3.1 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 [Reference Cahen, Leroy, Parker, Tricerri and Vanhecke8]. Otherwise, three-dimensional symmetric Lorentzian manifolds are of constant sectional curvature or (locally) a product of the form $\mathbb {R}\times N(c)$, 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 [Reference Brozos-Vázquez, Caeiro-Oliveira and García-Río6] that Cahen–Wallach symmetric spaces are also critical for all quadratic curvature functionals, whereas products of the form $\mathbb {R}\times N(c)$ are critical for $t=-\frac 12$.

3.2 Lie groups with left invariant metric

We work at the Lie algebra level. Let $\mathfrak {g}$ be a three-dimensional Lie algebra endowed with a non-degenerate scalar product $\langle \cdot,\, \cdot \rangle : \mathfrak {g} \times \mathfrak {g} \to \mathbb {R}$. Let $\times : \mathfrak {g} \times \mathfrak {g} \to \mathfrak {g}$ be the cross product satisfying $\langle e_i\times e_j,\, e_k \rangle = \det (e_i,\,e_j,\,e_k)$ for all orthonormal bases $\{e_1,\,e_2,\,e_3\}$. The endomorphism $L$ determined by $L(e_i\times e_j)=[e_i,\,e_j]$, $1 \leq i,\, j \leq 3$, is referred to as the structure operator of $\mathfrak {g}$.

3.2.1 Unimodular Lie groups

Three-dimensional unimodular Lie groups are characterized by the self-adjointness of the structure operator $L$ [Reference Milnor19, Reference Rahmani20]. Attending to the Jordan normal form of $L$, Rahmani showed in [Reference Rahmani20] that unimodular Lorentzian Lie algebras $(\mathfrak {g},\,\langle,\,\rangle )$ 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.

TABLE 1. Unimodular Lie algebras

3.2.2 Non-unimodular Lie groups

The Lie algebra of a non-unimodular Lie group is a semi-direct product $\mathfrak {g}=\mathfrak {u}\rtimes \mathbb {R}$, where $\mathfrak {u}=\{ x\in \mathfrak {g};\, \operatorname {tr}\operatorname {ad}(x)=0\}$ denotes the unimodular kernel, which is an abelian ideal of $\mathfrak {g}$ containing the commutator ideal $[\mathfrak {g},\,\mathfrak {g}]$ (see [Reference Milnor19]). These semi-direct products are determined by the endomorphism $-\operatorname {ad}(e_3)$, which does not depend on the choice of $e_3\notin \mathfrak {u}$. Thus, for a basis $\{e_1,\,e_2\}$ of $\mathfrak {u}$, the endomorphism $-\operatorname {ad}(e_3)$ is given by $-\operatorname {ad}(e_3)(e_1)=\alpha e_1 +\beta e_2$ and $-\operatorname {ad}(e_3)(e_2)=\gamma e_1+\delta e_2$. Let $A$ denote the matrix associated to $-\operatorname {ad}(e_3)$.

The study of Lorentzian non-unimodular Lie algebras splits into three different situations depending on the restriction of the scalar product $\langle,\,\rangle$ of $\mathfrak {g}$ to the unimodular kernel $\mathfrak {u}$, namely the cases in which $\langle,\,\rangle _{\mid _{\mathfrak {u}\times \mathfrak {u}}}$ 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.

TABLE 2. Non-unimodular Lie algebras

By rescaling $\{ e_1,\,e_2,\,e_3\}$ one may assume that $\operatorname {tr}\operatorname {ad}(e_3)=2$ and thus one works with a representative of the homothety class of the initial metric. Moreover, for type IV.3 Lie algebras, taking $\hat {u}_1=u_1$, $\hat {u}_2=\frac {\alpha +\delta }{2}u_2$ and $\hat {u}_3=\frac {2}{\alpha +\delta }u_3$, one has that $\operatorname {tr}\operatorname {ad}(u_3)=2$ and the new basis is still orthonormal, so one remains in the same isometry class.

In the Riemannian situation (see [Reference Milnor19]) one may rotate the orthonormal basis $\{ e_1,\,e_2\}$ so that $\operatorname {ad}(e_3)(e_1)$ is orthogonal to $\operatorname {ad}(e_3)(e_2)$. 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 $\alpha \gamma +\beta \delta =0$ and $\alpha +\delta =2$ in this case. This is due to the fact that the self-adjoint part of the endomorphism $\varphi$ 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 [Reference Cordero and Parker12] for cases IV.1 and IV.3 impose restrictions in the corresponding families of metrics.

3.3 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 ($\mathcal {CW}_\varepsilon$), 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 $g(\partial _y,\,\partial _y)=-2f(x,\,y)$, $g(\partial _y,\,\partial _{u})=g(\partial _x,\,\partial _x)=1$. The function $f$ determines the type of space, which is locally isometric to one of the following models [Reference García-Río, Gilkey and Nikčević13]:

• • $\mathcal {N}_b$ is defined by taking $f(x,\,y)=b^{-2}e^{bx}$, with $b\neq 0$,

• • $\mathcal {P}_c$ is defined by taking $f(x,\,y)=\tfrac {1}{2}x^{2} \alpha (y)$, with $\alpha ' =c\alpha ^{3/2}$ and $\alpha >0$,

• • $\mathcal {CW}_\varepsilon$ is defined by taking $f(x,\,y)=\varepsilon x^{2}$, with $\varepsilon =\pm 1$.

The geometries $\mathcal {P}_c$, and $\mathcal {CW}_\varepsilon$ are plane waves and, jointly with $\mathcal {N}_b$, cover all possible homogeneous $pp$-wave classes.

4.1 Type Ia critical metrics

$G$ is unimodular and there exists an orthonormal basis $\{e_1 (+),\, e_2(+),\, e_3(-)\}$ such that the structure constants of the Lie algebra $\mathfrak {g}$ are given by

(4.1)\begin{equation} [e_1,e_2]={-}\lambda_3 e_3, \quad [e_1,e_3]={-}\lambda_2 e_2, \quad [e_2,e_3]=\lambda_1 e_1, \end{equation}

with $\lambda _1,\, \lambda _2,\, \lambda _3 \in \mathbb {R}$. The Ricci operator is given by $2\operatorname {Ric}=\operatorname {diag}[(\lambda _2-\lambda _3)^{2}-\lambda _1^{2},\,(\lambda _1-\lambda _3)^{2}-\lambda _2^{2},\,(\lambda _1-\lambda _2)^{2}-\lambda _3^{2}]$, hence Einstein metrics in this family are those that satisfy $\lambda _1=\lambda _2=\lambda _3$ or $\lambda _i=\lambda _j$ and $\lambda _k=0$ for $\{i,\,j,\,k\}=\{1,\,2,\,3\}$. The scalar curvature for a metric given by (4.1) is $\tau =\frac {\lambda _1^{2}}{2}+\frac {\lambda _2^{2}}{2}+\frac {\lambda _3^{2}}{2}-\lambda _1 \lambda _2 -\lambda _1 \lambda _3 -\lambda _2 \lambda _3$. Note that, although this situation is very close to that discussed in [Reference Brozos-Vázquez, Caeiro-Oliveira and García-Río5], since the vector $e_3$ is timelike, it is not interchangeable with $e_1$ and $e_2$ as in the positive definite case.

Proof. Motivated by equation (2.3), we define the $(0,\,2)$-tensor field $\mathfrak {F}^{t} =\Delta \rho +2(R[\rho ]-\tfrac {1}{3}\Vert \rho \Vert ^{2} g)+2t\tau (\rho -\tfrac {1}{3}\tau g)$. Thus, $\mathfrak {F}^{t}$ vanishes if and only if $g$ is $\mathcal {F}_t$-critical. On the pseudo-orthonormal basis $\{e_1,\,e_2,\,e_3\}$, $\mathfrak {F}^{t}$ is given by the following non-vanishing expressions:

\begin{align*} 3\mathfrak{F}_{11}^{t}& = 2(3+t)\lambda_1^{4} -5(1+t)\lambda_1^{3} (\lambda_2 +\lambda_3) +(1+t)\lambda_1(\lambda_2-\lambda_3)^{2}(\lambda_2+\lambda_3) \\ & \quad-(\lambda_2-\lambda_3)^{2}( (3+t)\lambda_2^{2} +2(1-t)\lambda_2\lambda_3 +(3+t)\lambda_3^{2} ) \\ & \quad+\lambda_1^{2} ( (1+3t)\lambda_2^{2} +2(1+t)\lambda_2\lambda_3 +(1+3t)\lambda_3^{2} ),\\ 3\mathfrak{F}_{22}^{t}& ={-}(3+t)\lambda_1^{4} +(1+t)\lambda_1^{3}(\lambda_2+4\lambda_3) \\ & \quad+\lambda_1^{2}( (1+3t)\lambda_2^{2} -(1+t)\lambda_2\lambda_3 -2(1+3t)\lambda_3^{2} ) \\ & \quad-(1+t)\lambda_1(\lambda_2-\lambda_3)(5\lambda_2^{2}+3\lambda_2\lambda_3+4\lambda_3^{2}) \\ & \quad+(\lambda_2-\lambda_3)( 2(3+t)\lambda_2^{3} +2\lambda_2\lambda_3^{2} +(3+t)\lambda_3^{3} +(1-3t)\lambda_2^{2}\lambda_3 ),\\ \mathfrak{F}_{33}^{t}& =\mathfrak{F}_{11}^{t} +\mathfrak{F}_{22}^{t}. \end{align*}

Since equation (2.3) is invariant under homotheties, we normalize the constants $(\lambda _1,\,\lambda _2,\,\lambda _3)$ and divide the proof into the following cases, which cover all the non-Einstein possibilities:

1. (1) Two constants are equal and the third one is nonzero:

   1. (a) $(\lambda _1,\,\lambda _2,\,\lambda _3)=(1,\,0,\,0)$ or $(\lambda _1,\,\lambda _2,\,\lambda _3)=(0,\,0,\,1)$,

   2. (b) $(\lambda _1,\,\lambda _2,\,\lambda _3)=(1,\,\lambda,\,\lambda )$ or $(\lambda _1,\,\lambda _2,\,\lambda _3)=(\lambda,\,\lambda,\,1)$, with $\lambda \neq 0,\,1$.

2. (2) The three constants are different:

   1. (a) $(\lambda _1,\,\lambda _2,\,\lambda _3)=(0,\,\alpha,\,\beta )$ or $(\lambda _1,\,\lambda _2,\,\lambda _3)=(\alpha,\,\beta,\,0)$, with $0\neq \alpha \neq \beta \neq 0$.

   2. (b) $(\lambda _1,\,\lambda _2,\,\lambda _3)=(1,\,\alpha,\,\beta )$ or $(\lambda _1,\,\lambda _2,\,\lambda _3)=(\alpha,\,\beta,\,1)$, with $0,\,1\neq \alpha \neq \beta \neq 0,\,1$.

Case (1.a). We consider structure constants $(\lambda _1,\,\lambda _2,\,\lambda _3)=(1,\,0,\,0)$ and $(\lambda _1,\,\lambda _2,\,\lambda _3)=(0,\,0,\,1)$. Then, we get, respectively,

\[ \mathfrak{F}_{11}^{t}={-}2 \mathfrak{F}_{22}^{t}=2 \mathfrak{F}_{33}^{t}=\frac 23 (3+t) \text{ and } 2\mathfrak{F}_{11}^{t}=2 \mathfrak{F}_{22}^{t}= \mathfrak{F}_{33}^{t}={-}\frac 23 (3+t). \]

Therefore, these metrics are $\mathcal {F}_t$-critical for $t=-3$.

Case (1.b). For $(\lambda _1,\,\lambda _2,\,\lambda _3)=(1,\,\lambda,\,\lambda )$ and $(\lambda _1,\,\lambda _2,\,\lambda _3)=(\lambda,\,\lambda,\,1)$, with $\lambda \neq 0,\,1$, the expressions of $\mathfrak {F}_{ij}^{t}$ reduce, respectively, to

\[ \begin{array}{l} \mathfrak{F}_{11}^{t}={-}2\mathfrak{F}_{22}^{t}=2\mathfrak{F}_{33}^{t}=\dfrac{2}{3} (\lambda -1) (2 \lambda-3 +(4 \lambda -1) t), \text{ and }\\ 2\mathfrak{F}_{11}^{t}=2\mathfrak{F}_{22}^{t}=\mathfrak{F}_{33}^{t}={-}\dfrac{2}{3} (\lambda -1) (2 \lambda-3 +(4 \lambda -1) t).\end{array} \]

Hence, since $\lambda \neq 1$, these terms vanish if and only if $\lambda \neq \frac 14$ and $t=\frac {3-2\lambda }{-1+4\lambda }$.

Note that Case (1.a) corresponds to $\lambda =0$ in case (1.b), so this two cases constitute case (1).

Case (2). We work with structure constants $(\lambda _1,\,\lambda _2,\,\lambda _3)$, all of them different to each other. Then we compute

\[ \frac{1}{\lambda_2-\lambda_3}\left\{\frac{1}{\lambda_1-\lambda_2}(\mathfrak{F}^{t}_{11}-\mathfrak{F}^{t}_{22}) -\frac{1}{\lambda_1-\lambda_3}(\mathfrak{F}^{t}_{11}+\mathfrak{F}^{t}_{33}) \right\} =4(\lambda_1^{2}+\lambda_2^{2}+\lambda_3^{2})+4 t\tau. \]

Note that, if $\tau =0$, then the metric is not critical for any $t$, since $\mathfrak {F}^{t}_{ii}=0$ implies $\lambda _1=\lambda _2=\lambda _3=0$, contrary to our assumption. Hence, we have $\tau \neq 0$ and the value of $t$ is given by $t=-\frac {\lambda _1^{2}+\lambda _2^{2}+\lambda _3^{2}}{\tau }$. Taking the value of $t$ into account, the expressions of $\mathfrak {F}_{ii}^{t}$ reduce to

\begin{align*} 3\mathfrak{F}_{11}^{t}& =\left(2 \lambda _1-\lambda _2-\lambda _3\right)\! \left(\lambda _3^{3}-\lambda _1 \left(\lambda _2+\lambda _3\right){}^{2}-\lambda _1^{2} \left(\lambda _2+\lambda _3\right)+\left(\lambda _2-\lambda _3\right){}^{2} \left(\lambda _2+\lambda _3\right)\right)\!,\\ 3\mathfrak{F}_{22}^{t}& =\left(\lambda _1-2\lambda _2+\lambda _3\right)\! \left(\lambda _3^{3}-\lambda _1 \left(\lambda _2+\lambda _3\right){}^{2}-\lambda _1^{2} \left(\lambda _2+\lambda _3\right)+\left(\lambda _2-\lambda _3\right){}^{2} \left(\lambda _2+\lambda _3\right)\right)\!. \end{align*}

Since the values of $\lambda _1$, $\lambda _2$ and $\lambda _3$ are all distinct, the first factors in these expressions cannot vanish simultaneously. Hence, we have

(4.2)\begin{equation} \lambda _1^{3}+\lambda _2^{3}+\lambda _3^{3}-\lambda _2 \lambda _1^{2}-\lambda _3 \lambda _1^{2}-\lambda _2^{2} \lambda _1-\lambda _3^{2} \lambda _1-\lambda _2 \lambda _3^{2}-\lambda _2^{2} \lambda _3-2 \lambda _1 \lambda _2 \lambda_3=0. \end{equation}

Note that this equation and the value of $t$ are symmetric in the $\lambda$'s, so we work modulo permutation.

Case (2.a). Assume that $\lambda _1=0$. Then $(\lambda _2-\lambda _3)^{2} (\lambda _2+\lambda _3)=0$, so $\lambda _2=- \lambda _3$. We have two homothety classes: $(0,\,1,\,-1)$ and $(1,\,-1,\,0)$. In both cases the corresponding left invariant metrics are critical for $t=-1$ and the Ricci operator satisfies $\operatorname {Ric}=\operatorname {diag}[2 ,\,0,\,0]$ and $\operatorname {Ric}=\operatorname {diag}[0,\,0,\,2]$, respectively.

Case (2.b). Now we consider the case $\lambda _1\neq 0$. The obtained expressions coincide with those studied in the Riemannian analogue in [Reference Brozos-Vázquez, Caeiro-Oliveira and García-Río5]. Rescaling the metric so that $\lambda _1=1$ and setting $\lambda _2=\frac {-1+\mu _2+\mu _3}2$ and $\lambda _3=\frac {1+\mu _2-\mu _3}2$, expression (4.2) reduces to

\[ -\mu _2^{2}+\left(\mu _3-2\right) \mu _3 \mu _2+1=0. \]

Then, solving in $\mu _3$, one has

\[ \mu_3=1\pm\frac{\sqrt{\mu _2 \left(\mu _2^{2}+\mu _2-1\right)}}{\mu _2}. \]

Note that $\mu _3$ is well-defined if $\mu _2\in (-\varphi,\,0)$ or $\mu _2>\varphi ^{-1}$. Hence, set $\mu _2=\kappa$ to see that, for structure constants $\lambda _i\neq \lambda _j$ if $i\neq j$, the corresponding metric is $\mathcal {F}_t$-critical if and only if it is homothetic to a metric given by $\lambda _1=1$,

(4.3)\begin{equation} \lambda_2=\frac{\kappa}2 \pm\frac{\sqrt{\kappa \left(\kappa^{2}+\kappa-1\right)}}{2\kappa}, \text{ and } \lambda_3=\frac{\kappa}2 \mp\frac{\sqrt{\kappa \left(\kappa^{2}+\kappa-1\right)}}{2\kappa}. \end{equation}

The structure constants $(1,\,\lambda _2,\,\lambda _3)$ are distinct except if $\kappa =1$, $\kappa ={ \varphi ^{3}}$ or $\kappa ={-\varphi ^{-3}}$. Note that for $\kappa =- 1$ we get that $\lambda _3=0$. Thus, this case includes case (2.a).

If we choose to normalize $\lambda _3=1$, then we obtain another homothety family with values for $\lambda _1$ and $\lambda _2$ given by the expressions of $\lambda _2$ and $\lambda _3$ in (4.3). Also, for appropriate $\kappa$ we recover the case $\lambda _1=0$. This concludes case (2).

4.2 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 $\mathfrak {g}$ are given by

(4.4)\begin{equation} [e_1,e_2]={-}\beta e_2 -\alpha e_3,\quad [e_1,e_3]={-}\alpha e_2 +\beta e_3, \quad [e_2,e_3]=\lambda e_1, \end{equation}

with $\alpha,\, \beta,\, \lambda \in \mathbb {R}$, $\beta \neq 0$. Further observe that a replacement $e_1\mapsto -e_1$ provides an isometry interchanging $(\alpha,\,\beta,\,\lambda )$ with $(-\alpha,\,-\beta,\,-\lambda )$, whereas $e_2\mapsto -e_2$ or $e_3\mapsto -e_3$ interchanges $(\alpha,\,\lambda )$ with $(-\alpha,\,-\lambda )$, hence one may assume $\beta >0$.

Metrics defined by (4.4) are not Einstein and they have scalar curvature $\tau =\frac {1}{2}(\lambda (\lambda -4\alpha )-4\beta ^{2})$. Hence they are $\mathcal {S}$-critical if and only if $\lambda =2(\alpha \pm \sqrt {\alpha ^{2}+\beta ^{2}})$.

Proof. A straightforward computation shows that the possibly non-vanishing terms of $\mathfrak {F}^{t}$ with respect to the basis $\{e_1,\,e_2,\,e_3\}$ are the following:

\begin{align*} \mathfrak{F}_{11}^{t}& ={-}2 \mathfrak{F}_{22}^{t}=2\mathfrak{F}_{33}^{t}={-}\tfrac{2}{3}\left(2t \beta^{2} \lambda^{2} -(3+t)\lambda^{4} +(1+t)(8\beta^{4}+\alpha \lambda (4\beta^{2} +5\lambda^{2}))\right. \\ & \quad\left.-2\alpha^{2}(8\beta^{2} +(1+2t)\lambda^{2})\right),\\ \mathfrak{F}_{23}^{t}& = \beta\left( 8\alpha^{3} -8(1+t)\lambda \alpha^{2} -2(4(2+t)\beta^{2} -(1+3t)\lambda^{2})\alpha +(1+t)\lambda(4\beta^{2}-\lambda^{2}) \right). \end{align*}

We first consider the case in which the scalar curvature vanishes. If $\tau =4\beta ^{2} +(4\alpha -\lambda )\lambda =0$, then $\mathfrak {F}_{23}^{t}=-\beta (-8 \alpha ^{3} + 16 \alpha \beta ^{2} + 8 \alpha ^{2} \lambda - 4 \beta ^{2} \lambda - 2 \alpha \lambda ^{2} + \lambda ^{3})$ and there are two possibilities: either $\alpha =0$ and $\lambda = \pm 2\beta$, or $\alpha =-\tfrac {3}{2}\lambda$ and $\beta =\pm \tfrac {\sqrt {7}}{2}\lambda$. The term $\mathfrak {F}_{11}^{t}$ is given by $\mathfrak {F}_{11}^{t}= 80\beta ^{4}/3$ and $\mathfrak {F}_{11}^{t}= 2048 \beta ^{4}/147$, respectively. Hence $\mathfrak {F}_{11}^{t}$ does not vanish and these metrics are critical for $\mathcal {S}$, but they are not critical for any $\mathcal {F}_t$ functional.

Now, if $\tau \neq 0$, a direct analysis of the expression of $\mathfrak {F}_{23}^{t}$ shows that it vanishes if and only if one of the following assertions holds:

1. (i) $t=\frac {8\alpha ^{3} -8\alpha ^{2}\lambda +4\beta ^{2} \lambda -\lambda ^{3} +2\alpha (\lambda ^{2} -8\beta ^{2})}{(2\alpha -\lambda )\tau }$,

2. (ii) $\alpha =\lambda =0$.

In case (i), one has $(6\alpha -3\lambda )\mathfrak {F}_{11}^{t}=4((\alpha -\lambda )^{2}+\beta ^{2}) (4 \alpha ^{2} \lambda +2 \alpha (4 \beta ^{2}+\lambda ^{2})-\lambda ^{3})$. Since $\beta \neq 0$, the only possibility for $\mathfrak {F}_{11}^{t}$ to vanish is that $\beta ^{2}=\frac {\lambda }{8 \alpha }(\lambda ^{2} -2\alpha \lambda -4\alpha ^{2})$, which corresponds to case (1) in the theorem.

In case (ii), $\mathfrak {F}_{11}^{t}$ reduces to $\mathfrak {F}_{11}^{t}= -16(1+t)\beta ^{4}/3$. Hence $t=-1$. This is case(2).

4.3 Type II critical metrics

$G$ is unimodular and there exists a pseudo-orthonormal basis $\{u_1,\, u_2,\, u_3\}$, with $\langle u_1,\,u_2 \rangle =\langle u_3,\,u_3 \rangle =1$, such that the Lie brackets of the Lie algebra $\mathfrak {g}$ are given by

(4.5)\begin{equation} [u_1,u_2]=\lambda_2 u_3,\quad [u_1,u_3]={-}\lambda_1 u_1 -\varepsilon u_2, \quad [u_2,u_3]=\lambda_1 u_2, \end{equation}

with $\lambda _1,\, \lambda _2 \in \mathbb {R}$, $\varepsilon =\pm 1$. These metrics are Einstein if and only if they are Ricci flat, which occurs if and only if $\lambda _1=\lambda _2=0$. Furthermore, the scalar curvature is given by $\tau =\frac 12 \lambda _2 ( \lambda _2-4 \lambda _1 )$, so metrics given by (4.5) are $\mathcal {S}$-critical if and only if $\lambda _2=0$ or $\lambda _2=4\lambda _1$.

Proof. We compute $\mathfrak {F}^{t} =\Delta \rho +2(R[\rho ]-\tfrac {1}{3}\Vert \rho \Vert ^{2} g)+2t\tau (\rho -\tfrac {1}{3}\tau g)$ in the basis $\{u_1,\,u_2,\,u_3\}$, which is determined by

\begin{align*} \mathfrak{F}_{11}^{t}& = \varepsilon ( 8 \lambda_1^{3} -8(1+t)\lambda_1^{2}\lambda_2 +2(1+3t)\lambda_1\lambda_2^{2} -(1+t)\lambda_2^{3} ),\\ \mathfrak{F}_{12}^{t}& ={-}\frac 12 \mathfrak{F}_{33}^{t}={-}\tfrac{1}{3}( \lambda_1-\lambda_2 )\lambda_2^{2}( (2+4t)\lambda_1 -(3+t)\lambda_2 ). \end{align*}

$\mathfrak {F}_{12}^{t}$ vanishes if and only if $\lambda _2=0$, $\lambda _1=\lambda _2$ or $\lambda _2\neq 4\lambda _1$ and $t=\frac {-2\lambda _1 +3\lambda _2}{4\lambda _1 -\lambda _2}$.

If $\lambda _2=0$, then $\mathfrak {F}_{11}^{t}=8\varepsilon \lambda _1^{3}$. Thus $\lambda _1=\lambda _2=0$ and the manifold is Ricci-flat.

If $\lambda _1=\lambda _2$, it follows that $\mathfrak {F}_{11}^{t} =(1-3t)\varepsilon \lambda _2^{3}$. Hence, since $\lambda _1=\lambda _2=0$ corresponds to an Einstein metric, we conclude that $t= \tfrac {1}{3}$. This is case (1).

Now, assume $\lambda _2\neq 0$, $\lambda _1\neq \lambda _2$ and $\lambda _2\neq 4\lambda _1$. If $t=\frac {-2\lambda _1 +3\lambda _2}{4\lambda _1 -\lambda _2}$, then $\mathfrak {F}_{11}^{t}=2\varepsilon (\lambda _1-\lambda _2)(4\lambda _1^{2} +2\lambda _1\lambda _2-\lambda _2^{2})$. Hence $4\lambda _1^{2} +2\lambda _1\lambda _2-\lambda _2^{2}=0$, so $\lambda _1=-\frac {\varphi }{2}\lambda _2$, which corresponds to case (2), or $\lambda _1=- \varphi ^{-1}\lambda _2$, which corresponds to case (3).

4.4 Type III critical metrics

$G$ is unimodular and there exists a pseudo-orthonormal basis $\{u_1,\, u_2,\, u_3\}$, with $\langle u_1,\,u_2 \rangle =\langle u_3,\,u_3 \rangle =1$, such that the Lie brackets of the Lie algebra $\mathfrak {g}$ are given by

(4.6)\begin{equation} [u_1,u_2]=u_1 +\lambda u_3, \quad [u_1,u_3]={-}\lambda u_1, \quad [u_2,u_3]=\lambda u_2 +u_3, \end{equation}

with $\lambda \in \mathbb {R}$. There are no Einstein metrics in this family. Moreover, the scalar curvature is given by $\tau =-\frac 32 \lambda ^{2}$, so metrics defined by (4.6) are $\mathcal {S}$-critical if and only if $\lambda =0$.

Proof. With respect to the pseudo-orthonormal basis $\{u_1,\,u_2,\,u_3\}$, $\mathfrak {F}^{t}$ is determined by the following expressions:

\[ \mathfrak{F}_{22}^{t}= 6(2-t)\lambda^{2} \quad \text{ and }\quad\mathfrak{F}_{23}^{t}= (1-3t)\lambda^{3}. \]

Thus, $\mathfrak {F}^{t}$ vanishes identically if and only if $\lambda =0$.

4.5 Type IV.1: critical metrics with Lorentzian unimodular kernel

$G$ is non-unimodular and the induced metric on the unimodular kernel $\mathfrak {u}$ is Lorentzian. Then there exists an orthonormal basis $\{e_1(-),\, e_2(+),\, e_3(+)\}$ such that the Lie algebra $\mathfrak {g}$ is given by

(4.7)\begin{equation} [e_1,e_2]=0, \quad [e_1,e_3]=\alpha e_1 +\beta e_2, \quad [e_2,e_3]=\gamma e_1 +\delta e_2, \end{equation}

with $\alpha,\, \beta,\, \gamma,\, \delta \in \mathbb {R}$. We work with a representative of the homothety class so that $\alpha +\delta =2$. These metrics are Einstein for the following values of $\alpha$, $\beta$ and $\gamma$:

1. (i) $\alpha =1$ and $\gamma =\beta$,

2. (ii) $\beta =\pm \alpha$ and $\gamma =\pm (2-\alpha )$.

Note that the case $A=\operatorname {ad}(e_3)=\operatorname {\rm Id}$ is included in $(i)$. Also, note that the structure given by $(\beta,\,\gamma )=(\alpha,\,2-\alpha )$ is isometric to $(\beta,\,\gamma )=(-\alpha,\,\alpha -2)$ since the replacement $e_1\mapsto -e_1$ provides an isometric isomorphism interchanging $(\beta,\,\gamma )$ and $(-\beta,\,-\gamma )$. The scalar curvature has the expression $\tau =\frac {1}{2} (-4 (\alpha -2) \alpha +(\beta -\gamma )^{2}-16)$. Thus metrics given by (4.7) are $\mathcal {S}$-critical if and only if $\alpha =1\pm \frac {1}{2}\sqrt {(\beta -\gamma )^{2}-12}$ or the manifold is Einstein.

Proof. The symmetric (0,2)-tensor field $\mathfrak {F}^{t} =\Delta \rho +2(R[\rho ]-\tfrac {1}{3}\Vert \rho \Vert ^{2} g)+2t\tau (\rho -\tfrac {1}{3}\tau g)$ is given, with respect to the basis $\{e_1,\,e_2,\,e_3\}$, by the following components:

\begin{align*} \mathfrak{F}_{11}^{t}& ={-}\tfrac{1}{3}( 8\det A ^{2} +4( 6\alpha -\beta^{2} +4\beta\gamma +5\gamma^{2} -12 )\det A \\ & \quad+(6\alpha -\beta^{2} +\beta \gamma +2\gamma^{2} -8)(3(\beta +\gamma)^{2} -8) \\ & \quad+t((\beta+\gamma)^{2} +4(\det A -4))(2 \det A +6\alpha -\beta^{2} +\beta \gamma +2\gamma^{2} -8) ),\\ \mathfrak{F}_{12}^{t}& ={-}( (\alpha-2)\beta +\alpha\gamma ) ( 3(\beta+\gamma)^{2} -8 ) +4( (3-2\alpha)\beta +(1-2\alpha)\gamma )\det A \\ & \quad-t( (\alpha-2)\beta +\alpha\gamma )( (\beta+\gamma)^{2} +4(\det A -4) ),\\ \mathfrak{F}_{22}^{t}& = \tfrac{1}{3} ( 8\det A^{2} -4 \left(6 \alpha -5 \beta ^{2}-4 \beta \gamma +\gamma ^{2}\right)\det A \\ & \quad-(3 (\beta + \gamma )^{2}-8) \left(6 \alpha -2 \beta ^{2}-\beta \gamma +\gamma ^{2}-4\right) \\ & \quad+t \left( (\beta +\gamma )^{2}+4 \det A-16 \right) \left({-}6 \alpha +2 \beta ^{2}+\beta \gamma -\gamma ^{2}+2 \det A +4\right)),\\ \mathfrak{F}_{33}^{t}& ={-}\tfrac{1}{3} ( 4(\det A -1) + (\beta+\gamma)^{2} )\\ & \quad\times ( 4(\det A-2)+3(\beta+\gamma)^{2} +t(4(\det A-4) +(\beta+\gamma)^{2}) ). \end{align*}

A straightforward calculation shows that $\mathfrak {F}_{33}^{t}$ vanishes if and only if $\det A = 1-\frac {(\beta +\gamma )^{2}}{4}$ or $t=-\frac {4(\det A-2)+3(\beta +\gamma )^{2}}{4(\det A-4) +(\beta +\gamma )^{2}}$.

Firstly, we assume $\det A = 1-\frac {(\beta +\gamma )^{2}}{4}$, then $\alpha =1 \pm \tfrac {1}{2}(\beta -\gamma )$. Since $(\beta,\,\gamma )\to (-\beta,\,-\gamma )$ provides an isometry, we assume without loss of generality that $\alpha =1 + \tfrac {1}{2}(\beta -\gamma )$. The components of $\mathfrak {F}^{t}$ simplify to

\[ \mathfrak{F}^{t}_{11}={-}\mathfrak{F}^{t}_{12}=\mathfrak{F}^{t}_{22}={-}(2\det A +6t +\beta +\gamma)(\beta+\gamma-2)(\beta-\gamma). \]

If $\beta =\gamma$, then $\alpha =1$ and the metric is Einstein. If $\gamma =2-\beta$, then $\alpha =\beta$ and the metric is also Einstein. Hence $t=-\tfrac {1}{6}(2\det A +\beta +\gamma )$, which corresponds to case (1).

Secondly, we assume $t=-\frac {4(\det A-2)+3(\beta +\gamma )^{2}}{4(\det A-4) +(\beta +\gamma )^{2}}$. The components of $\mathfrak {F}^{t}$ reduce to

\[ \mathfrak{F}^{t}_{11}=\mathfrak{F}^{t}_{22}= 2\det A (\beta+\gamma)(\beta-\gamma), \text{ and } \mathfrak{F}^{t}_{12}={-}4\det A (\beta+\gamma)(\alpha-1). \]

Thus, the tensor $\mathfrak {F}^{t}$ vanishes if $\det A=0$, which corresponds to case (2), if $\beta =-\gamma$, which corresponds to case (3), or if $\alpha =1$ and $\beta =\gamma$, which is an Einstein metric.

Remark 4.13 Metrics in theorem 4.12 provide examples of $\mathcal {F}_t$-critical metrics for values of $t$ in $\mathbb {R} \setminus \{-1\}$. The attained values in each of the cases are illustrated in figure 2 and are described as follows:

• • Case (1) provides $\mathcal {F}_t$-critical metrics for $t \in (-\tfrac {5}{12},\,\infty )$ with energy $\|\rho \|^{2}+t\tau ^{2}=3 (\beta +\gamma )(\beta +\gamma -2)$.

• • Case (2) provides $\mathcal {F}_t$-critical metrics for $t \in (-\infty,\, -3) \cup (-\tfrac {1}{2},\, +\infty )$ with energy $\|\rho \|^{2}+t\tau ^{2}=6 (\beta +\gamma )^{2}$.

• • Case (3) provides $\mathcal {F}_t$-critical metrics for all $t \in \mathbb {R} \setminus \{-1\}$ with zero energy.

For any non-Einstein critical metric, observe that the energy of the corresponding functional $\mathcal {F}_t$ is zero if and only if $\beta =-\gamma$.

FIGURE 2. This diagram shows the values of $t$ for $\mathcal {F}_t$-critical metrics in each family of non-unimodular Lie groups following the notation in theorems 4.12, 4.16, 4.20.

4.6 Type IV.2: critical metrics with Riemannian unimodular kernel

$G$ is non-unimodular and the induced metric on the unimodular kernel $\mathfrak {u}$ is Riemannian. Then there exists an orthonormal basis $\{e_1(+),\, e_2(+),\, e_3(-)\}$ such that the Lie algebra $\mathfrak {g}$ is given by

(4.8)\begin{equation} [e_1,e_2]=0, \quad [e_1,e_3]=\alpha e_1 +\beta e_2, \quad [e_2,e_3]=\gamma e_1 +\delta e_2, \end{equation}

with $\alpha,\, \beta,\, \gamma,\, \delta \in \mathbb {R}$ satisfying $\alpha +\delta =2$. A direct calculation shows that these metrics are Einstein if and only if $\alpha =1$ and $\gamma =-\beta$ (which includes the case $\operatorname {ad}(e_3)=\operatorname {\rm Id}$). Moreover, the scalar curvature is given by $\tau =\frac {1}{2} (4 (\alpha -2) \alpha +(\beta +\gamma )^{2}+16)$ and satisfies $\tau >0$. Hence the only $\mathcal {S}$-critical metrics within this family are the Einstein ones.

Proof. On the pseudo-orthonormal basis $\{e_1,\,e_2,\,e_3\}$, the a priori non-vanishing components of $\mathfrak {F}^{t}$ are given by the following expressions:

\begin{align*} \mathfrak{F}_{11}^{t}& = \tfrac{1}{3}( 8 (\det A)^{2} +4(6 \alpha +\beta^{2} +4\beta \gamma -5\gamma^{2} -12)\det A \\ & \quad-(6\alpha +\beta^{2} +\beta\gamma -2\gamma^{2} -8)(3(\beta-\gamma)^{2}+8) \\ & \quad-t(2\det A +6\alpha +\beta^{2} +\beta \gamma -2\gamma^{2} -8)(4\det A -(\beta-\gamma)^{2} -16) ),\\ \mathfrak{F}_{12}^{t}& = \beta (8 \alpha ^{3}-28 \alpha ^{2}+\alpha (9 \gamma ^{2}+32)-2 (\gamma ^{2}+8))+3 (\alpha -2) \beta ^{3}-\alpha \beta ^{2} \gamma \\ & \quad+\alpha \gamma (4 \alpha -3 \gamma ^{2}-16 +8 \det A) +t((\alpha -2) \beta -\alpha \gamma ) ((\beta -\gamma) ^{2}-4 \det A+16),\\ \mathfrak{F}_{22}^{t}& = \tfrac{1}{3}( 8 (\det A)^{2} -4(6 \alpha +5 \beta ^{2}-4 \beta \gamma -\gamma ^{2})\det A \\ & \quad+(3(\beta-\gamma)^{2} +8)(6\alpha +2\beta^{2} -\beta \gamma -\gamma^{2} -4) )\\ & \quad+t(4 \det A-16 -(\beta-\gamma)^{2}) ({-}6 \alpha -2 \beta ^{2}+\beta \gamma +\gamma ^{2}+2 \det A+4),\\ \mathfrak{F}_{33}^{t}& = \tfrac{1}{3}( 4\det A -4 -(\beta-\gamma)^{2})\\ & \quad\times ( 4\det A -8 -3(\beta-\gamma)^{2} +t(4\det A -16 -(\beta-\gamma)^{2}) ). \end{align*}

A straightforward calculation shows that $\mathfrak {F}_{33}^{t}$ vanishes if and only if $\det A=1 +\tfrac {1}{4}(\beta -\gamma )^{2}$ or $t=-\frac {4\det A -8 -3(\beta -\gamma )^{2}}{4\det A -16 -(\beta -\gamma )^{2}}$. Since $\det A=-\alpha ^{2}+2 \alpha -\beta \gamma$, the equation $\det A=1 +\tfrac {1}{4}(\beta -\gamma )^{2}$ has real solutions only if $\beta =-\gamma$, in which case $\alpha =1$. Hence, this condition leads to an Einstein metric.

Hence, we assume $t=-\frac {4\det A -8 -3(\beta -\gamma )^{2}}{4\det A -16 -(\beta -\gamma )^{2}}$. Simplifying the components of $\mathfrak {F}^{t}$, we get:

\begin{align*} \mathfrak{F}_{11}^{t}& ={-}\mathfrak{F}_{22}^{t}={-}2 \det A (\beta -\gamma) (\beta +\gamma),\\ \mathfrak{F}_{12}^{t}& = 4\det A (\beta -\gamma)(\alpha-1). \end{align*}

Thus the tensor $\mathfrak {F}^{t}$ vanishes if $\det A=0$, which corresponds to case (1), if $\beta =\gamma$, which corresponds to case (2), or if $\alpha =1$ and $\beta =-\gamma$, which corresponds to an Einstein metric.

4.7 Type IV.3: critical metrics with degenerate unimodular kernel

$G$ is non-unimodular and the restriction of the metric to the unimodular kernel $\mathfrak {u}$ is degenerate. Then there exists a pseudo-orthonormal basis $\{u_1,\, u_2,\, u_3\}$ of the Lie algebra, with $\langle u_1,\,u_1 \rangle =\langle u_2,\,u_3 \rangle =1$, such that

(4.9)\begin{equation} [u_1,u_2]=0, \quad [u_1,u_3]=\alpha u_1 +\beta u_2, \quad [u_2,u_3]=\gamma u_1 +\delta u_2, \end{equation}

with $\alpha,\, \beta,\, \gamma,\, \delta \in \mathbb {R}$, and $\alpha +\delta =2$. These metrics are Einstein only if they are flat, which occurs if $\gamma =0$ and $\alpha =0,\,1$. Moreover, the scalar curvature is $\tau =\frac {\gamma ^{2}}2$, so metrics given by (4.9) are $\mathcal {S}$-critical if and only if $\gamma =0$.

Proof. The a priori non-vanishing components of the tensor field $\mathfrak {F}^{t}$ are given, with respect to the $\{u_1,\,u_2,\,u_3\}$ basis, by

\[ \begin{array}{ll} \mathfrak{F}_{11}^{t}= \tfrac{2}{3}(3+t)\gamma^{4}, & \mathfrak{F}_{13}^{t}={-}(3+t)\alpha\gamma^{3}, \\ \mathfrak{F}_{23}^{t}={-}\tfrac{1}{3}(3+t)\gamma^{4}, & \mathfrak{F}_{33}^{t}= \gamma^{2}( 3\alpha^{2} -\det A +t(\alpha^{2} -\det A) ). \end{array} \]

$\mathfrak {F}_{11}^{t}$ vanishes if and only if $\gamma =0$ or $t=-3$. If $\gamma =0$, the tensor field vanishes identically with independence of the value of $t$. This corresponds to case (2).

Now, we assume $\gamma \neq 0$ and $t=-3$. The only non-vanishing component is $\mathfrak {F}_{33}^{t}= 2\gamma ^{2} \det A$, that vanishes if and only if $\det A=0$. This corresponds to case (1).

Remark 4.23 The Ricci operator of any metric (4.9) satisfies

\[ \operatorname{Ric}= \left( \begin{array}{@{}ccc@{}} -\dfrac{\gamma^{2}}{2} & 0 & \alpha \gamma \\ \alpha \gamma & \dfrac{\gamma^{2}}{2} & \operatorname{det}A-\alpha^{2}\\ 0 & 0 & \dfrac{\gamma^{2}}{2} \\ \end{array} \right) \]

with eigenvalues $\left \{-\frac {\gamma ^{2}}2,\,\frac {\gamma ^{2}}2,\,\frac {\gamma ^{2}}2\right \}$. Considering the subfamily given by $\alpha ^{2}=\operatorname {det}A$ one has that the minimal polynomial of the Ricci operator is $(\lambda +\frac {\gamma ^{2}}2)(\lambda -\frac {\gamma ^{2}}2)$ if $\alpha \gamma =0$, but it is $(\lambda +\frac {\gamma ^{2}}2)(\lambda -\frac {\gamma ^{2}}2)^{2}$ if $\alpha \gamma \neq 0$. Hence there are metrics within this subfamily which are not isometric to any metric obtained with the normalization $\alpha \gamma =0$ proposed in [Reference Cordero and Parker12]. Therefore, the normalization $\operatorname {ad}(e_3)(e_1)\perp \operatorname {ad}(e_3)(e_2)$ (equivalently $\alpha \gamma =0$) in (4.9) cannot be applied if one intends to consider representatives of all metrics.

5.1 Critical metrics for the L$^{2}$-norm of the curvature tensor

In dimension three a metric is critical for the curvature functional $g\mapsto \int _M\| R_g\|^{2}\operatorname {dvol}_g$ if and only if it is $\mathcal {F}_t$-critical for $t=-\frac {1}{4}$. The first Riemannian homogeneous example of a non-Einstein $\mathcal {F}_{-1/4}$-critical metric was given by Lamontagne [Reference Lamontagne17], and it was shown in [Reference Brozos-Vázquez, Caeiro-Oliveira and García-Río5] that no other possibilities may occur in the positive definite setting. The Lorentzian situation allows other non-Einstein examples which, in addition to plane waves given in theorem 4.10 and theorem 4.20-(2), are homothetic to the following:

• • The unimodular Lie group $SL(2,\,\mathbb {R})$ with metric (4.1) given by $(\lambda _1,\,\lambda _2,\,\lambda _3)=(1,\,1,\,\frac {4}{11})$ or $(\lambda _1,\,\lambda _2,\,\lambda _3)=(\frac {4}{11},\,1,\,1)$ as in theorem 4.1-(1). The Ricci operators are $\operatorname {Ric}=-\frac { 4}{121}\operatorname {diag}[9,\,9,\,2]$ and $\operatorname {Ric}=-\frac { 4}{121}\operatorname {diag}[2,\,9,\,9]$, respectively.

• • The unimodular Lie group $SL(2,\,\mathbb {R})$ with metric (4.4) homothetic to a metric given by $\alpha =1$ and $\beta ^{2} =\frac {1}{12}(7\lambda ^{2}+4\lambda +16)$ where $\lambda$ is the only real solution of $3\lambda ^{3}-20\lambda ^{2}-20\lambda -32=0$ as in theorem 4.4-(1). The Ricci operator has complex eigenvalues in this case as shown in remark 4.6.

• • Non-unimodular Lie groups with Lorentzian unimodular kernel corresponding to theorem 4.12 as follows:

  • ◦ The non-unimodular Lie group with Lie algebra determined by $\operatorname {tr}A=2$ and $\operatorname {det}A=\frac {1}{4}(1-2\sqrt {2})$ equipped with a left-invariant metric are homothetic to those given in theorem 4.12-(1) with $\beta +\gamma =1+\sqrt {2}$.

  • ◦ The non-unimodular Lie group with Lie algebra determined by $\operatorname {tr}A=2$ and $\operatorname {det}A=0$ equipped with a left-invariant metric homothetic to those given in theorem 4.12-(2) with $\beta +\gamma = \frac {4}{\sqrt {11}}$.

  • ◦ The non-unimodular Lie group with a Lie algebra determined by $\operatorname {tr}A=2$ and $\operatorname {det}A=\frac 43$ equipped with a left-invariant metric homothetic to those given in theorem 4.12-(3) with $\alpha =1\pm \sqrt {\gamma ^{2}-\frac {1}{3}}$.

  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 $\mathcal {F}_{-3/8}$ or $\mathcal {F}_{-23/64}$.

5.2 Locally conformally flat homogeneous critical metrics

Homogeneous locally conformally flat three-dimensional manifolds were classified in [Reference Honda and Tsukada16], 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 $\alpha =-\frac {1}{2}$, $\beta =\frac {\sqrt {3}}{2}$ and $\lambda =1$.

• • A type III Lie group (4.6) with $\lambda =0$.

• • A type IV.1 Lie group (4.7) with $\beta +\gamma =1$ and $\alpha =\frac {1}{2}(2+\beta - \gamma )$, $\gamma \neq \frac {1}{2}$.

• • A type IV.3 Lie group (4.9) with $\gamma =0$ and $\alpha \notin \{0,\,1,\,2\}$.

Type Ib metrics above have Ricci curvatures $\{-2,\,1\pm \sqrt {3}\sqrt {-1}\}$. Hence they are $\mathcal {S}$-critical, since the scalar curvature vanishes, but they are not critical for any quadratic curvature functional $\mathcal {F}_t$. This is in sharp contrast with the curvature homogeneous case (see remarks 2.4 and 2.5). Type IV.1 metrics above are $\mathcal {F}_{-5/12}$-critical and have a single Ricci curvature $\mu =-2$, 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.

5.3 Homogeneous $\mathcal {F}_{-1/2}$-critical metrics and semi-symmetric spaces

All symmetric spaces are critical for the functional $\mathcal {F}_{-1/2}$ (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 $\mathcal {N}_b$ (of type II) or $\mathcal {P}_c$ (of type III).

Results in § 4 show that a unimodular non-Einstein Lie group is $\mathcal {F}_{-1/2}$-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 $\beta ^{2}=2 \alpha ^{2} +\alpha \lambda +\frac {3}{4}\lambda ^{2}$ and $16 \alpha ^{3}+12 \alpha ^{2} \lambda +8 \alpha \lambda ^{2}-\lambda ^{3}=0$. 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 $\gamma =0$ (in which case it is a $pp$-wave modelled on $\mathcal {P}_c$ or $\mathcal {CW}_\varepsilon$). Moreover, results in § 4 show that a non-unimodular Lie group is $\mathcal {F}_{-1/2}$-critical if and only if it is symmetric or type IV.3 given by (4.9) with $\gamma =0$.

A three-dimensional semi-symmetric homogeneous Lorentzian manifold which is critical for a quadratic curvature functional is $\mathcal {S}$-critical or $\mathcal {F}_{-1/2}$-critical. Conversely, any homogeneous $\mathcal {F}_{-1/2}$-critical metric is semi-symmetric unless it corresponds to the type Ib Lie group above with complex Ricci curvatures.

5.4 Critical metrics and algebraic Ricci solitons

A Ricci soliton is a triple $(M,\,\langle,\,\rangle,\,X)$, where $(M,\,\langle,\,\rangle )$ is a pseudo-Riemannian manifold and $X$ is a vector field on $M$ that satisfies the differential equation

(5.1)\begin{equation} \mathcal{L}_X \langle,\rangle +\rho = \mu \langle,\rangle, \end{equation}

where $\mathcal {L}$ denotes the Lie derivative and $\mu \in \mathbb {R}$. 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, $(G,\,\langle,\,\rangle )$ is said to be an algebraic Ricci soliton if the Ricci operator satisfies $\operatorname {Ric}=\mu \operatorname {\rm Id} +D$ for some derivation $D$ of the Lie algebra (see [Reference Lauret18]). This condition for the Ricci operator implies that equation (5.1) holds (see [Reference Lauret18]).

The aim of this section is to relate Lie groups endowed with a left-invariant $\mathcal {F}_t$-critical metric and those that are algebraic Ricci solitons. Following the classification in § 3.2 we consider unimodular Lie groups and analyse when $\operatorname {Ric}-\mu \operatorname {\rm Id}$ 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 [Reference Batat and Onda1]).

The corresponding analysis for the non-unimodular case gives the following.

The following result shows the relation between algebraic Ricci solitons and critical metrics for quadratic curvature functionals.