Homogeneous and curvature homogeneous Lorentzian critical metrics

We determine all three-dimensional homogeneous and $1$-curvature homogeneous Lorentzian metrics which are critical for a quadratic curvature functional. As a result, we show that any quadratic curvature functional admits different non-Einstein homogeneous critical metrics and that there exist homogeneous metrics which are critical for all quadratic curvature functionals without being Einstein.


Introduction
Einstein metrics are central in geometry and physics. Being critical for the Einstein-Hilbert functional EH : g → EH(g) = M τ g dvol g subject to a volume constraint, they provide optimal metrics in the sense that the scalar curvature τ is more evenly distributed about the manifold. Since the space of scalar curvature invariants of order one is generated by the scalar curvature, in the search of optimal metrics on a given manifold, it is natural to consider other functionals defined by integrating polynomial curvature invariants of higher order. The space of scalar curvature invariants of order two has dimension at most four and it is generated by {τ 2 , ρ 2 , R 2 , Δτ }. In dimension three, the curvature tensor R is totally determined by the Ricci tensor ρ and R 2 = 2 ρ 2 − 1 2 τ 2 . Thus, the space of quadratic curvature functionals in dimension three is generated by S : g → M τ 2 g dvol g and T : g → M ρ g 2 dvol g .
Hence, every quadratic curvature functional can be expressed as a multiple of S or F t = T + tS, for some t ∈ R.
There are many special cases of the functionals F t that can be found in the literature. We cite just a few examples. The functional defined by the L 2 -norm of the curvature tensor corresponds to F −1/4 . For t = − 1 3 one has the functional defined by the norm of the trace-free Ricci tensor ρ 0 = ρ − 1 3 τ g. The norm of the Schouten tensor S = ρ − 1 4 τ g is given by S 2 = ρ 2 − 5 16 τ 2 , thus defining the functional F −5/16 . The functional corresponding to t = − 3 8 is equivalent to the functional σ 2 : g → M σ 2 (S g ) dvol g defined by the second symmetric elementary function of the eigenvalues of the Schouten tensor (see [15]).
Some of these functionals also play a role in relativistic physics. The functional F −3/8 appears in three-dimensional massive gravity, which is a correction to Einstein's theory of gravity based on the equivalent functional EH − 1 m 2 F −3/8 , where m is the relative mass parameter [3]. Other critical gravity theories complement the Einstein-Hilbert functional with a conformally invariant term (as is the case of conformal gravity) or with a term that extends topological invariants which depend on the dimension (as in the Lovelock theory). The Branson Q-curvature gives rise to higher-curvature theories of gravity whose action is given by a series of dimensionally extended conformal invariants. The Q-curvature of a three-dimensional manifold is defined by (see [4]) Q = − 1 4 Δτ − 2 ρ 2 + 23 32 τ 2 . Hence the functional defined by the total Q-curvature is equivalent to F t for t = − 23 64 and the corresponding theories are based on the functional EH − 1 m 2 F −23/64 (see [11] and references therein).
Euler-Lagrange equations characterizing critical metrics for quadratic curvature functionals subject to a volume constraint were given in [2]. For the functional S and dimension three the equations read 2∇ 2 τ − 2 3 Δτ g − 2τ ρ − 1 3 τ g = 0. (1.1) Critical metrics for the functional T in dimension three are those satisfying where R[ρ] denotes the action of the curvature tensor on the Ricci tensor. From (1.1) and (1.2), the Euler-Lagrange equations for the functionals F t have the expression Although the functionals S and T were initially introduced in the compact setting, one extends them to non-compact manifolds as long as the integral exists. Alternatively one works on compact subsets K ⊂ M and the investigation focusses on the restriction of the metric to those subsets, considering variations with constant volume vanishing at the boundary. Variations of metrics with compact support and constant volume result in the Euler-Lagrange equations (1.1) and (1.2), which can be analysed without the compactness assumption [15]. It follows directly from equations (1.1), (1.2) and (1.3) that, if a metric is critical for two different quadratic curvature functionals, then it is critical for all quadratic curvature functionals. Einstein metrics are critical for all quadratic curvature functionals in dimension three, but there are also non-Einstein pp-waves which are S and F t -critical for all t ∈ R (see [6]).
The purpose of this work is to investigate three-dimensional Lorentzian critical metrics for quadratic curvature functionals with a high degree of symmetry. Hence we focus on the homogeneous and the 1-curvature homogeneous settings. Thus, we classify homogeneous metrics which are critical for quadratic curvature functionals defined on the whole space of metrics (not necessarily homogeneous) restricted to constant volume over compact subsets of M . In § 2 we consider the case of 1-curvature homogeneous spaces and show that they are F t -critical if and only if t = − 1 2 (in which case the underlying structure is a Brinkmann wave and the energy of the functional vanishes), or t − 5 2 (in which case there is a single Ricci curvature which is a double root of the minimal polynomial of the Ricci operator). Furthermore the energy of the functional is negative, zero or positive depending on the value of t.
We collect some basic facts about three-dimensional homogeneous Lorentzian 3manifolds in § 3 and, following the work [9], reduce the analysis to the context of left-invariant metrics on Lie groups. Three-dimensional Lie groups naturally split into the unimodular and the non-unimodular ones.
In § 4 we describe all the critical metrics on Lorentzian 3-dimensional Lie groups. In the unimodular case we obtain that the possible unimodular Lie groups admitting left-invariant F t -critical metrics are the following (modulo isomorphism): • The Heisenberg group, which only admits non-Einstein left invariant F t -critical metrics for the value t = −3.
• The Poincaré group E(1, 1), which admits non-Einstein F t -critical metrics for all t ∈ R.
• The Euclidean group E (2), which only admits non-Einstein left-invariant F tcritical metrics for the value t = −1.
• The Lie group SL(2, R), which admits non-Einstein F t -critical metrics for all t ∈ R.
While unimodular Lorentzian Lie groups were fully described by Rahmani in [20], the description of the non-unimodular ones given in [12] is not complete. This leads to new situations in the non-unimodular case that were not considered previously in the literature. On a non-unimodular Lie group, let A denote the matrix associated to the characteristic endomorphism of the unimodular kernel (see § 3.2.2). The analysis in § 4 shows that the Lorentzian signature framework is much richer than the Riemannian one, where Lie groups with A normalized by tr A = 2 admit critical metrics only if det A 1 (see [5]). Assuming the normalization tr A = 2, results for non-unimodular Lie groups can be summarized as follows: • Lie groups with det A = 0 admit critical metrics so that the restriction of the metric to the unimodular kernel is Lorentzian (theorem 4.12-(2)), Riemannian (theorem 4.16-(1)) or degenerate (theorem 4.20-(1)).
• For any value of det A there exist non-unimodular Lie groups that admit critical metrics such that the unimodular kernel is Lorentzian (theorem 4.12- (3)).
Finally, § 5 is devoted to analyse some special families of F t -critical metrics in more detail. Finally a relation is shown between algebraic Ricci solitons and critical metrics with zero energy.

Three-dimensional curvature homogeneous Lorentzian spaces
A Lorentzian manifold (M, g) is said to be k-curvature homogeneous if for any pair of points p, q ∈ M there exists a linear isometry Φ pq : Clearly any locally homogeneous Lorentzian manifold is k-curvature homogeneous for all k 0. However, the converse does not hold in general. A three-dimensional Lorentzian manifold is 0-curvature homogeneous if the Ricci operator has constant eigenvalues and the corresponding Jordan normal form does not change from point to point. In dimension three, 2-curvature homogeneity guarantees local homogeneity (see [14]), but there are exactly two classes of 1-curvature homogeneous Lorentzian manifolds which are not locally homogeneous. Attending to the Jordan normal form of the Ricci operator, these classes are described as follows [7].
(A) Diagonalizable Ricci operator. Let {u 1 , u 2 , u 3 } be a pseudo-orthonormal local frame field satisfying u 1 , u 1 = u 2 , u 3 = 1. The brackets given by where κ ∈ R and Φ is a function satisfying u 1 (Φ) = κΦ and u 2 (Φ) = − √ 2 2 b, b ∈ R, define 1-curvature homogeneous manifolds with Ricci tensor of the form (B) Non-diagonalizable Ricci operator. Let {u 1 , u 2 , u 3 } be a pseudo-orthonormal local frame field satisfying u 1 , u 1 = u 2 , u 3 = 1. The brackets where α, β ∈ R, α = β, β = 0 and Ψ is a function satisfying u 1 (Ψ) = 2ε − 2(α + β)Ψ and u 2 (Ψ) = 0, ε 2 = 1, define 1-curvature homogeneous manifolds with Ricci tensor given by Any k-curvature homogeneous manifold has constant scalar curvature, so equation (1.1) reduces to τ (ρ − 1 3 τ g) = 0. Therefore, for any k 0, a k-curvature homogeneous metric is S-critical if and only if it is Einstein or its scalar curvature vanishes. Equation (1.3) also simplifies if τ is constant and a k-curvature homogeneous metric is F t -critical if and only if it satisfies We classify F t -critical metrics which are 1-curvature homogeneous but not homogeneous as follows. (1) (M, g) belongs to the class (A) above with κ = 0 and b = 0. In this case, g Proof. Following [7], since (M, g) is a non-homogeneous 1-curvature homogeneous manifold, it either belongs to class (A) or class (B). Assume first that (M, g) belongs to class (A) and let {u 1 , u 2 , u 3 } be a pseudo-orthonormal local frame satisfying u 1 , u 1 = u 2 , u 3 = 1 with Lie brackets given by (2.1). A straightforward calculation shows that Hence, from equation In the former case the metric is Einstein, so it is locally homogeneous and must be excluded. If κ = 0, then equation (2.3) reduces to 4 3 . This corresponds to assertion (1). Now we assume that (M, g) belongs to class (B). A straightforward calculation shows that Hence, equation (2.3) reduces to 8{α 2 + αβ − (1 + 3t)β 2 }ε u 3 ⊗ u 3 = 0. Since ε, β = 0, we conclude that t = α 2 +αβ−β 2 3β 2 , which corresponds to assertion (2). Remark 2.2. Manifolds in theorem 2.1-(1) admit a parallel null line field L = span{u 2 }, so they are Brinkmann waves, but not pp-waves (see § 3.3). Since, moreover, they have non-vanishing constant scalar curvature, they are a particular family of the metrics described in theorem 5.1 of [6]. . Note that, since these metrics are F t -critical for the value t = − 1 2 , the functional has zero energy: ρ 2 − 1 2 τ 2 = 0. On the other hand, the Ricci operator of manifolds in theorem 2.1-(2) has a single eigenvalue λ = −2β 2 , which is a double root of the minimal polynomial. This family of manifolds provides F t -critical metrics for all t − 5 12 . Moreover, the energy of the F t functionals is given by 1 3 , and zero if t = − 1 3 . Remark 2.4. A three-dimensional Lorentzian manifold is locally conformally flat if and only if the Schouten tensor is Codazzi (i.e., ∇ X S Y Z = ∇ Y S XZ ) or, equivalently, if the Cotton tensor vanishes. A straightforward calculation shows that a three-dimensional non-homogeneous 1-curvature homogeneous Lorentzian manifold is locally conformally flat if and only if it corresponds to class (B) with β = −2α, in which case the metric is F t -critical for t = − 5 12 . An immediate calculation from the expression of t in theorem 2.1- (2) shows that a Lorentzian three-dimensional 1-curvature homogeneous (non-homogeneous) manifold is locally conformally flat if and only if it is F t critical for t = − 5 12 . Remark 2.5. Any locally conformally flat three-dimensional 0-curvature homogeneous Lorentzian manifold that is not 1-curvature homogeneous admits a local pseudo-orthonormal frame {u 1 , u 2 , u 3 } with u 1 , u 1 = u 2 , u 3 = 1 so that (see [10]) where Φ, Ψ, Ξ, Υ are smooth functions satisfying A straightforward calculation shows that the Ricci operator has a single eigenvalue κ + ε, which is a double root of the minimal polynomial. Thus, it is given by and thus equation (2.3) reduces to ε(κ + ε)(12t + 5) = 0. Hence, either κ + ε = 0 and (M, g) is critical for all quadratic curvature functionals or, otherwise, it is F t -critical for t = − 5 12 . Note that there exist locally conformally flat homogeneous metrics whose Ricci operator has complex eigenvalues which are not critical for any quadratic curvature functional F t . This analysis is carried out in § 5.2.
Remark 2.6. A Lorentzian manifold is said to be semi-symmetric if the curvature tensor satisfies R(X, Y ) · R = 0 for all vector fields, where R(X, Y ) acts as a derivation on R. Equivalently, the curvature tensor at each point coincides with that of a symmetric space (but possibly changing from point to point). Hence, if a nonhomogeneous 0-curvature homogeneous Lorentzian manifold is semi-symmetric, then the Ricci operator is two-step nilpotent, as in Cahen-Wallach symmetric spaces, or Ric = diag[0, λ, λ], as in direct products R × N (c) with N a surface of constant Gauss curvature. As a consequence, the only semi-symmetric 1-curvature homogeneous Lorentzian manifolds which are not locally homogeneous correspond to class (A) with κ = 0 and b = 0. This shows that a Lorentzian three-dimensional

1-curvature homogeneous (non-homogeneous) manifold is semi-symmetric if and only if it is
Note that there exist F −1/2 -critical homogeneous metrics whose Ricci operator has complex eigenvalues and thus they are not semi-symmetric (see § 5.3).

Three-dimensional homogeneous Lorentzian spaces
The set of three-dimensional homogeneous Lorentzian manifolds splits into two categories as follows.
Theorem 3.1 [9]. Let (M, g) be a three-dimensional connected, simply connected, complete homogeneous Lorentzian manifold. Then, either (M, g) is symmetric, or it is isometric to a three-dimensional Lie group equipped with a left-invariant Lorentzian metric.
The previous result is also true at the local level, therefore a locally homogeneous Lorentzian 3-manifold is either locally symmetric or locally isometric to a Lie group with a left-invariant Lorentzian metric.

Symmetric manifolds
Indecomposable but not irreducible Lorentzian symmetric spaces are locally isometric to Cahen-Wallach symmetric spaces and, hence, they are a particular family of plane waves [8]. Otherwise, three-dimensional symmetric Lorentzian manifolds are of constant sectional curvature or (locally) a product of the form R × N (c), where N (c) is a Riemannian or a Lorentzian surface of constant Gauss curvature.
Since manifolds of constant sectional curvature are Einstein, they are critical for all quadratic curvature functionals. It has been shown in [6] that Cahen-Wallach symmetric spaces are also critical for all quadratic curvature functionals, whereas products of the form R × N (c) are critical for t = − 1 2 .

Lie groups with left invariant metric
We work at the Lie algebra level. Let g be a three-dimensional Lie algebra endowed with a non-degenerate scalar product ·, · : g × g → R. Let × : g × g → g be the cross product satisfying e i × e j , e k = det(e i , e j , e k ) for all orthonormal bases {e 1 , e 2 , e 3 }. The endomorphism L determined by L(e i × e j ) = [e i , e j ], 1 i, j 3, is referred to as the structure operator of g.

Unimodular Lie groups
Three-dimensional unimodular Lie groups are characterized by the self-adjointness of the structure operator L [19,20]. Attending to the Jordan normal form of L, Rahmani showed in [20] that unimodular Lorentzian Lie algebras (g, , ) are equivalent to one of the four types given in table I, where {e 1 , e 2 , e 3 } and {u 1 , u 2 , u 3 } are orthonormal and pseudo-orthonormal bases as specified in each case.

Non-unimodular Lie groups
The Lie algebra of a non-unimodular Lie group is a semi-direct product g = u R, where u = {x ∈ g; tr ad(x) = 0} denotes the unimodular kernel, which is an abelian ideal of g containing the commutator ideal [g, g] (see [19]). These semi-direct products are determined by the endomorphism − ad(e 3 ), which does not depend on the choice of e 3 / ∈ u. Thus, for a basis {e 1 , e 2 } of u, the endomorphism − ad(e 3 ) is given by − ad(e 3 )(e 1 ) = αe 1 + βe 2 and − ad(e 3 )(e 2 ) = γe 1 + δe 2 . Let A denote the matrix associated to − ad(e 3 ).
The study of Lorentzian non-unimodular Lie algebras splits into three different situations depending on the restriction of the scalar product , of g to the unimodular kernel u, namely the cases in which , |u×u is Lorentzian, Riemannian or degenerate. In each of them there exists an adapted basis so that the Lie algebra is given as in table II. By rescaling {e 1 , e 2 , e 3 } one may assume that tr ad(e 3 ) = 2 and thus one works with a representative of the homothety class of the initial metric. Moreover, for type IV.3 Lie algebras, takingû 1 = u 1 ,û 2 = α+δ 2 u 2 andû 3 = 2 α+δ u 3 , one has that tr ad(u 3 ) = 2 and the new basis is still orthonormal, so one remains in the same isometry class.
In the Riemannian situation (see [19]) one may rotate the orthonormal basis {e 1 , e 2 } so that ad(e 3 )(e 1 ) is orthogonal to ad(e 3 )(e 2 ). A straightforward calculation shows that such normalization remains valid in case IV.2 (when the restriction of the inner product to the unimodular kernel is positive definite). Hence one may assume that the structure constants satisfy αγ + βδ = 0 and α + δ = 2 in this case. This is due to the fact that the self-adjoint part of the endomorphism ϕ is diagonalizable, a fact that cannot be assumed in the other two cases. The explicit calculations in § 4.5 (remark 4.15) and 4.7 (remark 4.23) show that the analogous normalizations considered in [12] for cases IV.1 and IV.3 impose restrictions in the corresponding families of metrics.

Homogeneous pp-waves
In the subsequent analysis of homogeneous critical metrics, we will see that some of the families that show up admit a parallel null line field, as occurs in theorem 2.1-(1). More specifically, some of these examples are pp-waves. Apart from Cahen-Wallach symmetric spaces (CW ε ), there are other two families of homogeneous pp-waves. A three-dimensional homogeneous pp-wave admits local adapted coordinates (u, x, y) where the metric is given by The function f determines the type of space, which is locally isometric to one of the following models [13]: The geometries P c , and CW ε are plane waves and, jointly with N b , cover all possible homogeneous pp-wave classes.
Theorem 3.2. Let (M, g) be a three-dimensional homogeneous pp-wave. Then one of the following holds: ) is a plane wave modelled on CW ε or P c then it is critical for all quadratic curvature functionals.
Proof. For metrics that belong to the families P c and CW ε , a direct calculation shows that Δρ, R[ρ], ρ and τ vanish identically. Hence, equation (2.3) holds for all t ∈ R and these metrics are critical for all quadratic curvature functionals. For manifolds modelled on N b , the terms R[ρ], ρ and τ vanish, but Δρ(∂ y , ∂ y ) = b 2 e bx = 0, since b = 0. Therefore, equation (2.3) is not satisfied for any t.

Critical metrics on Lorentzian Lie groups
Based on the classification given in § 3.2, we study Lie groups which admit Lorentzian metrics that are critical for quadratic curvature functionals.
Surprisingly, the Golden number ϕ = 1+ plays a distinguished role in describing F t -critical unimodular Lie groups of types Ia, Ib and II. The golden ratio appears not only in the description of left-invariant metrics, as in theorem 4.1 and theorem 4.7, but also on the range of the real parameter t, as in remark 4.5 (see also Fig. 1). with λ 1 , λ 2 , λ 3 ∈ R. The Ricci operator is given by
If we choose to normalize λ 3 = 1, then we obtain another homothety family with values for λ 1 and λ 2 given by the expressions of λ 2 and λ 3 in (4.3). Also, for appropriate κ we recover the case λ 1 = 0. This concludes case (2).
. This is in sharp contrast with the Riemannian setting, where the Euclidean group does not admit any F t -critical metric which is not Einstein.

Type Ib critical metrics
G is unimodular and there exists an orthonormal basis {e 1 (+), e 2 (+), e 3 (−)} such that the Lie brackets of the Lie algebra g are given by

.4. Let G be a type Ib Lie group with a left-invariant metric g. Then g is F t -critical if and only if it is isometric to a metric given by (4.4) with structure constants satisfying one of the following conditions:
(2) α = λ = 0. In this case, G is isomorphic to E(1, 1) and the metric g is F tcritical for t = −1.
Proof. A straightforward computation shows that the possibly non-vanishing terms of F t with respect to the basis {e 1 , e 2 , e 3 } are the following: We first consider the case in which the scalar curvature vanishes.
and there are two possibilities: either α = 0 and λ = ±2β, or α = − 3 2 λ and β = ± √ 7 2 λ. The term F t 11 is given by F t 11 = 80β 4 /3 and F t 11 = 2048β 4 /147, respectively. Hence F t 11 does not vanish and these metrics are critical for S, but they are not critical for any F t functional.

Type III critical metrics
G is unimodular and there exists a pseudo-orthonormal basis {u 1 , u 2 , u 3 }, with u 1 , u 2 = u 3 , u 3 = 1, such that the Lie brackets of the Lie algebra g are given by with λ ∈ R. There are no Einstein metrics in this family. Moreover, the scalar curvature is given by τ = − 3 2 λ 2 , so metrics defined by (4.6) are S-critical if and only if λ = 0.  g is F t -critical if and only if G is isomorphic to E(1, 1) and g is isometric to a metric given by (4.6) with λ = 0. In this case, the metric is critical for all t ∈ R.
Proof. With respect to the pseudo-orthonormal basis {u 1 , u 2 , u 3 }, F t is determined by the following expressions: Thus, F t vanishes identically if and only if λ = 0.
Remark 4.11. Left-invariant metrics in theorem 4.10 have a null parallel line field L = span{u 1 }. Moreover, the Ricci operator is two-step nilpotent, and thus they are pp-waves. Since no metric in theorem 4.10 is locally symmetric, they correspond to metrics given in theorem 3.2-(1), so they are locally isometric to plane waves in the family P c .
For any non-Einstein critical metric, observe that the energy of the corresponding functional F t is zero if and only if β = −γ.
Remark 4.14. The Ricci operator of critical metrics in theorem 4.12 is determined as follows: • Critical metrics corresponding to case (1) have a single Ricci curvature equal to −2, which is a double root of the minimal polynomial.
• Critical metrics in case (2) have diagonalizable Ricci operator with two-distinct   [12] is not always possible. Indeed, left-invariant metrics in theorem 4.12-(3), where A is self-adjoint, satisfy the above normalization only if A diagonalizes. Thus, the cases with complex eigenvalues and a double root of the minimal polynomial are missed if one assumes the possibility of normalizing as above. Indeed, if one considers metrics given by (4.7) with αγ − βδ = 0, no F t -critical metric is found with t ∈ (−3, − 1 2 ).

Lie group. A non-Einstein left-invariant metric g on G is F t -critical if and only if it is homothetic to a metric given by
Proof. On the pseudo-orthonormal basis {e 1 , e 2 , e 3 }, the a priori non-vanishing components of F t are given by the following expressions: A straightforward calculation shows that Hence, we assume t = − 4 det A−8−3(β−γ) 2 4 det A−16−(β−γ) 2 . Simplifying the components of F t , we get: Thus the tensor F t vanishes if det A = 0, which corresponds to case (1), if β = γ, which corresponds to case (2), or if α = 1 and β = −γ, which corresponds to an Einstein metric.
Remark 4.19. The Lie algebra given by (4.8) corresponds to a semi-direct product R r 2 where the metric restricted to r 2 has positive definite signature. Therefore, the results are analogous to those obtained for three-dimensional Riemannian nonunimodular Lie groups (cf. [5]).

Type IV.3: critical metrics with degenerate unimodular kernel
G is non-unimodular and the restriction of the metric to the unimodular kernel u is degenerate. Then there exists a pseudo-orthonormal basis {u 1 , u 2 , u 3 } of the Lie algebra, with u 1 , u 1 = u 2 , u 3 = 1, such that with α, β, γ, δ ∈ R, and α + δ = 2. These metrics are Einstein only if they are flat, which occurs if γ = 0 and α = 0, 1. Moreover, the scalar curvature is τ = γ 2 2 , so metrics given by (4.9) are S-critical if and only if γ = 0.
Theorem 4.20. Let G be a type IV.3 Lie group. A non-Einstein left-invariant metric g on G is F t -critical if and only if it is homothetic to a metric given by (4.9) with structure constants as follows: (1) det A = 0. In this case, g is F t -critical for t = −3.
(2) The derivation ad(e 3 ) is self-adjoint, i.e., γ = 0. In this case, g is F t -critical for all t ∈ R.
Proof. The a priori non-vanishing components of the tensor field F t are given, with respect to the {u 1 , u 2 , u 3 } basis, by F t 11 vanishes if and only if γ = 0 or t = −3. If γ = 0, the tensor field vanishes identically with independence of the value of t. This corresponds to case (2). Now, we assume γ = 0 and t = −3. The only non-vanishing component is F t 33 = 2γ 2 det A, that vanishes if and only if det A = 0. This corresponds to case (1).
Remark 4.21. The Ricci operator of metrics corresponding to theorem 4.20-(1) is diagonalizable with eigenvalues {− γ 2 2 , γ 2 2 , γ 2 2 } from where it also follows that the energy vanishes in this case ( ρ 2 − 3τ 2 = 0).  2) have a parallel degenerate line field L = span{u 2 }. Moreover, the Ricci operator is two-step nilpotent, so they are pp-waves. In the non-flat case, if α = 2, then they are locally symmetric and, hence, a Cahen-Wallach symmetric space CW ε . Otherwise, they are plane waves P c (see theorem 3.2). Furthermore, since the Ricci operator is two-step nilpotent, all functionals F t have zero energy in this case.
The non-unimodular Lie group with Lie algebra determined by tr A = 2 and det A = 0 equipped with a left-invariant metric homothetic to those given in theorem 4.12-(2) with β + γ = 4 √ 11 . The non-unimodular Lie group with a Lie algebra determined by tr A = 2 and det A = 4 3 equipped with a left-invariant metric homothetic to those given in theorem 4.12-(3) with α = 1 ± γ 2 − 1 3 . The Ricci operator corresponding to the above non-unimodular groups has been discussed in remark 4.14.
Proceeding in an analogous way one may explicitly give all homogeneous metrics which are critical for functionals with a geometric or physical meaning, such as F −3/8 or F −23/64 .

Locally conformally flat homogeneous critical metrics
Homogeneous locally conformally flat three-dimensional manifolds were classified in [16], from where it follows that they are locally symmetric or, otherwise, they correspond to one of the following homothetic classes: • A type Ib Lie group (4.4) with α = − 1 2 , β = √ 3 2 and λ = 1. • A type III Lie group (4.6) with λ = 0.
Type Ib metrics above have Ricci curvatures {−2, 1 ± √ 3 √ −1}. Hence they are S-critical, since the scalar curvature vanishes, but they are not critical for any quadratic curvature functional F t . This is in sharp contrast with the curvature homogeneous case (see remarks 2.4 and 2.5). Type IV.1 metrics above are F −5/12critical and have a single Ricci curvature μ = −2, which is a double root of the minimal polynomial. Metrics corresponding to types III and IV.3 are critical for all quadratic curvature functionals and have two-step nilpotent Ricci operator.

Homogeneous F −1/2 -critical metrics and semi-symmetric spaces
All symmetric spaces are critical for the functional F −1/2 (see subsection 3.1). These spaces are generalized by the so-called semi-symmetric spaces. They are manifolds whose curvature tensor coincides with that of a symmetric space at each point, where the symmetric model may change from point to point. Consequently, a unimodular Lie group is semi-symmetric if and only if it is symmetric or it corresponds to a pp-wave locally modelled on N b (of type II) or P c (of type III).
Results in § 4 show that a unimodular non-Einstein Lie group is F −1/2 -critical if and only if it is of type III as in theorem 4.10 (and thus semi-symmetric) or it is a type Ib Lie group determined by (4.4) with β 2 = 2α 2 + αλ + 3 4 λ 2 and 16α 3 + 12α 2 λ + 8αλ 2 − λ 3 = 0. Furthermore, the Ricci operator has complex eigenvalues in this case and, hence, it does not correspond to any semi-symmetric space.
A non-unimodular Lie group is semi-symmetric if and only if it is symmetric or it corresponds to a type IV.3 Lie group given by (4.9) with γ = 0 (in which case it is a pp-wave modelled on P c or CW ε ). Moreover, results in § 4 show that a nonunimodular Lie group is F −1/2 -critical if and only if it is symmetric or type IV.3 given by (4.9) with γ = 0.
A three-dimensional semi-symmetric homogeneous Lorentzian manifold which is critical for a quadratic curvature functional is S-critical or F −1/2 -critical. Conversely, any homogeneous F −1/2 -critical metric is semi-symmetric unless it corresponds to the type Ib Lie group above with complex Ricci curvatures.

Critical metrics and algebraic Ricci solitons
A Ricci soliton is a triple (M, , , X), where (M, , ) is a pseudo-Riemannian manifold and X is a vector field on M that satisfies the differential equation where L denotes the Lie derivative and μ ∈ R. Ricci solitons are not only generalizations of Einstein metrics, but they correspond to self-similar solutions of the Ricci flow. A Ricci soliton is said to be trivial if the pseudo-Riemannian metric g is Einstein. In the context of Lie groups with left-invariant metric, a Ricci soliton is said to be left-invariant if equation (5.1) holds for a left-invariant vector field X. Moreover, (G, , ) is said to be an algebraic Ricci soliton if the Ricci operator satisfies Ric = μ Id +D for some derivation D of the Lie algebra (see [18]). This condition for the Ricci operator implies that equation (5.1) holds (see [18]). The aim of this section is to relate Lie groups endowed with a left-invariant F t -critical metric and those that are algebraic Ricci solitons. Following the classification in § 3.2 we consider unimodular Lie groups and analyse when Ric −μ Id is a derivation of the Lie algebra. After some straightforward calculations (we omit details in the interest of brevity), we obtain the following (see also [1]).
shows that Ric −μId acts as a derivation only if γ = 0 (see lemma 5.2), so they are algebraic Ricci solitons only in this case, which is the intersection with the subfamily in theorem 4.20-(2).