x i is a non null vector i, and only i,, and h are given as in Theorems 1.4 or 1.5. The ollowing theorem shows that there are ininitely many warped products M = (R n, g) F m steady gradient Ricci solitons with h as potential unction, which are invariant under the action o an (n 1) dimensional group acting on R n, when α = ni=1 α i / x i is null vector. Theorem 1.7 Let (ξ) and (ξ) be any positive dierentiable unctions, where ξ = ni=1 α i x i and n i=1 ε i αi 2 = 0. Then the unction h(ξ) given by ( [ h(ξ) = 2 (m 2 + 2m ]) (n 2) )dξ + c 4 dξ + c 5 ; c 4, c 5 R (11) satisies (6) and M = (R n, g) F m is a steady gradient Ricci soliton with h as potential unction. Beore proving our main results, we present two examples illustrating Theorem 1.7. Let (ξ) = (ξ) = ke Aξ, where k > 0, A 0, ξ = n i=1 α i x i and n i=1 ε i α 2 i (11) we get h(ξ) = (3m (n 2)) A 2 ξ c 4 2Ak 2e 2Aξ + c 5 ; c 4, c 5 R. = 0. Solving It ollows rom Theorem 1.7 that M = (R n, g) F m is a steady gradient Ricci soliton, with h as potential unction. Similarly, i we choose (ξ) = e ξ2 ollows rom Theorem 1.7 that h(ξ) = 2m ( 3 ξ3 2mξ 2 + m n 2 2 and (ξ) = e ξ, then it ) ξ 1 2 c 4e 2ξ + c 5 ; c 4, c 5 R, is the potential unction o the steady gradient Ricci soliton g = ḡ + 2 g F. 2 Proos o the Main Results Proo o Theorem 1.1: Let M = B F be a gradient Ricci soliton with potential h, then Ric g + Hess g (h) = ρ g, ρ R. 7

8 Considering X 1,... X n L(B) Y 1..., Y m L(F), where L(B) and L(F) are respectively the lit o a vector ield on B and F to B F, we have Ric g (X i, X j ) = Ric gb (X i, X j ) m Hess g B (X i, X j ), i, j = 1,... n Ric g (X i, Y j ) = 0, i = 1,... n, j = 1,... m (12) Ric g (Y i, Y j ) = Ric gf (Y i, Y j ) g(y i, Y j ) ( gb + (m 1) g(, ) ), i, j = 1,... m 2 How g(x i, Y j ) = 0 or i = 1... n and j = 1... m (see [15]), we have that Note that Hess g (h)(x i, Y j ) = 0. (13) Hess g (h)(x i, Y j ) = X i Y j (h) ( Xi Y j )(h), i = 1... n, j = 1... m, where is the connection o M. Since (see [15]), we have Using (14) and (13) we obtain: Xi Y j = X i() Y j Hess g (h)(x i, Y j ) = h, xi y j, x i h, y j. (14) h, xi y j, x i h, y j = 0. (15) I h = h(x 1,..., x n ) then the equation (15) is trivially satisied. Suppose that there is at least one y j, with 1 j m such that h, yj 0. In this case, it ollows rom (15) that integrating (16) in relation to x i we obtain: h, xi y j =, x i, i = 1... n, j = 1... m. (16) h, yj ln h, yj = ln + l( ˆx i ) this is, h, yj = e l( ˆx i). (17) Fixing i and j in ( 17) and deriving in relation to x k with k i, we obtain h, yj x k =, xk e l( ˆx i) + l, xk e l( ˆx i) which is equivalent to 8

9 and again using (16) we have: h xk y j h yj = x k + l x k, l xk = 0, This means that l does not depend x k, this is l = l( ˆx i, ˆx k ). Repeating this process we obtain that l depends only on the iber, i.e. l = l(y 1,..., y m ), thereore, h yj = e l(y 1,...,y m) with 1 j m. (18) Integrating (18) in relation to y j, we obtain: h(x 1,..., x n ; y 1... y m ) = (x 1,..., x n ) e l(y 1,...,y m) dy j + m(ŷ j ). (19) Using the irst equation (12), we have, Hess g (h)(x i, X k ) = ρg B (X i, X k ) Ric gb (X i, X k ) + m Hess g B (X i, X k ), i, k = 1,... n, proving that Hess g (h)(x i, X k ), i, k = 1,... n, depends only on the base. Thus, considering j ixed in equation (19) we have: y j Hess g (h)(x i, X k ) = 0, i, k = 1,... n. (20) On the other hand i, k = 1,... n we have Hess g (h)(x i, X k ) = ( Hess g ()(X i, X k ) y j y j ) e l(y 1,...,y m) dy j + Hess g (m)(x i, X k ) Since m = m(ŷ j ), and using the deinition o the Hessian get that y j Hess g (m)(x i, X k ) = 0, i, k = 1,... n, (21) and as Hess g ()(X i, X k ) = Hess gb ()(X i, X k ) ollow that y j Hess g (h)(x i, X k ) = Hess gb ()(X i, X k )e l(y 1,...,y m), i, k = 1,... n. (22) 9

10 Using (20) and (22) we obtain that: Hess gb ()(X i, X k )e l(y 1,...,y m) = 0, i, k = 1,... n, We have or hypothesis that there is at least one pair o vector (X i, X k ) o the base, such that Hess gb ()(X i, X k ) 0 then e l(y 1,...,y m) = 0, but this is impossible. Thereore h, yj = 0 j = 1... m. Consequently h depends only on the base. This concludes the proo o the Theorem 1.1. Proo o Corollary 1.1 Let M = B F be a gradient Ricci soliton with potential h deined only on the base, then by the equation (1) we have, Ric g + Hess g (h) = ρ g, ρ R. Considering Y, Z L(F), we have that Ric g (Y, Z) + Hess g (h)(y, Z) = ρ g(y, Z), (23) but, and g(y, Z) = 2 g F (Y, Z) Ric g (Y, Z) = Ric gf (Y, Z) ( gb + (m 1) grad gb 2 )g F (Y, Z) (see or example [15]). Replacing Ric g (Y, Z) in (23), we have Ric gf (Y, Z) = ( 2 + gb + (m 1) grad gb 2 )g F (Y, Z) Hess g (h)(y, Z). (24) It ollows rom (24) that F is Einstein i, and only i, Hess g (h)(y, Z) = λg F (Y, Z). Indeed, as by hypotheses h depends only on the base, thus grad g h = grad gb h, then H(grad g h) = grad gb h, V(grad g h) = 0 10

11 and Thereore Y (grad g h) = grad g B h() Y. Hess g (h)(y, Z) = grad g B h() g(y, Z) = grad gb h()g F (Y, Z). (25) This concludes the proo Corollary 1.1. Proo o Theorem 1.2: Assume initially that m > 1. It ollows rom [15] that i X 1, X 2,..., X n L(R n ) and Y 1, Y 2,..., Y m L(F) (L(R n ) and L(F) are respectively the lit o a vector ield on R n and F to R n F ), then Ric g (X i, X j ) = Ric g (X i, X j ) mhess g(x i, X j ), i, j = 1,... n Ric g (X i, Y j ) = 0, i = 1,... n, j = 1,... m (26) Ric g (Y i, Y j ) = Ric gf (Y i, Y j ) g(y i, Y j )( g + (m 1) g(, ) ), i, j = 1,... m 2 It is well known (see, e.g., [3]) that i g = 1 2 g, then Ric g = 1 2 { (n 2)Hessg () + [ g (n 1) g 2 ]g }. Since g(x i, X j ) = ε i δ ij, we have Ric g (X i, X j ) = 1 {(n 2)Hess g()(x i, X j )} i j = 1,... n Ric g (X i, X i ) = 1 2 { (n 2)Hessg ()(X i, X i ) + [ g (n 1) g 2 ]ε i } i = 1,... n. Since Hess g ()(X i, X j ) =,xi x j, g = ε k,xk x k and g 2 = ε k 2,x k, we have Ric g (X i, X j ) = (n 2),x i x j i j : 1... n n (n 2),xi x i + ε i ε k (27),xk x k n ε k 2,x Ric g (X i, X i ) = (n 1)ε k i 2 Recall that Hess g ()(X i, X j ) =,xi x j k Γ k ij,x k, where Γ k ij are the Christoel symbols o the metric g. For i, j, k distinct, we have Γ k ij = 0 Γi ij =,x j Γ k ii = ε iε k,xk Γ i ii =,x j 11

12 thereore, Hess g ()(X i, X j ) =,xi x j +,x j,x i +,x i,x j, i j = 1... n Hess g ()(X i, X i ) =,xi x i + 2,x i,x i ε i Substituting (27) and (28) in the irst equation o the (26) we obtain and Ric g (X i, X j ) = (n 2),x i x j m [,xi x j +,x j,x i +,x i,x j ε k,xk,x k (28) ], i j (29) Ric g (X i, X i ) = n (n 2),xi x i + ε i ε k,xk x k (n 1)ε i m [,xi x i + 2 ],x i,xk,x i ε i ε k,x k. n ε k 2,x k 2 (30) On the other hand, Ric gf (Y i, Y j ) = λ F g F (Y i, Y j ) g(y i, Y j ) = 2 g F (Y i, Y j ) g = 2 n ε k,xk x k (n 2) n ε k,xk,xk g(, ) = 2 n ε k 2,x k (31) Substituting (31) in the third equation o (26), we have where, Ric g (Y i, Y j ) = γ ij g F (Y i, Y j ) (32) n γ ij = λ F 2 ε k,xk x k + (n 2) ε k,xk,xk (m 1) 2 n ε k,x 2 k. We want to ind h satisying Ric g (X, Y ) + Hess g (h)(x, Y ) = ρ g(x, Y ), X, Y X (M). (33) On the other hand, since h : R n R, we have that Hess g (h)(x i, X j ) = Hess g (h)(x i, X j ), 1 i, j n 12

13 i.e., Hess g (h)(x i, X j ) = h,xi x j +,x j h,x i +,x i h,x j, i j = 1... n Hess g (h)(x i, X i ) = h,xi x i + 2,x i h,x i ε i ε k,xk h,x k. By substituting (29) and the irst equation o (34) into (33), we obtain (2). Again replacing (30) and the second equation o (34) into (33) we get (3). Now or X i L(R n ) and Y j L(F) (1 i n and 1 j m) we get Hess g (h)(x i, Y j ) = 0 by Theorem 1.1. In this case the equation (33) is trivially satisied. Taking Y i, Y j L(F) with 1 i, j m by using the equation (25) we have: but, Then we have, that Hess g (h)(y i, Y j ) = grad gh() g(y i, Y j ) = grad g h()g F (Y i, Y j ) grad g h() = g(grad gb h, grad gb ) = 2 Hess g (h)(y i, Y j ) = 2 n By substituting (32) and (35) into (33) we obtain (4) n ε k,xk h,xk (34) ε k,xk h,xk g F (Y i, Y j ). (35) The reciprocal o this theorem can be easily veriied. In the case m = 1 just remember Ric g (X i, X j ) = Ric g (X i, X j ) 1 Hess g(x i, X j ), i, j = 1,... n Ric g (X i, Y ) = 0, i = 1,... n Ric g (Y, Y ) = g(y, Y ) g. In this case the equation (2) and (3) remain the same and the equation (4) reduces to 2 n ε k,xk x k + (n 2) ε k,xk,xk + 2 This concludes the proo Theorem 1.2. n ε k,xk h,xk = ρ. Proo o Theorem 1.3 Assume initially that m > 1. Let g = 2 g be a conormal metric o g. We are assuming that (ξ), (ξ) and h(ξ) are unctions o ξ, where ξ = n α i x i, 13 i=1

14 α i R and i ε i αi 2 = ε i 0 or i ε i αi 2 = 0. Hence, we have,xi = α i,,xi x j = α i α j,,xi = h α i,,xi x j = α i α j, and h,xi = h α i, h,xi x j = h α i α j. Substituting these expressions into (2), we get (n 2) α i α j + h α i α j m α i α j 2m α i α j + 2 h = 0, i j. I there exist i j such that α i α j 0, then this equation reduces to (n 2) + h m 2m + 2 h = 0 (36) Similarly, considering equation (3), we get α 2 i [(n 2) + h m 2m + 2 h ] + ε i k ε k α 2 k [ (n 1)( ) 2 + m h ] = ε i ρ. (37) Due to the relation between, and h given in (36), the equation (37) reduces to [ ε k αk 2 (n 1)( ) 2 + m h ] = ρ. (38) k Analogously to equation (4) reduces to [ ε k αk (n 2) (m 1) h ] = ρ. 2 λ F (39) k Thereore, i k ε k αk 2 = ε i 0, we obtain the equations o the system (5). I k ε k αk 2 = 0, we have (36) satisied and (38) implies ρ = 0, hence λ F = 0 i.e., (6) holds. I or all i j, we have α i α j = 0, then ξ = x i0, and equation (2) is trivially satisied or all i j. Considering (3) or i i 0, we get ε k αk[ 2 (n 1) 2 + m h ] = ρ and hence, the second equation o (5) is satisied. Considering i = i 0 in (3) we get that the irst equation o (5) is satisied. Considering i = i 0 or i i 0 in (4), we get that the third equation (5) is satisied. When m = 1 the irst and the second equation o the system (5) are the same, and the third equation reduces to 14

15 ε k αk 2 [ 2 + (n 2) + 2 h ] = ρ. This concludes the proo Theorem 1.3. In order to prove Theorems 1.4 and 1.5, we consider unctions (ξ), (ξ) and h(ξ), where ξ = α i x i, α i R, ε i αi 2 = ±1. It ollows rom Theorem 1.3 that g = g + 2 g F i=1 i=1 is a steady gradient Ricci soliton with Ricci-lat F and h as potential unction i, and only i,, and h satisy i.e., ( ) (n 2) + (n 2) ( ) ( (n 2) ( ) ( m (n 2) + h m 2m + 2 h = 0, ( ) 2 (n 1) + m h = 0 ( ) 2 + (n 2) (m 1) + h = 0, ( ) 2 ( ) + h ( m m ) 2 + m h = 0 ) 2 + (n 2) + h = 0. Then, it ollows rom the second equation o (41) that (40) ) 2 2m + 2 h = 0, (41) h = ( ) (n 2) + m, (42) on the other hand, by using the third equation o (41) we have hence substituting equation (43) into (42), we get h = ( ) (n 2) + m, (43) ( ) Integrating the equation (44) we obtain: = ( ). (44) = k, k > 0. (45) 15

16 Substituting (45) in the system (41), we get ( ) h ( ) + [(n 2)k m] = [2mk + m (n 2)k 2 2 ] 2k ( ) ( ) 2 = [(n 2)k m] + h h (46) Considering x(ξ) = ( /)(ξ) and y = h + [(n 2)k m] the system o equations (46) is equivalent to y = [m + (n 2)k 2 ]x 2 2kxy x = xy (47) (48) It ollows rom (48) that xydy {[m + (n 2)k 2 ]x 2 2kxy}dx = 0 (49) In order to study this equation (see e.g. [12] p. 37), we take y(ξ) = x(ξ)z(ξ) (50) Theorem 1.4 and 1.5 are obtained by considering z to be a non zero constant and a nonconstant unction, respectively. Proo o Theorem 1.4. We consider solutions o the system (48), as in (50) where z(ξ) = N and N is a nonzero constant, i.e., y(ξ) = Nx(ξ). By substituting y in the irst and second equations o (48), we get x = m + (n 2)k2 2kN N x 2 and x = Nx 2 respectively. Comparing the two expressions we conclude that N must satisy Thereore, we get two values o N, given by N 2 + 2kN (m + (n 2)k 2 ) = 0 N ± = k ± m + (n 1)k 2 (51) 16

17 Going back to the second equation, we have that Hence we have x ± (ξ) = 1 N ± ξ + b x x 2 = N ± and y ± (ξ) = N ± N ± ξ + b. From (45) and (47) we have = k, x(ξ) = ( /)(ξ) and y = h + [(n 2)k m], hence we get, and h given by (7). This concludes the proo Theorem 1.4. Proo o Theorem 1.5. We now consider solutions o the system (48), where y(ξ) = x(ξ)z(ξ) and z(ξ) is a smooth nonconstant unction. Substituting y into (49) and assuming, without loss o generality, x 0 on an open subset, we get dx x = z dz (52) z 2 + 2kz m (n 2)k2 Integrating this equation, we get x in terms o z as given by equation (8). Since y = xz, it ollows rom the irst equation o (48) that x z + xz = [m + (n 2)k 2 ]x 2 2kx 2 z, and second equation o (48) that x = x 2 z Hence we get, z = x[m + (n 2)k 2 2kz z 2 ]. (53) Substituting the equation (8) into (53), we conclude that z must satisy the dierential equation (9). Now a straight orward computation shows that both equations o (48) are satisied. Since we have determined z(ξ) and x(ξ), we can obtain (ξ), (ξ) and h(ξ) integrating (10). This concludes the proo Theorem 1.5. Proo o Theorem 1.6. When ρ = λ F = 0, by introducing the auxiliary unctions (ξ) = k (ξ), (ξ) = x(ξ) and h (ξ) = [z(ξ) + m k(n 2)]x(ξ), we have seen that (5) is equivalent to the system (48) or x and y. The solutions o this system can be written as y(ξ) = x(ξ)z(ξ), where z(ξ) is a nonzero unction. 17

18 I z(ξ) is a non zero constant, then the proo o Theorem 1.4 shows that the solutions o (48) are given by (7). I the unction is not constant, then proo o Theorem 1.5 shows that z is determined by (9) and x is given algebraically in terms o z by (8). Then one gets the unctions, and the potential h by integrating the ordinary dierential equations given by (10). This concludes the proo Theorem 1.6. Proo o Theorem 1.7. Let (R n, g) be a pseudo-euclidean space, n 3 with coordinates x = (x 1,, x n ) and g ij = δ ij ε i. Consider M = (R n, g) F m a warped product where g = 1 2g, F a semi Riemannian maniold Ricci lat. Let (ξ) and (ξ) be any non-vanishing dierentiable unctions invariant by the translation o (n 1)-dimensional translation group, whose basic invariant is ξ = α i x i, where α i R and ε i αi 2 = 0,. Then it i=1 ollows rom Theorem 1.3 that the warped product metric g = g + 2 g F is a gradient Ricci soliton with h as a potential i, and only i, ρ = 0 and h satisies the linear ordinary dierential equation (6) determined by and. Then is easy to see that [ h (ξ) = 2 (m 2 + 2m i=1 (n 2) )dξ + c 4 ] and hence is given by (11) This concludes the proo Theorem 1.7. Reerences [1] E. Barbosa, R. Pina, K. Tenenblat - On Gradient Ricci Solitons conormal to pseudo- Euclidean space, Israel J. Math. 200 (2014), nº 1, [2] W. Batat, M. Brozos Vásquez, E. Garcia Rio, S. Gavino-Fernández - Ricci solitons on Lorentzian maniolds with large isometry groups, Bull. London Math. Soc., 43 (2011), nº 6, [3] A. L. Besse - Einstein Maniolds, Spring-Verlag, Berlin, [4] M. Brozos Vásquez, G. Calvaruso, E. Garcia Rio, S. Gavino-Fernández - Three dimension Lorentzian homogeneous Ricci solitons, Israel. J. Math.s, 188 (2012),

19 [5] M. Brozos Vásquez, E. Garcia Rio, S. Gavino-Fernández - Locally conormally lat Lorentzian gradient Ricci solitons, Journal o Geometric Analysis, 23 (2013), no. 3, [6] M. Brozos Vásquez, E. Garcia Rio, R. Vásquez Lorenzo - Some remarks on locally conormally lat static spacetimes, J. Math. Phys. 46 (2), (2005), 11 pp. [7] Kim, Byung Hak; Lee, Sang Deok; Choi, Jin Hyuk; Lee Young Ok On warped product spaces with a certain Ricci condition. Bull. Korean Math. Soc. 50 (2013), no. 5, [8] M. Brozos Vásquez, E. Garcia Rio, R. Vásquez Lorenzo - Warped product metrics and locally conormally lat structures, Matemática Contemporânea, SBM, vol 28, [9] W. Bryant - Local existence o gradient Ricci solitons, unpublished. [10] H. D. Cao, Q. Chen - On locally conormally lat steady gradient solitons, Trans. Amer. Math. Soc., 364 (2012), no. 5, [11] Chow, B., Chu, S.-C., Glickenstein, D., Guenther, C., Isenberg, J., Ivey, T., Knop, D., Lu, P., Luo, F., Ni, L. - The Ricci low: techniques and applications. Part I: Geometric Aspects., Math. Surveys and Monographs, vol 135, AMS, Providence, RI, [12] H. T. Davis - Introduction to nonlinear dierential and integral equations. Dover Publications, 1962, New York. [13] M. Fernández López, E. Garcia Rio - Rigidity o shrinking Ricci solitons, Math. Z., 269 (2011), no. 1-2, [14] Kim, B.H., Lee, S.D., Choi, J.H., Lee, Y.O. - On warped product spaces with a certain Ricci condition. Bull. Korean Math. Soc., 50, (2013), No. 5, [15] B. O neil - Semi Riemannian Geometry with Applications to Relativity (Academic Press, New York) [16] K. Onda - Lorentzian Ricci solitons on 3-dimensional Lie groups, Geom. Dedicata, 147 (2010),