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Three-dimensional homogeneous generalized Ricci solitons

We study three-dimensional generalized Ricci solitons, both in Riemannian and Lorentzian settings. We shall determine their homogeneous models, classifying left-invariant generalized Ricci solitons on three-dimensional Lie groups.


Introduction
Generalized Ricci solitons were recently introduced in [12]. A generalized Ricci soliton is a pseudo-Riemannian manifold (M, g) admitting a smooth vector field X, such that (1.1) L X g + 2αX ♭ ⊙ X ♭ + 2βRic = 2λg, for some real constants α, β, λ, where L X denotes the Lie derivative in the direction of X, X ♭ denotes a 1-form such that X(Y ) = g(X, Y ) and Ric is the Ricci tensor.
For particular values of the constants α, β, λ, several important equations occur as special cases of equation (1.1). In particular, one has: (K) the Killing vector field equation when α = β = λ = 0; (H) the homothetic vector field equation when α = β = 0; (RS) the Ricci soliton equation when α = 0 and β = 1 [6]. (E-W) a special case of the Einstein-Weyl equation in conformal geometry when α = 1 and β = − 1 n−2 (n > 2) [2]; (PS) the equation for a metric projective structure with a skew-symmetric Ricci tensor representative in the projective class when α = 1, β = − 1 n−1 and λ = 0 [14]; (VN-H) the vacuum near-horizon geometry equation of a spacetime when α = 1 and β = 1 2 , with λ playing the role of the cosmological contant [9]. Equation (1.1) corresponds to an overdetermined system of partial differential equations of finite type. The study of this system was undertaken in the fundamental paper [12]. Explicit solutions were determined in [12] in the two-dimensional case. For the threedimensional case, the authors restricted in [12] to the case with α = 0. Note that as already pointed out in [12], for α = 0 = β, rescaling the vector field X to − 1 β X, equation (1.1) reduces to the Ricci soliton equation. We also observe that a trivial solution of (1.1) is given by X = 0 and β = λ = 0, so we shall always exclude this solution. The aim of this paper is to determine the three-dimensional homogeneous models of generalized Ricci solitons. A connected, complete and simply connected three-dimensional homogeneous manifold, if not symmetric, is isometric to some Lie group equipped with a left-invariant metric (see [15] for the Riemannian case and [3] for the Lorentzian one). Moreover, with the obvious exceptions of R × S 2 (Riemannian) and R 1 × S 2 (Lorentzian), three-dimensional connected simply connected symmetric spaces are also realized in terms of suitable left-invariant metrics on Lie groups [4]. For this reason, we shall consider three-dimensional Lie groups, equipped with a left-invariant metric (either Riemannian or Lorentzian). We shall specify our study to solutions of (1.1) determined by a left-invariant vector field X. In this way, (1.1) will transform into a system of algebraic equations, which we can solve, obtaining a complete classification of three-dimensional left-invariant generalized Ricci solitons, and determining several new solutions of (1.1). We recall that the study of three-dimensional Ricci solitons already showed some interesting differences arising between the Riemannian case (for which left-invariant solutions do not occur [7]) and the Lorentzian one, where several left-invariant solutions exist [1].

3D Riemannian left-invariant generalized Ricci solitons
Three-dimensional Riemannian Lie groups were classified in [11]. We shall treat separately the unimodular and non-unimodular cases.
2.1. Unimodular case. Let G be a connected three-dimensional Lie group with a leftinvariant Riemannian metric. Choose an orientation for the Lie algebra g of G, so that the cross product × is defined on g. The Lie algebra g is unimodular if and only if the endomorphism L, defined by [Z, Y ] = L(Z × Y ), is self-adjoint [11]. Therefore, g admits an orthonormal basis {e 1 , e 2 , e 3 } of eigenvectors for L, so that for some real constants A, B, C. Explicitly, depending on the sign of A, B, C, the Lie group G is isomorphic to one of the cases listed in the following Table I. Lie group (A, B, C) In the above Table I and throughout the paper, SL(2, R) will denote the universal covering of SL(2, R), E(2) the universal covering of the group of rigid motions in the Euclidean two-space, E(1, 1) the group of rigid motions of the Minkowski two-space and H 3 the Heisenberg group. The description of the Ricci curvature with respect to the basis {e 1 , e 2 , e 3 } is well known (see again [11]). We have: In particular, it is easily seen that the left-invariant metric is of constant sectional curvature if and only if either We now consider an arbitrary vector field X, that is, X = X i e i ∈ g, for some real constants X 1 , X 2 , X 3 . Then, by (2.1), we get with respect to the basis {e 1 , e 2 , e 3 }. Moreover, since {e 1 , e 2 , e 3 } is orthonormal, for any vector X = X i e i ∈ g we have X ♭ ⊙ X ♭ (e i , e j ) = X i X j . Therefore, equation (1.1) becomes the following system of algebraic equations: By standard calculations, we obtain the solutions of (2.2), proving the following result. (2) A = B = C, λ = − 1 2 βA 2 , X = 0, for all α, β: when A = B = C, the metric is Einstein.
Solutions (1)-(3) are somewhat "trivial", as they correspond to cases of metrics of constant sectional curvature and could be expected. On the other hand, by (4) we have solutions when just two between A, B, C coincide. By Table I [7]. Finally, we remark that λ = 0 in solution (4). This excludes the possibility of solutions of (PS). On the other hand, if (E-W) holds for a three-dimensional manifold, then αβ = −1. Therefore, condition αβA(A − C) > 0 in (4) yields A(A − C) < 0. Similarly, in the case of the vacuum near-horizon geometry equation (VN-H) (considered in [12] both for Riemannian and Lorentzian two-manifolds), we have αβ = 1, so that condition αβA(A − C) > 0 in (4) yields A(A − C) > 0. Taking into account the above Table I, we then have the following.

2.2.
Non-unimodular case. Let now g denote a three-dimensional non-unimodular Riemannian Lie algebra. Then, its unimodular kernel u is two-dimensional. Choosing an orthonormal basis {e l , e 2 , e 3 } so that e 1 is orthogonal to u and [e 1 , e 2 ], [e 1 , e 3 ] are mutually orthogonal [11], the bracket product is described by for some real constants A, B, C, D.
With respect to the basis {e 1 , e 2 , e 3 }, the Ricci curvature is described by (see [11])

In particular, the left-invariant metric is of constant sectional curvature if and only if
For an arbitrary left-invariant vector field X = X i e i ∈ g, we have with respect to the basis {e 1 , e 2 , e 3 }, and we have again We now solve (2.4) and list its different solutions, proving the following result. Theorem 2.3. Let g denote a three-dimensional non-unimodular Riemannian Lie algebra, as described by (2.3) with respect to a suitable orthonormal basis {e 1 , e 2 , e 3 }. Then, up to a renumeration of e 1 , e 2 , e 3 , the nontrivial left-invariant generalized Ricci solitons on g are the following: , for any α = 0 and β, with X 2 = X 3 = 0 and It is easy to check that the above solutions (1) and (5) are compatible with (PS) (since for α = −1 and β = − 1 2 we have λ = 0 in both (1) and (5)), while all solutions (1), (4), (5) are compatible with (E-W). Thus, we have the following.

3D Lorentzian left-invariant unimodular generalized Ricci solitons
Let now × denote the Lorentzian vector product on the Minkowski space R 3 1 , induced by the product of the para-quaternions (e 1 × e 2 = e 3 , e 2 × e 3 = e 1 , e 3 × e 1 = e 2 , for a pseudo-orthonormal basis e 1 , e 2 , e 3 , with e 3 time-like. The Lie bracket [, ] defines the corresponding Lie algebra g, which is unimodular if and only if the endomorphism L, [13]. Differently from the Riemannian case, in Lorentzian settings L can assume four different standard forms (Segre types), giving rise to four classes of three-dimensional unimodular Lorentzian Lie algebras: g 1 : L is of Segre type {3}, that is, its minimal polynomial has a triple root. g 2 : L is of Segre type {1zz}, that is, it has two complex conjugate eigenvalues. g 3 : L is of Segre type {11, 1}, that is, diagonalizable. g 4 : L is of Segre type {21}, that is, its minimal polynomial has a double root. We shall treat these cases separately.
Solution (2) in Theorem 3.1 corresponds to the existence of Ricci solitons on this class of Lorentzian Lie algebras [1]. Solution (3), requiring that αβ ≤ 1 8 and λ = 0, is incompatible with (VN-H), but compatible with (E-W) and (PS). It is also worth to remark that for B = 0, vector X = X 2 (e 2 + e 3 ) occurring in solution (3) spans a parallel degenerate line field, so showing that the Lorentzian Lie group is then a three-dimensional Walker manifold [8]. These observations yield the following result. In this case, With respect to {e i }, the Ricci tensor is given by (see [4]) and the left-invariant metric is never Einstein. The Lie derivative L X g with respect to a vector X = X i e i ∈ g is described by with respect to the basis {e 1 , e 2 , e 3 }, and again X ♭ ⊙ X ♭ (e i , e j ) = ε i ε j X i X j for all indices i, j. Thus, equation (1.1) becomes We then solve (3.4), obtaininig the following classification result.  The following Table II lists all the Lie groups G which admit a Lie algebra g 3 , according to the different possibilities for A, B and C: Following [4] (or by direct calculation), the Ricci curvature of Lorentzian Lie algebra g 3 , with respect to {e i }, is described by

and the left-invariant metric is Einstein (equivalently, of constant sectional curvature) if and only if either
In the last three cases, the metric is flat. For a left-invariant vector field X = X i e i ∈ g, we have with respect to the basis {e 1 , e 2 , e 3 }. Finally, X ♭ ⊙ X ♭ (e i , e j ) = ε i ε j X i X j for all indices i, j. Hence, equation (1.1) now yields Although system (3.7) is rather similar to (2.2), the different signs, due to the different signature of the metric, are responsible for the existence of many more solutions for (3.7). Solving (3.7), we prove the following.  (4) A = B = C, λ = 1 2 βA 2 , X = 0, for all α, β: when A = B = C, the metric is Einstein.
With regard to solution (8), we observe that equation (3.8) admits real solutions for all values of A, B.
Because of condition αβ = − 3 8 , solutions (7) and (8) are not compatible with none of equations (RS), (E-W), (PS) and (VN-H). We now focus our attention to solutions (5) and (6). Note that these solutions are very similar to one another. However, we listed both of them, since for (5) the vector satisfying (3.7) is space-like, while for (6) is timelike. Moreover, we observe that conditions A = B and B = C are also related to the classification of three-dimensional naturally reductive Lorentzian Lie groups [5].

and the left-invariant metric is Einstein if and only if
For a left-invariant vector field X = X i e i ∈ g, we have with respect to the basis {e 1 , e 2 , e 3 }, and X ♭ ⊙ X ♭ (e i , e j ) = ε i ε j X i X j . Therefore, equation (1.1) now becomes (3.10) We then solve (3.10) and prove the following.
Theorem 3.6. Consider the three-dimensional unimodular Lorentzian Lie algebra g 4 , as described by (3.9) with respect to a suitable pseudo-orthonormal basis {e 1 , e 2 , e 3 }, with e 3 time-like. Then, the nontrivial left-invariant generalized Ricci solitons on g 4 are the following: , for all α, β satisfying ηAαβ < 0.
In the above Theorem 3.6, solution (2) corresponds to the existence of Ricci solitons for Lorentzian Lie algebras of the form g 4 [1]. Solution (1) requires that the sign of αβ is opposite to the one of ηA. Consequently, taking into account the above Table III,

3D Lorentzian left-invariant non-unimodular generalized Ricci solitons
Let now g denote a three-dimensional non-unimodular Lorentzian Lie algebra. Differently from the Riemannian case, we must now consider three distinct cases, depending on whether the two-dimensional unimodular kernel u is either space-like, time-like or degenerate [10]. These three cases were listed in [3] as Lorentzian Lie algebras g 5 , g 6 and g 7 . With respect to the basis {e 1 , e 2 , e 3 }, the Ricci curvature is described by (see [4])

In particular,the left-invariant metric is of constant sectional curvature if and only if
For an arbitrary left-invariant vector field X = X i e i ∈ g, we have With respect to the basis {e 1 , e 2 , e 3 }, we have again X ♭ ⊙ X ♭ (e i , e j ) = ε i ε j X i X j . Hence, equation (1.1) now gives We then solve (4.2) and obtain the following. ( (3) A − D = B + C = 0, λ = −2βA 2 , X = 0, for all α, β: the metric is of constant sectional curvature.