1 Introduction

We recall that a seven-dimensional smooth manifold \(M\) admits a \(G_2\)-structure if the structure group of the frame bundle reduces to the exceptional Lie group \(G_2\). The existence of a \(G_2\)-structure is equivalent to the existence of a non-degenerate 3-form \(\varphi \) defined on the whole manifold (see for example [26]) and using this 3-form it is possible to define a Riemannian metric \(g_\varphi \) on \(M\).

If \(\varphi \) is parallel with respect to the Levi–Civita connection, i.e., \(\nabla ^{LC} \varphi =0\), then the holonomy group is contained in \(G_2\), the \(G_2\)-structure is called parallel and the corresponding manifolds are called \(G_2\)-manifolds. In this case, the induced metric \(g_\varphi \) is Ricci-flat. The first examples of complete metrics with holonomy \(G_2\) were constructed by Bryant and Salamon [6]. Compact examples of manifolds with holonomy \(G_2\) were obtained first by Joyce [24–26] and then by Kovalev [28] and by Corti, Haskins, Nordström, Pacini [12]. Incomplete Ricci-flat metrics of holonomy \(G_2\) with a 2-step nilpotent isometry group \(N\) acting on orbits of codimension 1 were obtained in [9, 20]. It turns out that these metrics are locally isometric (modulo a conformal change) to homogeneous metrics on solvable Lie groups, which are obtained as rank one extensions of a six-dimensional nilpotent Lie group endowed with an invariant \(\mathrm{SU}(3)\)-structure of a special kind, known in the literature as half-flat [10].

Examples of compact and non-compact manifolds endowed with non-parallel \(G_2\)-structures were given for instance in [1, 7–9, 14, 15, 17, 19, 27, 38]. In particular, in [9] conformally parallel \(G_2\)-structures on solvmanifolds, i.e., on simply connected solvable Lie groups, were studied. More in general, in [23] it was shown that a seven-dimensional compact Riemannian manifold \(M\) admits a locally conformal parallel \(G_2\)-structure if and only if it has as covering a Riemannian cone over a compact nearly Kähler 6-manifold such that the covering transformations are homotheties preserving the corresponding parallel \(G_2\)-structure.

By [5, 11, 18], it is evident that the Riemannian scalar curvature of a \(G_2\)-structure may be expressed in terms of the \(3\)-form \(\varphi \) and its derivatives. More precisely, in [5] an expression of the Ricci curvature and the scalar curvature in terms of the four intrinsic torsion forms \(\tau _i, i =0, \ldots , 3\), and their exterior derivatives was given. Moreover, using this it is possible to show that the scalar curvature has a definite sign for certain classes of \(G_2\)-structures.

If \(d\varphi = 0\), the \(G_2\)-structure is called calibrated or closed. The geometry of this family of \(G_2\)-structures was studied in [11]. Furthermore, Bryant proved in [5] that if the scalar curvature of a closed \(G_2\)-structure is non-negative then the \(G_2\)-structure is parallel.

We say that a \(G_2\)-structure \(\varphi \) is Einstein if the underlying Riemannian metric \(g_{\varphi }\) is Einstein. In [5, 11] it was proved, as an analogous of Goldberg conjecture for almost-Kähler manifolds, that on a compact manifold an Einstein (or, more in general, with divergence-free Weyl tensor [11]) calibrated \(G_2\)-structure has holonomy contained in \(G_2\). In the non-compact case, Cleyton and Ivanov showed that the same result is true with the additional assumption that the \(G_2\)-structure is \(*\)-Einstein, but it still an open problem to see if there exist (even incomplete) Einstein metrics underlying calibrated \(G_2\)-structures. Recently, some negative results were proved in the case of non-compact homogeneous spaces in [16]. In particular, the authors showed that a seven-dimensional solvmanifold cannot admit any left-invariant calibrated \(G_2\)-structure inducing an Einstein metric \(g_\varphi \) unless \(g_\varphi \) is flat.

In the present paper, we are mainly interested in the geometry of locally conformal calibrated \(G_2\)-structures, i.e., \(G_2\)-structures whose associated metric is conformally equivalent (at least locally) to the metric induced by a calibrated \(G_2\)-structure.

In Sect. 3, we prove that a compact manifold endowed with an Einstein locally conformal calibrated \(G_2\)-structure has non-positive scalar curvature (and then has either zero or negative curvature if it is also connected) and we show that a compact homogeneous 7-manifold cannot admit an invariant Einstein locally conformal calibrated \(G_2\)-structure unless the underlying metric is flat.

In the last section, we give a non-compact example of a homogeneous manifold endowed with an Einstein locally conformal calibrated \(G_2\)-structure. The homogeneous manifold is a solvmanifold, thus this example and the aforementioned result of [16] highlight a different behaviour of calibrated and locally conformal calibrated \(G_2\)-structures. Moreover, the homogeneous Einstein metric is a rank-one extension of a Ricci soliton on the complex Heisenberg group induced by a coupled \(\mathrm{SU}(3)\)-structure \((\omega , \Psi )\) such that \(d \omega = - \mathrm{Re} (\Psi )\). Recall that a half-flat \(\mathrm{SU}(3)\)-structure is said to be coupled if \(d\omega \) is proportional to \(\mathrm{Re} (\Psi )\) at each point (see [37]). Finally, we classify nilpotent Lie groups admitting a left-invariant coupled \(\mathrm{SU}(3)\)-structure, showing that the complex Heisenberg group is, up to isomorphisms, the only nilpotent Lie group admitting a coupled \(\mathrm{SU}(3)\)-structure \((\omega , \Psi )\) whose associated metric is a Ricci soliton.

2 Preliminaries on \(G_2\) and \(\mathrm{SU}(3)\)-structures

Let \(\left( e_1, \ldots , e_7\right) \) be the standard basis of \({\mathbb {R}}^7\) and \(\left( e^1, \ldots , e^7\right) \) be the corresponding dual basis. We set

$$\begin{aligned} \varphi = e^{123} + e^{145} + e^{167} + e^{246} - e^{257} - e^{347} - e^{356}, \end{aligned}$$

where for simplicity \(e^{ijk}\) stands for the wedge product \(e^i \wedge e^j \wedge e^k\) in \(\Lambda ^3 ({({\mathbb {R}}^7)}^*)\). The subgroup of \(\mathrm{GL}(7, {\mathbb {R}})\) fixing \(\varphi \) is \(G_2\). The basis \((e^1,\ldots , e^7)\) is an oriented orthonormal basis for the underlying metric and the orientation is determined by the inclusion \(G_2\subset \mathrm{SO}(7)\). The group \(G_2\) also fixes the 4-form

$$\begin{aligned} * \varphi = e^{4567} + e^{2367} + e^{2345} + e^{1357} - e^{1346} - e^{1256} - e^{1247}, \end{aligned}$$

where \(*\) denotes the Hodge star operator determined by the associated metric and orientation.

We recall that a \(G_2\)-structure on a \(7\)-manifold \(M\) is characterized by a positive 3-form \(\varphi \). Indeed, it turns out that there is a \(1-1\) correspondence between \(G_2\)-structures on a 7-manifold and 3-forms for which the bilinear form \(B_{\varphi }\) defined by

$$\begin{aligned} B_{\varphi } (X, Y) = \frac{1}{6} \, i_{X} \varphi \wedge i_Y\varphi \wedge \varphi \end{aligned}$$

is positive definite, where \(i_{X}\) denotes the contraction by \(X\). A 3-form \(\varphi \) for which \(B_{\varphi }\) is positive definite defines a unique Riemannian metric \(g_{\varphi }\) and volume form \(dV_\varphi \) such that for any couple of vectors \(X\) and \(Y\) on \(M\) the following relation holds

$$\begin{aligned} g_{\varphi } (X, Y) dV_\varphi = \frac{1}{6} \, i_{X} \varphi \wedge i_Y\varphi \wedge \varphi . \end{aligned}$$

As in [11], we let

$$\begin{aligned} \varphi = \frac{1}{6} \varphi _{ijk} e^{ijk} \end{aligned}$$

and define the \(*\)-Ricci tensor of the \(G_2\)-structure as

$$\begin{aligned} \rho ^*_{sm} {:}= R_{ijkl} \varphi _{ijs} \varphi _{klm}. \end{aligned}$$

A \(G_2\)-structure is said to be \(*\)-Einstein if the traceless part of the \(*\)-Ricci tensor vanishes, i.e., if \(\rho ^* = \frac{s^*}{7} g\), where \(s^*\) is the trace of \(\rho ^*\).

On a \(7\)-manifold endowed with a \(G_2\)-structure, the action of \(G_2\) on the tangent spaces induces an action of \(G_2\) on the exterior algebra \({\Lambda }^p(M)\), for any \(p \ge 2\). In [4], it was shown that there are irreducible \(G_2\)-module decompositions

$$\begin{aligned} \Lambda ^2 ({({\mathbb {R}}^7)}^*)= & {} \Lambda ^2_{7} ({({\mathbb {R}}^7)}^*) \oplus \Lambda ^2_{14} ({({\mathbb {R}}^7)}^*),\\ \Lambda ^3 ({({\mathbb {R}}^7)}^*)= & {} \Lambda ^3_{1} ({({\mathbb {R}}^7)}^*)\oplus \Lambda ^3_7 ({({\mathbb {R}}^7)}^*) \oplus \Lambda ^3_{27} ({({\mathbb {R}}^7)}^*), \end{aligned}$$

where \(\Lambda ^p_k ({({\mathbb {R}}^7)}^*)\) denotes an irreducible \(G_2\)-module of dimension \(k\). Using the previous decomposition of \(p\)-forms, in [5] a simple expression of \(d\varphi \) and \(d*\varphi \) was obtained, where \(*\) denotes the Hodge operator defined by the metric \(g_\varphi \) and the volume form \(dV_\varphi \). More precisely, for any \(G_2\)-structure \(\varphi \) there exist unique differential forms \(\tau _0 \in \Lambda ^0 (M),\) \(\tau _1 \in \Lambda ^1 (M),\) \(\tau _2 \in \Lambda ^2_{14} (M),\) \(\tau _3 \in \Lambda ^3_{27} (M),\) such that

$$\begin{aligned} d \varphi= & {} \tau _0 * \varphi + 3 \tau _1 \wedge \varphi + * \tau _3,\\ d * \varphi= & {} 4 \tau _1 \wedge * \varphi + \tau _2 \wedge \varphi , \end{aligned}$$

where \(\Lambda ^p_k (M)\) denotes the space of sections of the bundle \(\Lambda ^p_k (T^* M)\).

In the case of a closed \(G_2\) structure we have

$$\begin{aligned} d \varphi= & {} 0,\\ d*\varphi= & {} \tau _2 \wedge \varphi . \end{aligned}$$

By the results of [5], the scalar curvature is given by

$$\begin{aligned} \mathrm{Scal} (g_{\varphi }) = - \frac{1}{2} | \tau _2|^2 \end{aligned}$$

and from this it is clear that it cannot be positive.

For a locally conformal calibrated \(G_2\)-structure \(\varphi \) one has \(\tau _0\equiv 0\) and \(\tau _3\equiv 0\), so

$$\begin{aligned} d \varphi= & {} 3 \tau _1 \wedge \varphi ,\\ d*\varphi= & {} 4 \tau _1 \wedge * \varphi + \tau _2 \wedge \varphi , \end{aligned}$$

and taking the exterior derivative of the former it is easy to show that \(\tau _1\) is a closed 1-form. Moreover, in this case the scalar curvature has not a definite sign as one can check from its expression

$$\begin{aligned} \mathrm{Scal} (g_{\varphi }) = 12 \delta \tau _1 + 30 | \tau _1 |^2 - \frac{1}{2} | \tau _2 |^2, \end{aligned}$$

where \(\delta \) denotes the adjoint of the exterior derivative \(d\) with respect to the metric \(g_\varphi \).

If the only nonzero intrinsic torsion form is \(\tau _1\), we have the so called locally conformal parallel \(G_2\)-structures. They are named in this way since a conformal change of the metric \(g_\varphi \) associated to a \(G_2\)-structure of this kind gives (at least locally) the metric induced by a parallel \(G_2\)-structure. In this case

$$\begin{aligned} d \varphi= & {} 3 \tau _1 \wedge \varphi ,\\ d*\varphi= & {} 4 \tau _1 \wedge * \varphi . \end{aligned}$$

We will give an example of such a structure at the end of Sect. 4.

We recall that a six-dimensional smooth manifold admits an \(\mathrm{SU}(3)\)-structure if the structure group of the frame bundle can be reduced to \(\mathrm{SU}(3)\). It is possible to show that the existence of an \(\mathrm{SU}(3)\)-structure is equivalent to the existence of an almost Hermitian structure \((h, J, \omega )\) and a unit \((3,0)\)-form \(\Psi \).

Since \(\mathrm{SU}(3)\) is the stabilizer of the transitive action of \(G_2\) on the 6-sphere \(S^6,\) it follows that a \(G_2\)-structure on a 7-manifold induces an \(\mathrm{SU}(3)\)-structure on any oriented hypersurface. If the \(G_2\)-structure is parallel, then the \(\mathrm{SU}(3)\)-structure is half-flat [10]. In terms of the forms \((\omega , \Psi )\) this means \(d (\omega \wedge \omega ) =0, d (\mathrm{Re}(\Psi ))=0\).

In our computations we will use another characterization of \(\mathrm{SU}(3)\)-structures which follows from the results of [22, 36]. We describe it here. Consider a six-dimensional oriented real vector space \(V\), a \(k\)-form on \(V\) is said to be stable if its GL(V)-orbit is open. Let \(A:\Lambda ^5(V^*) \rightarrow V\otimes \Lambda ^6(V^*)\) denote the canonical isomorphism given by \(A(\gamma ) = w \otimes \Omega \), where \(i_w\Omega = \gamma \), and define for a fixed 3-form \(\sigma \in \Lambda ^3(V^*)\)

$$\begin{aligned} K_\sigma : V \rightarrow V\otimes \Lambda ^6(V^*),\ \ K_\sigma (w) = A((i_w \sigma )\wedge \sigma ) \end{aligned}$$

and

$$\begin{aligned} \lambda : \Lambda ^3(V^*) \rightarrow (\Lambda ^6(V^*))^{\otimes 2},\ \ \lambda (\sigma ) = \frac{1}{6}\mathrm{tr}K^2_\sigma . \end{aligned}$$

A 3-form \(\sigma \) is stable if and only if \(\lambda (\sigma )\ne 0\) and whenever this happens it is possible to define a volume form by \(\sqrt{|\lambda (\sigma )|} \in \Lambda ^6(V^*)\), where the positively oriented root is chosen, and an endomorphism

$$\begin{aligned} J_\sigma = \frac{1}{\sqrt{|\lambda (\sigma )|}}K_\sigma , \end{aligned}$$

which is a complex structure when \(\lambda (\sigma )<0\).

A pair of stable forms \((\omega ,\sigma )\in \Lambda ^2(V^*)\times \Lambda ^3(V^*)\) is called compatible if \(\omega \wedge \sigma =0\) and normalized if \(J^*_\sigma \sigma \wedge \sigma = \frac{2}{3}\omega ^3\) (the latter identity is non-zero since a 2-form \(\omega \) is stable if and only if \(\omega ^3\ne 0\)). Such a pair defines a (pseudo) Euclidean metric \(h(\cdot ,\cdot ) = \omega (J_\sigma \cdot ,\cdot )\). As a consequence, on a six-dimensional smooth manifold \(N\) there is a one to one correspondence between \(\mathrm{SU}(3)\)-structures and pairs \((\omega ,\sigma )\in \Lambda ^2(N)\times \Lambda ^3(N)\) such that for each point \(p\in N\) the pair of forms defined on \(T_pN\) \((\omega _p,\sigma _p)\) is stable, compatible, normalized, has \(\lambda (\sigma _p)<0\) and induces a Riemannian metric \(h_p(\cdot ,\cdot ) = \omega _p(J_{\sigma _p}\cdot ,\cdot )\). In this case we have \(\Psi = \sigma + \mathrm{i}J^*_\sigma \sigma \) and, then, \(\sigma = \mathrm{Re}(\Psi )\). We refer to \(h\) as the associated Riemannian metric to the \(\mathrm{SU}(3)\)-structure \((\omega ,\sigma )\).

An \(\mathrm{SU}(3)\)-structure \((\omega ,\sigma )\) on a \(6\)-manifold \(N\) is called coupled if \(d \omega = c \sigma \), with \(c\) a non-zero real number. Note that in particular a coupled \(\mathrm{SU}(3)\)-structure is half-flat since \(d (\omega ^2) =0\) and \(d \sigma =0\) and its intrinsic torsion belongs to the space \({{\mathcal {W}}_1}^- \oplus {{\mathcal {W}}_2}^- \), where \({{\mathcal {W}}_1}^- \cong {\mathbb {R}}\) and \({{\mathcal {W}}_2}^- \cong \mathfrak {su} (3)\) (see [10]).

It is interesting to notice that the product manifold \(N \times \mathbb {R}\), where \(N\) is a \(6\)-manifold endowed with a coupled \(\mathrm{SU}(3)\)-structure \((\omega , \sigma )\), has a natural locally conformal calibrated \(G_2\)-structure defined by

$$\begin{aligned} \varphi = \omega \wedge dt + \sigma . \end{aligned}$$

Indeed,

$$\begin{aligned} d \varphi = c \sigma \wedge dt = c \varphi \wedge dt, \end{aligned}$$

since in local coordinates the components of \(\sigma \) are functions defined on \(N\) and thus they do not depend on \(t\). Then, \(\tau _0\equiv 0, \tau _3\equiv 0\) and \(\tau _1= \left( -\frac{1}{3}c\right) dt\).

3 Einstein locally conformal calibrated \(G_2\)-structures on compact manifolds

We will show now that a seven-dimensional, compact, smooth manifold \(M\) endowed with an Einstein locally conformal calibrated \(G_2\)-structure \(\varphi \) has \(\mathrm{Scal} (g_{\varphi }) \le 0\). It is worth observing here that, up to now, there are no known examples of smooth manifolds endowed with a locally conformal calibrated \(G_2\)-structure whose associated metric is Ricci-flat (and then has zero scalar curvature).

First of all recall that given a Riemannian manifold \((M,g)\) of dimension \(n \ge 3\) it is possible to define the so called conformal Yamabe constant \(Q(M,g)\) in the following way: set \(a_n {:}= \frac{4(n-1)}{n-2}, p_n {:}= \frac{2n}{n-2}\) and let \(C^\infty _c(M)\) denote the set of compactly supported smooth real valued functions on \(M\). Then

$$\begin{aligned} Q(M,g){:}= \underset{u\in C^\infty _c(M), u \not \equiv 0}{\mathrm{inf}} \left\{ \frac{\int _M(a_n|du|_g^2 + u^2 {\mathrm{Scal}}(g))dV_g}{(\int _M|u|^{p_n}dV_g)^{\frac{2}{p_n}}} \right\} \!. \end{aligned}$$

The sign of \(Q(M,g)\) is a conformal invariant, in particular the following characterization holds:

Proposition 3.1

If \((M,g)\) is a compact Riemannian manifold of dimension \(n \ge 3\), then \(Q(M,g)\) is negative/zero/positive if and only if \(g\) is conformal to a Riemannian metric of negative/zero/positive scalar curvature.

Using the conformal Yamabe constant it is possible to prove the following

Theorem 3.2

Let \(M\) be a seven-dimensional, compact, smooth manifold endowed with an Einstein locally conformal calibrated \(G_2\)-structure \(\varphi \). Then \(\mathrm{Scal} (g_{\varphi })\le 0\). Moreover, if \(M\) is connected, \(\mathrm{Scal}(g_\varphi )\) is either zero or negative.

Proof

Suppose that \(\mathrm{Scal}(g_\varphi ) >0\), then the 1-form \(\tau _1\) is exact. Indeed, since \(d\tau _1=0\), we can consider the de Rham class \([\tau _1] \in H^1_{\mathrm{dR}} (M)\) and take the harmonic 1-form \(\xi \) representing \([\tau _1]\), that is, \(\tau _1= \xi + df\), where \(\Delta \xi = 0\) and \(f\in C^\infty (M)\). \(\xi \) has to vanish everywhere on \(M\) since it is compact, oriented and has positive Ricci curvature. Then \(\tau _1=df\). Let us consider \(\tilde{\varphi }{:}= e^{-3f}\varphi \), it is clear that \(\tilde{\varphi }\) is a \(G_2\)-structure defined on \(M\). Moreover

$$\begin{aligned} d\tilde{\varphi }= & {} d(e^{-3f}\varphi ) \\= & {} -3e^{-3f}df\wedge \varphi + e^{-3f}d\varphi \\= & {} -3e^{-3f}\tau _1\wedge \varphi + e^{-3f}(3\tau _1\wedge \varphi ) \\= & {} 0, \end{aligned}$$

so \(\tilde{\varphi }\) is a closed \(G_2\)-structure and \(\mathrm{Scal}(g_{\tilde{\varphi }}) \le 0\) by [5]. We have \(g_{\tilde{\varphi }} = e^{-2f}g_\varphi \), that is, \(g_{\tilde{\varphi }}\) is conformal to the Riemannian metric \(g_\varphi \) of positive scalar curvature, then the conformal Yamabe constant \(Q(M, g_{\tilde{\varphi }})\) is positive by the previous characterization.

Since \(M\) is compact, it has finite volume and is complete as a consequence of the well known Hopf–Rinow Theorem. Then, by [34], Corollary 2.2] we have that \(Q(M, g_{\tilde{\varphi }}) \le 0\), which is in contrast with the previous result. \(\square \)

As a consequence of the previous proposition we have the

Corollary 3.3

A seven-dimensional, compact, homogeneous, smooth manifold \(M\) cannot admit an invariant locally conformal calibrated Einstein \(G_2\)-structure \(\varphi \), unless the underlying metric \(g_{\varphi }\) is flat.

Proof

Recall that a homogeneous Einstein manifold with negative scalar curvature is not compact [3]. Thus, every seven-dimensional, compact, homogeneous, smooth manifold \(M\) with an invariant \(G_2\)-structure \(\varphi \) whose associated metric is Einstein has \(\mathrm{Scal}(g_{\varphi }) \ge 0\). Combining this result with the previous proposition we have \(\mathrm{Scal} (g_{\varphi }) = 0\) and, in particular, \(g_\varphi \) is Ricci-flat. The statement then follows recalling that in the homogeneous case Ricci flatness implies flatness [2]. \(\square \)

4 Noncompact homogeneous examples and coupled \(\mathrm{SU}(3)\)-structures

In this section, after recalling some facts about noncompact homogeneous Einstein manifolds, we first study the classification of coupled \(\mathrm{SU}(3)\)-structures on nilmanifolds and then we construct an example of a locally conformal calibrated \(G_2\)-structure \(\varphi \) inducing an Einstein (non Ricci-flat) metric on a noncompact homogeneous manifold.

All the known examples of noncompact homogeneous Einstein manifolds are solvmanifolds, i.e., simply connected solvable Lie groups \(S\) endowed with a left-invariant metric (see for instance the recent survey [32]). D. Alekseevskii conjectured that these might exhaust the class of non-compact homogeneous Einstein manifolds (see [3], 7.57]).

Lauret in [33] showed that every Einstein solvmanifold is standard, i.e., it is a solvable Lie group \(S\) endowed with a left-invariant metric such that the orthogonal complement \({\mathfrak {a}}=[{\mathfrak {s}},{\mathfrak {s}}]^{{\perp }}\), where \({\mathfrak {s}}\) is the Lie algebra of \(S\), is abelian. We recall that given a metric nilpotent Lie algebra \({\mathfrak {n}}\) with an inner product \(\langle \cdot , \cdot \rangle _\mathfrak {n}\), a metric solvable Lie algebra \(({\mathfrak {s}} = {\mathfrak {n}} \oplus {\mathfrak {a}}, \langle \cdot , \cdot \rangle _\mathfrak {s})\) is called a metric solvable extension of \(({\mathfrak {n}}, \langle \cdot , \cdot \rangle _\mathfrak {n})\) if \([\mathfrak {s},\mathfrak {s}] = \mathfrak {n}\) and the restrictions to \({\mathfrak {n}}\) of the Lie bracket of \({\mathfrak {s}}\) and of the inner product \(\langle \cdot , \cdot \rangle _\mathfrak {s}\) coincide with the Lie bracket of \({\mathfrak {n}}\) and with \(\langle \cdot , \cdot \rangle _\mathfrak {n}\), respectively. The dimension of \({\mathfrak {a}}\) is called the algebraic rank of \({\mathfrak {s}}\).

In [21], 4.18], it was proved that the study of standard Einstein metric solvable Lie algebras reduces to the rank-one metric solvable extension of a nilpotent Lie algebra (i.e., those for which \(\dim ({\mathfrak {a}}) = 1\)). Indeed, by [21] the metric Lie algebra of any \((n + 1)\)-dimensional rank-one solvmanifold can be modelled on \(({\mathfrak {s}} = \mathfrak {n}\oplus \mathbb {R}H, \langle \cdot , \cdot \rangle _\mathfrak {s})\) for some nilpotent Lie algebra \(\mathfrak {n}\), with the inner product \(\langle \cdot , \cdot \rangle _\mathfrak {s}\) such that \( \langle H, {\mathfrak {n}} \rangle _\mathfrak {s} = 0\), \(\langle H,H\rangle _\mathfrak {s} = 1\) and the Lie bracket on \(\mathfrak {s}\) given by

$$\begin{aligned}{}[H, X]_\mathfrak {s} = D X, \quad [X, Y]_\mathfrak {s}= [X, Y]_{\mathfrak {n}}, \end{aligned}$$

where \([\cdot , \cdot ]_{\mathfrak {n}}\) denotes the Lie bracket on \({\mathfrak {n}}\) and \(D\) is some derivation of \(\mathfrak {n}\). By [30], a left-invariant metric \(h\) on a nilpotent Lie group \(N\) is a Ricci soliton if and only if the Ricci operator satisfies \(\mathrm{Ric}(h) = \mu I + D\), for some \(\mu \in {\mathbb {R}}\) and some derivation \(D\) of \(\mathfrak {n}\), when \(h\) is identified with an inner product on \(\mathfrak {n}\) or, equivalently, if and only if \((N, h)\) admits a metric standard extension whose corresponding standard solvmanifold is Einstein. The inner product \(h\) is also called nilsoliton.

Using the results of [29, 31], in [39] all the seven-dimensional rank-one Einstein solvmanifolds were determined, proving that each one of the 34 nilpotent Lie algebras \(\mathfrak {n}\) of dimension 6 admits a rank-one solvable extension which can be endowed with an Einstein inner product.

Six-dimensional nilpotent Lie algebras admitting a half-flat \(\mathrm{SU}(3)\)-structure were classified in [13]. For coupled \(\mathrm{SU}(3)\)-structures we can show the following

Theorem 4.1

Let \(\mathfrak {n}\) be a six-dimensional, non-abelian, nilpotent Lie algebra admitting a coupled \(\mathrm{SU}(3)\)-structure. Then \(\mathfrak {n}\) is isomorphic to one of the following

$$\begin{aligned} \mathfrak {n}_9 = (0,0,0,e^{12},e^{14}-e^{23},e^{15}+e^{34}), \quad \mathfrak {n}_{28}=(0,0,0,0,e^{13}-e^{24},e^{14}+e^{23}), \end{aligned}$$

where for instance \( \mathfrak {n}_9 = (0,0,0,e^{12},e^{14}-e^{23},e^{15}+e^{34})\) means that there exists a basis \((e^1, \ldots , e^6)\) of \(\mathfrak {n}_9^*\) such that

$$\begin{aligned} d e^j =0, j = 1,2,3, \quad d e^4 = e^{12}, \quad d e^5 = e^{14}-e^{23}, \quad d e^6 = e^{15}+e^{34}. \end{aligned}$$

Moreover, the only nilpotent Lie algebra admitting a coupled \(\mathrm{SU}(3)\)-structure inducing a nilsoliton is \(\mathfrak {n}_{28}\).

Proof

By the results in [13], the generic nilpotent Lie algebra \({\mathfrak {n}}\) admitting a half-flat \(\mathrm{SU}(3)\)-structure is isomorphic to one of the 24 Lie algebras described in Table 1. Consider on \({\mathfrak {n}}\) a generic 2-form

$$\begin{aligned} \omega= & {} b_1e^{12} + b_2e^{13}+b_3e^{14}+b_4e^{15} + b_5e^{16} + b_6e^{23} + b_7 e^{24} +b_8e^{25}\\&+\,b_9e^{26} + b_{10}e^{34} + b_{11}e^{35} + b_{12}e^{36} +b_{13}e^{45} + b_{14} e^{46} + b_{15}e^{56}, \end{aligned}$$

where \(b_i\in \mathbb {R}, i=1,\ldots ,15\), and the 3-form

$$\begin{aligned} \sigma = c(d\omega ), \quad c\in \mathbb {R}-\{0\}. \end{aligned}$$

The expression of \(\lambda (\sigma )\) for each nilpotent Lie algebra considered is given in Table 1.

Table 1 Expression of \(\lambda (\sigma )\) for the six-dimensional nilpotent Lie algebras admitting a half-flat \(\mathrm{SU}(3)\)-structure
Full size table

We observe that among the 24 nilpotent Lie algebras admitting a half-flat \(\mathrm{SU}(3)\)-structure we have:

  • 1 case \(({\mathfrak {n}}_{28})\) for which \(\lambda (\sigma )<0\) if \(b_{15}\ne 0\),

  • 2 cases \(({\mathfrak {n}}_4\) and \({\mathfrak {n}}_9)\) for which the sign of \(\lambda (\sigma )\) depends on \(\omega \),

  • 21 cases for which \(\lambda (\sigma )\) cannot be negative.

Therefore, the 21 algebras having \(\lambda (\sigma ) \ge 0\) do not admit any coupled \(\mathrm{SU}(3)\)-structure.

Consider \(\mathfrak {n}_4\), it has structure equations

$$\begin{aligned} (0,0,e^{12},e^{13}, e^{14}+e^{23}, e^{24}+e^{15}). \end{aligned}$$

First of all, observe that if \(b_{15}=0\) then \(\lambda (\sigma )=0\). So if we want to find an \(\mathrm{SU}(3)\)-structure we have to look for \(\omega \) with \(b_{15}\ne 0\). Moreover, \(\sigma \) induces an almost complex structure if and only if \(\lambda (\sigma )\) is negative, then we have to suppose in addition that \(b_{15}(b_{12}+b_{13})>b^2_{14}\). Since we want \(\omega \) to be the 2-form associated to an \(\mathrm{SU}(3)\)-structure, it must be a form of type \((1,1)\) and this happens if and only if \(\omega (\cdot ,\cdot )=\omega (J\cdot ,J\cdot )\), where \(J = J_{\sigma }\). Computing the previous identity with respect to the considered frame, we have that the following equations have to be satisfied by the components of \(\omega \):

$$\begin{aligned} \omega _{ab} =\sum _{k,m=1}^6 J^k_aJ^m_b\omega _{km},\quad 1\le a < b \le 6 \end{aligned}$$

(observe that \(\omega _{12}=b_1\), \(\omega _{13}=b_2\) and so on). Using these equations it is possible to write four of the \(b_i\) in terms of the remaining and obtain a new expression for \(\omega \). We can now compute the matrix associated to \(h(\cdot ,\cdot ) = \omega (J\cdot ,\cdot )\) with respect to the basis \((e_1,\ldots ,e_6)\) and observe that for the nonzero vector \(v = e_4 -\frac{b_{14}}{b_{15}}e_5 +\frac{b_{13}}{b_{15}}e_6\) we have \(h(v,v)=0\). Therefore, \(h\) cannot be positive definite and, as a consequence, it is not possible to find a coupled \(\mathrm{SU}(3)\)-structure on \(\mathfrak {n}_4\).

For the Lie algebras \(\mathfrak {n}_9\) and \(\mathfrak {n}_{28}\) we can give an explicit example of coupled \(\mathrm{SU}(3)\)-structure. Consider on \(\mathfrak {n}_9\) the forms

$$\begin{aligned} \omega= & {} -\frac{3}{2}e^{12}-\frac{1}{4}e^{14}-e^{15}-e^{24}+\frac{1}{2}e^{26}-\frac{1}{2}e^ {35}-e^{36}+e^{56}, \\ \sigma= & {} \frac{\sqrt{15}\root 4 \of {2}}{4}e^{123} +\frac{\sqrt{15}\root 4 \of {2}}{8}e^{234} -\frac{\sqrt{15}\root 4 \of {2}}{8}e^{125} +\frac{\sqrt{15}\root 4 \of {2}}{8}e^{134}\\&+\,\frac{\sqrt{15}\root 4 \of {2}}{4}e^{135} -\frac{\sqrt{15}\root 4 \of {2}}{4}e^{146} +\frac{\sqrt{15}\root 4 \of {2}}{4}e^{236} +\frac{\sqrt{15}\root 4 \of {2}}{4}e^{345}. \end{aligned}$$

We have

$$\begin{aligned} \omega \wedge \sigma =0, \quad \omega ^3\ne 0, \quad \lambda (\sigma ) =-\frac{225}{64}, \quad d\omega = -\frac{4}{\sqrt{15}\root 4 \of {2}}\sigma , \end{aligned}$$

in particular \((\omega ,\sigma )\) is a compatible pair of stable forms. The associated almost complex structure \(J = J_{\sigma }\) has the following matrix expression with respect to the basis \((e_1,\ldots ,e_6)\):

$$\begin{aligned} J= \left[ \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} 0&{}0&{}-\sqrt{2}&{}0&{}0&{}0 \\ \sqrt{2}&{}0&{}0&{}-\sqrt{2}&{}0&{}0\\ \frac{\sqrt{2}}{2}&{}0&{}0&{}0&{}0&{}0\\ 0&{}\frac{\sqrt{2}}{2}&{}-\sqrt{ 2}&{}0&{}0&{}0\\ \sqrt{2}&{}0&{}\frac{\sqrt{2}}{2}&{}\frac{\sqrt{2}}{2} &{}0&{}\sqrt{2}\\ -\frac{\sqrt{2}}{4}&{}-\frac{\sqrt{2}}{4}&{}\frac{ 3\sqrt{2}}{2}&{}0&{}-\frac{\sqrt{2}}{2}&{}0\end{array} \right] \end{aligned}$$

and it is easy to check that \(J^*\sigma \wedge \sigma = \frac{2}{3}\omega ^3\), i.e., the pair \((\omega ,\sigma )\) is normalized.

The inner product \(h(\cdot ,\cdot ) = \omega (J\cdot ,\cdot )\) is given with respect to the basis \((e_1,\ldots ,e_6)\) by

$$\begin{aligned} h= \left[ \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} \frac{5\sqrt{2}}{2}&{}\frac{\sqrt{2}}{8}&{} \frac{\sqrt{2}}{4}&{}-\sqrt{2}&{}0&{}\sqrt{2}\\ \frac{\sqrt{2}}{8}&{}\frac{5\sqrt{2}}{8}&{}-\frac{\sqrt{2}}{4}&{}0&{}\frac{\sqrt{2}}{4}&{}0\\ \frac{\sqrt{2}}{4}&{}-\frac{\sqrt{2}}{4}&{}\frac{7\sqrt{2}}{4}&{}\frac{\sqrt{2}}{4}&{}-\frac{ \sqrt{2}}{2}&{}\frac{\sqrt{2}}{2}\\ -\sqrt{2}&{}0&{}\frac{\sqrt{ 2}}{4}&{}\sqrt{2}&{}0&{}0\\ 0&{}\frac{\sqrt{2}}{4}&{}-\frac{\sqrt{2}}{2}&{}0 &{}\frac{\sqrt{2}}{2}&{}0\\ \sqrt{2}&{}0&{}\frac{\sqrt{2}}{2}&{}0&{}0&{} \sqrt{2}\end{array} \right] . \end{aligned}$$

and it is positive definite. Therefore, we can conclude that \((\omega ,\sigma )\) is a coupled \(\mathrm{SU}(3)\)-structure on \(\mathfrak {n}_9\).

For \(\mathfrak {n}_{28}\) consider the pair of compatible, normalized, stable forms

$$\begin{aligned} \left( \omega = e^{12} + e^{34} - e^{56}, \quad \sigma = e^{136}-e^{145}-e^{235}-e^{246}\right) . \end{aligned}$$
(1)

This pair defines a coupled \(\mathrm{SU}(3)\)-structure with \(d \omega = -\sigma \). Moreover, the associated inner product

$$\begin{aligned} h = (e^1)^2 + \cdots + (e^6)^2 \end{aligned}$$

is a nilsoliton with

$$\begin{aligned} \mathrm{Ric}(h) = -3I + 2 \, {\text{ diag }} (1,1,1,1,2,2). \end{aligned}$$

Summarizing our results, we can conclude that \(\mathfrak {n}_9\) and \(\mathfrak {n}_{28}\) are, up to isomorphisms, the only six-dimensional nilpotent Lie algebras admitting a coupled \(\mathrm{SU}(3)\)-structure.

We have just provided a coupled \(\mathrm{SU}(3)\)-structure on \(\mathfrak {n}_{28}\) whose associated inner product is a nilsoliton, we claim that this is the unique case among all six-dimensional nilpotent Lie algebras. It is clear that to prove the previous assertion it suffices to show that \(\mathfrak {n}_9\) does not admit any coupled \(\mathrm{SU}(3)\)-structure inducing a nilsoliton inner product. In order to do this, we consider an orthonormal basis \((e_1,\ldots ,e_6)\) of \(\mathfrak {n}_9\) whose dual basis satisfies the structure equations

$$\begin{aligned} \left( 0,0,0,\frac{\sqrt{5}}{2}e^{12},e^{14}-e^{23}, \frac{\sqrt{5}}{2}e^{15}+e^{34}\right) \end{aligned}$$

(by the results of [30] and [39], these are, up to isomorphisms, the structure equations for which the considered inner product on \(\mathfrak {n}_9\) is a nilsoliton). As we did before, consider a generic 2-form \(\omega \), the 3-form \(\sigma = c(d\omega )\), evaluate \(\lambda (\sigma )\) and impose that it is negative. Then compute \(J_\sigma \) and the matrix associated to \(h(\cdot ,\cdot ) = \omega (J_\sigma \cdot ,\cdot )\) with respect to the considered basis. Since \(h\) has to be the restriction to \(\mathfrak {n}_9\) of an Einstein inner product defined on \(\mathfrak {n}_9 \oplus \mathbb {R}e_7\) and since the latter is unique up to scaling, we have to impose that the symmetric matrix associated to \(h\) is a multiple of the identity. Solving the associated equations we find that \(\lambda (\sigma )\) has to be zero, which is a contradiction. \(\square \)

Starting from a six-dimensional nilpotent Lie algebra \(\mathfrak {n}\) endowed with a coupled \(\mathrm{SU}(3)\)-structure, it is possible to construct a locally conformal calibrated \(G_2\)-structure on the rank-one solvable extension \(\mathfrak {s} = \mathfrak {n} \oplus \mathbb {R}e_7\) under some extra hypothesis. Let \(\hat{d}\) denote the exterior derivative on \(\mathfrak {n}\) and \(d\) denote the exterior derivative on \(\mathfrak {s}\). Observe that given a \(k\)-form \(\theta \in \Lambda ^k(\mathfrak {n}^*)\) we have

$$\begin{aligned} d\theta = \hat{d}\theta + \rho \wedge e^7 \end{aligned}$$

for some \(\rho \in \Lambda ^k(\mathfrak {n}^*)\).

Proposition 4.2

Let \(\mathfrak {n}\) be a six-dimensional, nilpotent Lie algebra endowed with a coupled \(\mathrm{SU}(3)\)-structure \((\omega ,\sigma )\) with \(\hat{d} \omega = c \sigma \), \(c \in {\mathbb {R}}- \{ 0 \}\). Consider on its rank one solvable extension \(\mathfrak {s} = \mathfrak {n} \oplus \mathbb {R}e_7\) the \(G_2\)-structure defined by \(\varphi = \omega \wedge e^7 + \sigma \), where the closed 1-form \(e^7\) is the dual of \(e_7\). Then the \(G_2\)-structure is locally conformal calibrated with \(\tau _1=\frac{1}{3}ce^7\) if and only if \(d\sigma = -2c\sigma \wedge e^7\).

Proof

Suppose that \(d\sigma = -2c\sigma \wedge e^7\), we can write \(d\omega = \hat{d}\omega + \gamma \wedge e^7\) for some 2-form \(\gamma \in \Lambda ^2 (\mathfrak {n}^*)\). We obtain \(d \varphi = ce^7\wedge \varphi .\) Then, \(\varphi \) is locally conformal calibrated with \(\tau _1= \frac{1}{3}ce^7\).

Conversely, suppose that \(\varphi \) is locally conformal calibrated with \(\tau _1=\frac{1}{3}ce^7\). Then we have \(d\varphi = ce^7 \wedge \varphi \). Moreover, we know that \(d\sigma = \hat{d}\sigma + \beta \wedge e^7 = \beta \wedge e^7\) for some 3-form \(\beta \in \Lambda ^3 (\mathfrak {n}^*)\), since \(\sigma \) is \(\hat{d}\)-closed. We then have

$$\begin{aligned} d\varphi = d\omega \wedge e^7 + d\sigma = e^7\wedge (-c\sigma -\beta ) \end{aligned}$$

and comparing this with the previous expression of \(d\varphi \) we obtain

$$\begin{aligned} e^7\wedge (-c\sigma -\beta ) = c e^7 \wedge \varphi = e^7 \wedge (c\sigma ) \end{aligned}$$

from which follows \(\beta = -2c\sigma \). \(\square \)

Now we will construct an Einstein locally conformal calibrated \(G_2\)-structure on a rank-one extension of the Lie algebra \(\mathfrak {n}_{28}\) (Lie algebra of the \(3\)-dimensional complex Heisenberg group) endowed with the coupled \(\mathrm{SU}(3)\)-structure (1).

Example 4.3

Consider \(\mathfrak {n}_{28}\) and the metric rank-one solvable extension \(\mathfrak {s} = \mathfrak {n}_{28} \oplus \mathbb {R}e_7\) with structure equations

$$\begin{aligned} \left( \frac{1}{2}e^{17},\frac{1}{2}e^{27},\frac{1}{2}e^{37},\frac{1}{2}e^{47}, e^{13} - e^{24}+e^{57}, e^{14} + e^{23}+e^{67},0\right) . \end{aligned}$$

The associated solvable Lie group \(S\) is not unimodular and so it does not admit any compact quotient [35]. Consider on \(\mathfrak {n}_{28}\) the coupled \(\mathrm{SU}(3)\)-structure \((\omega , \sigma )\) given by (1) with the nilsoliton associated inner product

$$\begin{aligned} h = (e^1)^2 + \cdots + (e^6)^2. \end{aligned}$$

Then the inner product on \(\mathfrak {s}\)

$$\begin{aligned} g = (e^1)^2 + \cdots + (e^7)^2 \end{aligned}$$

is Einstein with Ricci tensor Ric\((g) = - 3 g\).

Since \(d\sigma = 2\sigma \wedge e^7\), by the previous proposition we have a locally conformal calibrated \(G_2\)-structure on \(\mathfrak {s}\) given by

$$\begin{aligned} \varphi = \omega \wedge e^7 +\sigma = e^{127} + e^{347} - e^{567} + e^{136} - e^{145} - e^{235} - e^{246} \end{aligned}$$

and it is easy to show that \(g_{\varphi } = g\). Then the corresponding solvmanifold \((S, \varphi )\) is an example of non-compact homogeneous manifold endowed with an Einstein (non-flat) locally conformal calibrated \(G_2\)-structure.

Observe that the \(G_2\)-structure \(\varphi \) satisfies the conditions

$$\begin{aligned} d \varphi= & {} - e^7 \wedge \varphi ,\\ d * \varphi= & {} - e^7 \wedge (3 e^{1256} + 2 e^{1234} + 3 e^{3456}). \end{aligned}$$

Then

$$\begin{aligned} \tau _1 = - \frac{1}{3} e^7, \end{aligned}$$

as we expected from Proposition 4.2, and

$$\begin{aligned} \tau _2 = -\left( \frac{5}{3} e^{12} + \frac{5}{3} e^{34} + \frac{10}{3} e^{56} \right) . \end{aligned}$$

Moreover, the \(G_2\)-structure is not \(*\)-Einstein, since by direct computation with respect to the orthonormal basis \(\left( e_1, \ldots , e_7\right) \) one has

$$\begin{aligned} \rho ^* = \left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} 1&{}0&{}0&{}0&{}0&{}0&{}0\\ 0&{} 1&{}0&{}0&{}0&{}0&{}0\\ 0&{}0&{} 1&{}0&{}0&{}0&{}0\\ 0&{}0&{}0&{}1&{}0&{}0&{}0\\ 0&{}0&{}0&{}0&{}22&{}0&{}0\\ 0&{}0&{}0&{}0&{}0&{}22&{}0\\ 0&{}0&{}0&{}0&{}0&{}0&{}-6 \end{array} \right) . \end{aligned}$$

It is worth emphasizing here that, by [16], on seven-dimensional solvmanifolds there are no left-invariant calibrated \(G_2\)-structures inducing an Einstein non-flat metric. The previous example shows that the situation is different in the case of locally conformal calibrated \(G_2\)-structures.

We provide now a non-compact example of homogeneous manifold admitting an Einstein (non-flat) locally conformal parallel \(G_2\)-structure.

Example 4.4

The Einstein rank-one solvable extension of the six-dimensional abelian Lie algebra is the solvable Lie algebra with structure equations

$$\begin{aligned} (ae^{17},ae^{27},ae^{37},ae^{47},ae^{57},ae^{67},0), \end{aligned}$$

where \(a\) is a nonzero real number. The Riemannian metric

$$\begin{aligned} g = (e^1)^2 + \cdots + (e^7)^2 \end{aligned}$$

is Einstein with Ricci tensor given by Ric\((g) = - 6 a^2 g\).

The 3-form

$$\begin{aligned} \varphi = -e^{125} - e^{136}- e^{147} + e^{237} - e^{246} + e^{345} -e^{567} \end{aligned}$$

has stabilizer \(G_2\), is such that \(g_\varphi = g\) and satisfies the conditions

$$\begin{aligned} d \varphi= & {} -3 a e^{2467} + 3 a e^{3457} - 3 a e^{1257} - 3 a e^{1367},\\ d * \varphi= & {} 4 a e^{23567} + 4 a e^{12347} - 4 a e^{14567}. \end{aligned}$$

It is immediate to show that \(\tau _1= -ae^7 \) and \(\tau _0\equiv 0, \tau _2\equiv 0, \tau _3\equiv 0\), that is, the \(G_2\)-structure \(\varphi \) is locally conformal parallel.