Critical metrics for quadratic curvature functionals on some solvmanifolds

We prove the existence of four-dimensional compact manifolds admitting some non-Einstein Lorentzian metrics, which are critical points for all quadratic curvature functionals. For this purpose, we consider left-invariant semi-direct extensions GS=H⋊exp(RS)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_{\mathcal S}=H \rtimes \exp ({\mathbb {R}}S)$$\end{document} of the Heisenberg Lie group H, for any S∈sp(1,R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal S \in {\mathfrak {s}}{\mathfrak {p}}(1,\mathbb R)$$\end{document}, equipped with a family ga\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_a$$\end{document} of left-invariant metrics. After showing the existence of lattices in all these four-dimensional solvable Lie groups, we completely determine when ga\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_a$$\end{document} is a critical point for some quadratic curvature functionals. In particular, some four-dimensional solvmanifolds raising from these solvable Lie groups admit non-Einstein Lorentzian metrics, which are critical for all quadratic curvature functionals.

Let R, , τ respectively denote the Riemann curvature tensor, the Ricci tensor and the scalar curvature of a given metric g. The Euler-Lagrange equations associated to the Einstein-Hilbert functional g → M τ dvolg are given by = λg, for some real constant λ, so that critical metrics on M 1 for the Einstein-Hilbert functional coincide with the Einstein ones.
The next natural step is to consider functionals defined by scalar quadratic curvature invariants. In Riemannian settings this topic started in [1], has been intensively studied and is a very active field of research (see for example [4, 10, 12, 17-22, 27, 28, 30] and references therein). We report here some essential information on the study of quadratic curvature invariants, referring to [10] and [30] for excellent surveys on the topic. A basis for the space of quadratic curvature invariants is given by { τ, τ 2 , || || 2 , ||R|| 2 } so that, correspondingly, an arbitrary quadratic curvature functional has the form g → M a||R|| 2 + b|| || 2 + cτ 2 dvol g , for some real constants a, b, c. In this general framework, the four-dimensional case carries some special features and interest, as these functionals also arise naturally in some gravitational field theories (see for example [11]). When dim M = 4, the Gauss-Bonnet Theorem yields Consequently, in dimension four, all quadratic curvature functionals are equivalent to || || 2 + tτ 2 dvol g , t ∈ R.
As already observed in [2], Einstein metrics are critical for F t on M 1 for every t ∈ R. In general, critical metrics for quadratic curvature functionals need not be Einstein. For example, Bach-flat metrics are critical points for F −1/3 , and Weyl metrics of vanishing scalar curvature are critical points for F −1/4 (and for S). Moreover, it may be observed that although this problem has been extensively studied for Riemannian metrics, its formulation is also possible in different signatures, and it leads to the same Euler-Lagrange equations. These remarks lead naturally to the following questions: (1) Do there exist conditions under which a critical metric for F t is necessarily Einstein? (2) Do there exist non-Einstein critical metrics for all quadratic curvature functionals? (3) What happens considering metrics of different signatures?
The first two of the above questions have been studied by several authors. In particular, suitable curvature conditions forcing critical metrics to be Einstein were found in [10]. A positive answer to the second of the above questions was obtained in [4]: there exist four-dimensional Riemannian metrics which are critical for all quadratic curvature invariants but are not Einstein. The examples investigated in [4] are non-flat cones R + × r N , where N is a three-dimensional Einstein manifold of constant sectional curvature −3. We may observe that by their own construction, these examples are not compact.
On the other hand, up to our knowledge, the third question is just starting to attract the interest of researchers (see for example [5]) and has not been extensively investigated yet. The aim of this paper is to contribute to this general topic, with particular regard to the third of the above questions, proving that some compact four-dimensional manifolds admit non-Einstein Lorentzian metrics, which are critical points for all quadratic curvature functionals. These compact manifolds naturally arise as solvmanifolds, i.e., compact quotients of solvable Lie groups with respect to some lattices.
Following [26], where semi-direct extensions of the Heisenberg group (of arbitrary dimension) were introduced, let H denote the three-dimensional Heisenberg group Each matrix S belonging to the Lie algebra sp(1, R) of the symplectic group Sp(1, R) on R 2 , defines a derivation S of h, given by and so, a corresponding one-dimensional semi-direct extension g = h (RS) of h.
We denote by G S = H exp(RS) the connected, simply connected Lie group corresponding to h (RS).
It is easy to check that S ∈ R 2,2 satisfies for some real constants α, β, γ and the Lie algebra h (RS)=span{U , X , Y , S} is completely described by the following brackets: In particular, for we find the well known oscillator algebra. The corresponding (four-dimensional) oscillator group [29] admits a bi-invariant metric g 0 , which has been generalized to a one-parameter family g a , a 2 < 1, of left-invariant metrics. Setting U = e 1 , X = e 2 , Y = e 3 and S = e 4 , this family of metrics is described by Since its introduction in [16] as a generalization of previous examples studied in [24], this family of metrics has been intensively studied in different contexts (see for example [6], [8] and references therein). We remark that the above equations (1.3) define a metric for any real constant a satisfying a 2 = 1. In particular, these metrics are Lorentzian if a 2 < 1 and Riemannian when a 2 > 1.
Generalizing this example, one can consider G S = H exp(RS), equipped with an arbitrary left-invariant metric g a of the form (1.3) (see [7], [9]).
In this paper we shall completely classify left-invariant metrics g a on G S , which are critical points for some quadratic curvature functionals. In particular, we will prove that whatever the form of the defining matrix S, the Lorentzian metric g 0 is a critical point for all quadratic curvature functionals. We will also show that all Lie groups G S admit some lattices, that is, discrete co-compact subgroups, so that they give rise to solvmanifolds. As a consequence, we prove the existence of some non-Einstein Lorentzian critical metrics for all quadratic curvature invariants on some compact four-manifolds.
The paper is organized in the following way. In Sect. 2 we shall provide some basic information on Lie groups G S and their left-invariant metrics (1.3). In Sect. 3 we shall investigate the existence of lattices on all these solvable Lie groups. In Sect. 4 we shall consider the Euler-Lagrange equations for the quadratic curvature functionals and solve them for metrics of the form (1.3).

Preliminaries
Let G S = H exp(RS) denote the semi-direct extension of the three-dimensional Heisenberg group corresponding to an Hamiltonian matrix S ∈ sp(1, R). The following result holds (see [7] for more details).
Proposition 1 [7] Given S ∈ sp(1, R) described as in (1.1), the semi-direct extension G S = H exp(RS) of the Heisenberg group determined by S can be realized as the following four-dimensional subgroup of GL(4, R): and, depending on whether = − det(S) = α 2 + βγ is positive, null or negative, we have: Let ∂ j := ∂/∂ x j denote the coordinate vector field corresponding to the x j -coordinate. Then, vector fields determine a basis of left-invariant vector fields on G S , with (e j ) I = (∂ x j ) I . Explicitly, we find We observe that there is a natural distinction between the properties of the Lie algebra g, and the ones of the left-invariant metric g a we are equipping g with. At the Lie algebra level, g = h (RS) is uniquely determined up to isomorphisms, and different choices of S ∈ sp(1, R) can lead to isomorphic Lie algebras. On the other hand, the specific S ∈ sp(1, R) plays a fundamental role, via the Koszul formula, in determining the connection and curvature properties of (G S , g a ). We shall take advantage of this simple remark in the following sections.

Lattices on G S = H exp(RS)
It is easy to check that for any prescribed S ∈ sp(1, R), the corresponding fourdimensional Lie group G S = H exp(RS) is solvable. C. Bock [3] classified solvable Lie groups, up to dimension six, which admit some lattices. We shall now prove that any Lie group G S appears in Bock's classification and so, it gives rise to some solvmanifolds.
We start from the three-dimensional Heisenberg Lie algebra h =span{X , Y , U }, with [X , Y ] = U . Following [7], we remark that for any real constants a, b, c, d such that ad − bc = 0, if we consider the linearly independent vectorsX = a X + bY ,Ỹ = cX + dY and we setŨ = (ad − bc)U , we have [X ,Ỹ ] =Ũ , so that the description of the Heisenberg Lie algebra h is exactly the same with respect to both bases {X , Y , U } and {X ,Ỹ ,Ũ }.
Substitution of {X , Y } by {X ,Ỹ } corresponds to substitute a given S ∈ sp(1, R) by a similar matrixS ∈ sp(1, R), and Lie algebras h (RS) and h (RS) (whence, the corresponding simply connected Lie groups G S and GS ) are isomorphic.
As trace(S) = 0, the characteristic equation of S is completely determined by = − det(S). It is then natural to consider separately three cases, depending on the sign of . By standard linear algebra arguments one then concludes that for an arbitrary S ∈ sp(1, R), the corresponding one-dimensional extension h (RS) is isomorphic to the Lie algebra g = span{e 1 =Ũ , e 2 =X , e 3 =Ỹ , e 4 =S}, completely described by [e 2 , e 3 ] = e 1 and

Remark 1
The Lie algebra described in Case (C) corresponds to the oscillator group. Case (A) is known is literature as Boidol's group or split oscillator group. Case (B) is nilpotent.
Then, with respect to {X 1 , X 2 , X 3 , X 4 }, the Lie algebra g is completely determined by brackets which is exactly case g −1 4.8 in Table A.1 of [3]. With regard to case (B), if μ = 0 then we already have case g 3.1 ⊕ g 1 in Table A Then, with respect to {X 1 , X 2 , X 3 , X 4 }, the Lie algebra g is described by which is case g 4.1 in Table A.1 of [3]. Finally, in case (C) we set and the Lie algebra g takes the form which is case g 0 4.9 in Table A.1 of [3]. Thus, we have the following.
Proposition 2 For any S ∈ sp(1, R), the corresponding one-dimensional extension g = h (RS) of the Heisenberg Lie algebra h admits some lattices and so, it gives rise to some solvmanifolds.
We now report some explicit examples of lattices and corresponding solvmanifolds for the connected, simply connected Lie groups corresponding to the Lie algebras described above with respect to bases of the form {X 1 , .., X 4 }.
Case (A). Following [3], elements 2 ). Then, the connected, simply connected Lie group corresponding to case (A) can be described as H ρ A R. Since ρ A (1) preserves , one obtains a lattice ρ A Z and the corresponding solvmanifold H ρ A R/ ρ A Z for case (A). Case (B)-1: μ = 0 As shown in [3], the corresponding connected, simply connected Lie group can be described as Case (B)-2: μ > 0 In this case, again by [3], the connected, simply connected Lie group can be described as Case (C). Following [13] and referring to the notations we used in Sect. 2, for the oscillator group defines a lattice for any r ∈ N, determining a corresponding solvmanifold H exp(RS)/ r for case (C).

Remark 2
The classification of lattices of a given solvable Lie group is a very interesting topic, which has been investigated by several authors. With regard to the Lie groups we considered above: complete classifications of lattices of the oscillator Lie group are known (see [13], [25]) and have been recently applied to harmonic analysis of solvmanifolds [14]; lattices for Lie group corresponding to case (A) have been recently classified [15].

Critical metrics on G S for quadratic curvature functionals
The Euler-Lagrange equations of quadratic curvature functional are well known. They have been calculated in the Riemannian case ( [2], [17]). As the argument does not depend on the signature of the involved metrics, they extend to pseudo-Riemannian settings.
The gradients of functionals S(g) = M τ 2 dvol g and F t (g) = M || || 2 + tτ 2 dvol g are respectively given by and g is critical for F t if and only if (∇F t ) = cg for some real constant c. Since the trace of this equation yields with λ = F t (g) (see [10]). It follows that Einstein metrics are critical for F t for all values of t. Moreover, by the above expression of ∇S, metrics which are either Einstein or of vanishing scalar curvature are critical points for S.
It is worthwhile to observe that the above Euler-Lagrange equations simplify remarkably in the case of a four-dimensional metric of constant scalar curvature, which is the case for any four-dimensional homogeneous metric (in particular, for left-invariant metrics on four-dimensional Lie groups). In fact, in such a case, (∇S) i j = −2τ i j − 1 4 τ g i j , equation (4.2) is automatically satisfied, while equation (4.1) reduces to where R denotes the tensor defined by components kl R ik jl . We shall now completely classify solutions to equation (4.3) among left-invariant metrics g a , a 2 = 1 on The description of the Levi-civita connection and curvature of (G S = H exp(RS), g a ) was obtained in [7]. Since the metrics are left-invariant, it suffices to work at the Lie algebra level. With respect to the basis {e 1 = U , e 2 = X , e 3 = Y , e 4 = S} of left-invariant vector fields, the Levi-Civita connection ∇ is completely determined by the following possibly non-vanishing covariant derivatives:  The Riemann-Christoffel curvature tensor R of (G S , g a ) is then completely determined by components R i jkh = g a (R(e i , e j )e k , e h ), where R(e i , e j )e k = ∇ e i ∇ e j e k − ∇ e j ∇ e i e k − ∇ [ei ,e j ] e k for all indices i, j, k. Explicitly, we find The Ricci tensor (X , Y ) = tr(Z → R(Z , X )Y ) of g a is then described, with respect to the basis {e i }, by the symmetric matrix In particular, it is easily seen that (G S = H exp(RS), g a ) is Einstein if and only if a = 0 and its Lie algebra, as described in (1.2), satisfies 4α 2 = 1 − (β + γ ) 2 (in this case, g a is Ricci-flat [7]). Again from (4.6), it is easy to check that the scalar curvature of g a is given by Finally, the above descriptions of R, and τ yield that Weyl conformal curvature tensor W is completely determined by the following possibly non-vanishing components W i jkh with respect to {e i }: In particular, as the vanishing of the Weyl tensor characterizes the (locally) conformally flat metrics in dimension four, from the above components we easily conclude that (G S = H exp(RS), g a ) is locally conformally flat if and only if a = 0 and its Lie algebra, as described in (1.2), satisfies either β = γ − 1 or α = β + γ = 0 (which is the well-known case of the bi-invariant metric of the oscillator group).
We are now ready to determine when metrics g a are critical for some curvature invariants.
First, g a is a critical point for functional S if and only if either g a is Einstein, or its scalar curvature vanishes. Since the scalar curvature also vanishes when g a is Einstein (indeed, Ricci-flat), by (4.7) we directly obtain the following characterization. We now turn our attention to equation (4.3), which, by our previous analysis, completely characterizes g a being a critical point for functional F t .
If β = 0 then the components of F t reduce to: Therefore, g a is critical if and only if a 2 − 1 − γ 2 = 0. Observe that for β = a 2 − 1 − γ 2 = 0, we get t = − 3 2 . We may also remark that a 2 − 1 − γ 2 = 0 yields a = ± 1 + γ 2 , for any real value of γ . On the other hand, it also yields γ = ± √ a 2 − 1, which impies that a 2 > 1, that is, g a is Riemannian. In the case where γ = 0, the argument and calculations are very similar to the above ones for case β = 0 and we shall omit them. In this case we conclude that g a is critical for F t if and only if t = − 3 2 and a 2 − 1 − β 2 = 0. Remark 3 We observe that equation (4.16) is also satisfied, for all values of t, if 1 − (β + γ ) 2 − a 2 = −2βγ − 3γ 2 − 3β 2 = 0. However, starting from (4.11), a standard calculation yields that in this case necessarily a = 0, which we already considered in the above Case (i). So, this concludes subcase (iii)-B.
We substitute this expression of α into (4.11) and we find that F ti j = 0 for all indices i, j if ad only if whence, a = ± (β − γ ) 2 + 1, for all values of β = γ . Substituting the expressions of a and α in the above expression of t, we conclude that in this case t = − 3 2 . We remark that this case includes subcase (iii)-B, which is the special case obtained when βγ = 0.
The above calculations and results are summarized in the following main result.
Theorem 2 For any S ∈ sp(1, R) and real constant a with a 2 = 1, consider the semidirect extension G S = H exp(RS) of the Heisenberg group, equipped with the left-invariant metric g a described by (1.3). Then, g a is a critical point for functional F t (g) = M || || 2 + tτ 2 dvol g if and only if one of the following cases occurs: (1) a = 0. In this case, g a is a critical point for F t for all t ∈ R.
We may observe that case (3) in the above Theorem 2 corresponds to the oscillator group. So, all left-invariant metrics g a (both Riemannian and Lorentzian) on the oscillator group are critical for the quadratic curvature functional F −3 .

Remark 4
Metric g 0 carries several special properties. In fact, for any S ∈ sp(1, R), among left-invariant metrics g a , a 2 = 1 on the corresponding semi-direct extension G S = H exp(RS) of the Heisenberg group, the metric g 0 is the only one: • Admitting a null parallel vector field (namely, e 1 ) and so, being a Walker metric [7]. In particular, g 0 is also a pp-wave, as it follows from [23] because g 0 admits the null parallel vector field e 1 and, by (4.5), is transversally flat (R(x, y) = 0 whenever x, y are orthogonal to e 1 );  • Being Ricci-parallel [7]; • Satisfying the Ricci soliton equation L X g + = λg for all real values of λ, and defining a Yamabe soliton [6], [7]; • Conformally Einstein [9].
Investigation of critical metrics for quadratic curvature functionals emphasizes once more the special role of g 0 . In fact, Proposition 2 and Theorems 1 and 2 yield the following consequence. The examples we studied suggest the possible existence of some interplays between properties listed above and critical metrics for quadratic curvature functionals. It would be very interesting to investigate further these possible interplays.