Special types of locally conformal closed G$_2$-structures

Motivated by analogous results in locally conformal symplectic geometry, we study different classes of G$_2$-structures defined by a locally conformal closed 3-form. In particular, we give a complete characterization of invariant exact locally conformal closed G$_2$-structures on simply connected Lie groups, and we present examples of compact manifolds with different types of locally conformal closed G$_2$-structures.


Introduction
Over the last years, the study of smooth manifolds endowed with geometric structures defined by a differential form which is locally conformal to a closed one has attracted a great deal of attention. Particular consideration has been devoted to locally conformal Kähler (LCK ) structures and their non-metric analogous, locally conformal symplectic (LCS ) structures, see [3,12,28,31] and the references therein. In both cases, the condition of being locally conformal closed concerns a suitable non-degenerate 2-form ω, and is encoded in the equation dω = θ ∧ ω, where θ is a closed 1-form called the Lee form. LCK structures belong to the pure class W 4 of Gray-Hervella's celebrated sixteen classes of almost Hermitian manifolds, see [19]. They are, in particular, Hermitian structures and their understanding on compact complex surfaces is related to the global spherical shell conjecture of Nakamura. As pointed out in [31], LCS geometry is intimately related to Hamiltonian mechanics. Very recently, Eliashberg and Murphy used h-principle arguments to prove that every almost complex manifold M with a non-zero [θ] ∈ H 1 dR (M ) admits a LCS structure whose Lee form is (a multiple of) θ, see [13].
In odd dimensions, 7-manifolds admitting G 2 -structures provide a natural setting where the locally conformal closed condition is meaningful. Recall that G 2 is one of the exceptional Riemannian holonomy groups resulting from Berger's classification [5], and that a G 2structure on a 7-manifold M is defined by a 3-form ϕ with pointwise stabilizer isomorphic to G 2 . Such a 3-form gives rise to a Riemannian metric g ϕ and to a volume form dV ϕ on M, with corresponding Hodge operator * ϕ . A G 2 -structure ϕ satisfying the conditions for some closed 1-form θ, is locally conformal to one which is both closed and coclosed. G 2 -structures fulfilling (1.1) correspond to the class W 4 in Fernández-Gray's classification [15], and they are called locally conformal parallel (LCP ), as being closed and coclosed for a G 2 -form ϕ is equivalent to being parallel with respect to the associated Levi Civita connection, see [15]. It was proved by Ivanov, Parton and Piccinni in [22,Thm. A] that a compact LCP G 2 -manifold is a mapping torus bundle over S 1 with fibre a simply connected nearly Kähler manifold of dimension six and finite structure group. This shows that LCP G 2 -structures are far from abundant. Relaxing the LCP condition by ruling out the second equation in (1.1) leads naturally to locally conformal closed, a.k.a. locally conformal calibrated (LCC ), G 2 -structures. Also in this case, the unique closed 1-form θ for which dϕ = θ ∧ ϕ is called Lee form. LCC G 2structures have been investigated in [14,16,17]; in particular, in [14] the authors showed that a result similar to that of Ivanov, Parton and Piccinni holds for compact manifolds with a suitable LCC G 2 -structure. Roughly speaking, they are mapping tori bundle over S 1 with fibre a 6-manifold endowed with a coupled SU(3)-structure, of which nearly Kähler structures constitute a special case. We refer the reader to Theorem 4.3 below for the relevant definitions and the precise statement.
In LCS geometry, one distinguishes between structures of the first kind and of the second kind (see [4,31]); the distinction depends on whether or not one can find an infinitesimal automorphism of the structure which is transversal to the foliation defined by the kernel of the Lee form. Another way to distinguish LCS structures is according to the vanishing of the class of ω in the Lichnerowicz cohomology defined by the Lee form. This leads to the notions of exact and non-exact LCS structures. A LCS structure of the first kind is always exact, but the converse is not true (see e.g. [4,Ex. 5.4]). The LCS structures constructed by Eliashberg and Murphy in [13] are exact.
The purpose of this note is to bring ideas of LCS geometry into the study of LCC G 2structures. In Sections 3 and 4, after recalling the notion of conformal class of a LCC G 2 -structure, we consider exact structures, and we distinguish between structures of the first and of the second kind. As it happens in the LCS case, the difference between first and second kind depends on the existence of a certain infinitesimal automorphism of the LCC G 2 -structure ϕ which is everywhere transversal to the kernel of the Lee form. As for exactness, every LCC G 2 -structure ϕ defines a class [ϕ] θ in the Lichnerowicz cohomology H • θ (M ) associated with the Lee form θ; ϕ is said to be exact if [ϕ] θ = 0 ∈ H 3 θ (M ). As we shall see, LCC G 2 -structures of the first kind are always exact, but the opposite needs not to be true (cf. Example 6.4).
In the literature, there exist many examples of left-invariant LCP and LCC G 2 -structures on solvable Lie groups, see e.g. [9,14,17]. In the LCC case, the examples exhibited in [14] admit a lattice, hence provide compact solvmanifolds endowed with an invariant LCC G 2 -structure. In Section 5, we completely characterize left-invariant exact LCC G 2structures on simply connected Lie groups: their Lie algebra is a rank-one extension of a six-dimensional Lie algebra with a coupled SU(3)-structure by a suitable derivation (see Theorem 5.4). Moreover, using the classification of seven-dimensional nilpotent Lie algebras which carry a closed G 2 -structure by Conti and Fernández [10], we prove that no such nilpotent Lie algebra admits a LCC G 2 -structure (Proposition 5.5). Finally, in Section 6 we show that there exist solvable Lie groups admitting a left-invariant LCC G 2 -structure which is not exact (see Example 6.2). This is not true on nilpotent Lie groups, as every leftinvariant LCC G 2 -structure must be exact by a result of Dixmier [11] on the Lichnerowicz cohomology. We also show that, unlike in the LCS case, there exist exact LCC structures on unimodular Lie algebras which are not of the first kind (see Remark 6.6).

Preliminaries
Let M be a seven-dimensional manifold. A G 2 -reduction of its frame bundle, i.e., a G 2structure, is characterized by the existence of a 3-form ϕ ∈ Ω 3 (M ) which can be pointwise written as ϕ| p = e 127 + e 347 + e 567 + e 135 − e 146 − e 236 − e 245 , with respect to a basis (e 1 , . . . , e 7 ) of the cotangent space T * p M. Here, the notation e ijk is a shorthand for e i ∧ e j ∧ e k . A G 2 -structure ϕ gives rise to a Riemannian metric g ϕ with volume form dV ϕ via the identity for all vector fields X, Y ∈ X(M ). We shall denote by * ϕ the corresponding Hodge operator. When a G 2 -structure ϕ on M is given, the G 2 -action on k-forms (cf. [6, Sect. 2]) induces the following decompositions: The decompositions of Ω k (M ), for k = 4, 5, are obtained from the previous ones via the Hodge operator.
By the above splittings, on a 7-manifold M endowed with a G 2 -structure ϕ there exist unique differential forms τ 0 ∈ C ∞ (M ), [7,Prop. 1]. Such forms are called intrinsic torsion forms of the G 2 -structure ϕ, as they completely determine its intrinsic torsion. In particular, ϕ is torsion-free if and only if all of these forms vanish identically, that is, if and only if ϕ is both closed (dϕ = 0) and coclosed (d * ϕ ϕ = 0). When this happens, g ϕ is Ricci-flat and its holonomy group is isomorphic to a subgroup of G 2 .
In this paper, we shall mainly deal with the G 2 -structures defined by a 3-form which is locally conformal equivalent to a closed one. As we will see in Section 3, this condition corresponds to the vanishing of the intrinsic torsion forms τ 0 and τ 3 . For the general classification of G 2 -structures, we refer the reader to [15].
In a similar way as in the case of G 2 -structures, the intrinsic torsion of an SU(3)-structure (ω, ψ) is encoded in the exterior derivatives dω, dψ, dψ (see [8]). According to [8,Def. 4 We shall refer to c as the coupling constant.
If h : N ֒→ M is an oriented hypersurface of a 7-manifold M endowed with a G 2 -structure ϕ, and V is a unit normal vector field on N , then the SU(3)-structure on N induced by ϕ is defined by the differential forms The reader may refer to [27] for more details on the relationship between G 2 -and SU(3)structures in this setting.

Locally conformal closed G 2 -structures
A G 2 -structure ϕ on a 7-manifold M is said to be locally conformal closed or locally conformal calibrated (LCC for short) if for some θ ∈ Ω 1 (M ). Notice that such a 1-form is unique and closed, as the map is injective for k = 1, 2. Moreover, it can be written in terms of ϕ as follows Definition 3.1. The unique closed 1-form θ fulfilling (3.1) is called the Lee form of the LCC G 2 -structure ϕ.
Henceforth, we denote a LCC G 2 -structure ϕ with Lee form θ by (ϕ, θ). As the name suggests, a LCC G 2 -structure (ϕ, θ) is locally conformal equivalent to a closed one. Indeed, since dθ = 0, each point of M admits an open neighborhood U ⊆ M where θ = df , for some f ∈ C ∞ (U ), and the 3-form e −f ϕ defines a closed G 2 -structure on U with associated metric e − 2 3 f g ϕ and orientation e − 7 3 f dV ϕ . Moreover, a LCC G 2 -structure is globally conformal equivalent to a closed one when θ is exact, and it is closed if and only if θ vanishes identically.
Simple examples of manifolds admitting a LCC G 2 -structure can be obtained as follows. Start with a 6-manifold N endowed with a coupled SU(3)-structure (ω, ψ) such that dω = cψ (various examples can be found, for instance, in [17,18,30]). Then, the product manifold N × R admits a LCC G 2 -structure given by the 3- More generally, if (ω, ψ) is coupled and ν ∈ Diff(N ) is a diffeomorphism such that ν * ω = ω, then the quotient of N × R by the infinite cyclic group of diffeomorphisms generated by (p, t) → (ν(p), t + 1) is a smooth seven-dimensional manifold N ν endowed with a LCC G 2 -structure ϕ (see [14,Prop. 3.1]). N ν is called the mapping torus of ν, and the natural projection In [16], Fernández and Ugarte proved that the LCC condition (3.1) can be characterized in terms of a suitable differential subcomplex of the de Rham complex. In detail: whered denotes the restriction of the differential d to B k (M ), for k = 3, 4.
As the Lee form θ of a LCC G 2 -structure ϕ is closed, it is also possible to introduce the Lichnerowicz (or Morse-Novikov) cohomology of M relative to θ. This is defined as the It is clear that the condition (3.1) is equivalent to d θ ϕ = 0. Thus, ϕ defines a cohomology , then the LCC G 2 -structure ϕ is said to be d θ -exact or exact. Notice that being exact is a property of the conformal class of ϕ.
More generally, if a G 2 -structure ϕ is d θ -exact with respect to some closed 1-form θ, then it is LCC with Lee form θ. The converse might not be true, as we shall see in Example 6.2.

LCC G 2 -structures of the first and of the second kind
A special class of exact LCC G 2 -structures can be introduced after some considerations on the infinitesimal automorphisms.
Recall that the automorphism group of a seven-dimensional manifold M endowed with a G 2 -structure ϕ is Aut(M, ϕ) := {F ∈ Diff(M ) | F * ϕ = ϕ} . Clearly, Aut(M, ϕ) is a closed Lie subgroup of the isometry group Iso(M, g ϕ ) of the Riemannian manifold (M, g ϕ ). Moreover, its Lie algebra is given by and every infinitesimal automorphism X ∈ aut(M, ϕ) is a Killing vector field for g ϕ .
Let us now focus on the case when ϕ is LCC with Lee form θ not identically vanishing. For every infinitesimal automorphism X ∈ aut(M, ϕ), we have whence we see that L X θ = 0. Consequently, θ(X) is constant and the map is a well-defined morphism of Lie algebras. This suggests that various meaningful ideas of locally conformal symplectic geometry (e.g. [1,2,4,31]) make sense for LCC G 2 -structures, too. In particular, as the map ℓ θ is either identically zero or surjective, we can give the following G 2 -analogue of a definition first introduced by Vaisman in [31]. If there exists at least one point p of M where θ| p = 0, then the LCC G 2 -structure ϕ is necessarily of the second kind. As a consequence, if ϕ is a LCC G 2 -structure with Lee form θ such that θ| p = df | p for some smooth function f ∈ C ∞ (M ), then the 3-form e −f ϕ defines a LCC G 2 -structure of the second kind, as the corresponding Lee form is θ − df . Hence, being of the first kind is not an invariant of the conformal class of ϕ.
Assume now that ϕ is a LCC G 2 -structure of the first kind. Then, its Lee form θ is nowhere vanishing and, consequently, Thus, a LCC G 2 -structure of the first kind is always exact. More precisely, it belongs to the image of the restriction of d θ to Ω 2 7 (M ). We shall say that an exact G 2 -structure ϕ is of the first kind if it can be written as ϕ = d θ (ι X ϕ) with θ(X) = −1. Proof. The first assertion follows from the identity The second assertion is an immediate consequence of the above definition.
Some examples of LCC G 2 -structures of the first and of the second kind will be discussed in Section 6. In particular, we will see that there exist exact G 2 -structures of the form ϕ = d θ σ with σ ∈ Ω 2 7 (M ). In [14,Thm. 6.4], the structure of compact 7-manifolds admitting a LCC G 2 -structure satisfying suitable properties was described. In view of the definitions introduced in this section, we can rewrite the statement of this structure theorem as follows.

Theorem 4.3 ([14]
). Let M be a compact seven-dimensional manifold endowed with a LCC G 2 -structure (ϕ, θ) of the first kind. If the g ϕ -dual vector field θ ♯ of θ belongs to aut(M, ϕ), then 1) M is the total space of a fibre bundle over S 1 , and each fibre is endowed with a coupled SU(3)-structure; 2) M has a LCC G 2 -structureφ such that dφ =θ ∧φ, whereθ is a 1-form with integral periods.
Motivated by the structure results for locally conformal symplectic structures of the first kind obtained in [2,4], we state the following more general problem.

Question 1.
What can one say about the structure of a (compact) 7-manifold M endowed with a LCC G 2 -structure of the first kind?
We conclude this section mentioning a mild issue related to the above statement. In order to prove Theorem 4.3, one deforms the given LCC G 2 -structure to one which gives M the claimed structure of a bundle over S 1 whose fibres are equipped with a coupled SU(3)structure, and the deformed structure has nothing to do with the given one. This kind of issue appears also in cosymplectic and in locally conformal symplectic geometry; results similar in spirit to Theorem 4.3 were obtained by Li [25] in the cosymplectic case, and by Banyaga [2] in the locally conformal symplectic case. A different approach, which does not deform the given structure, was taken in [20] for the cosymplectic case and in [4] for the locally conformal symplectic case: the same structure result holds, with the given structure, provided that the codimension-one foliation defined by the 1-form θ has one compact leaf.

Lie algebras with a LCC G 2 -structure
We begin this section recalling a few basic facts on Lie algebras, in order to introduce some notations. Then, we focus on the construction of Lie algebras admitting a LCC G 2structure, and we show a structure result for Lie algebras with an exact LCC G 2 -structure. All Lie algebras considered in this section are assumed to be real.

5.1.
Rank-one extension of Lie algebras. Let h be a Lie algebra of dimension n, denote by [·, ·] h its Lie bracket, and by d h the corresponding Chevalley-Eilenberg differential. The structure equations of h with respect to a basis (e 1 , . . . , e n ) are given by with c k ij ∈ R, c k ij = −c k ji , and n r=1 c r ij c s rk + c r jk c s ri + c r ki c s rj = 0. Equivalently, if (e 1 , . . . , e n ) is the dual basis of (e 1 , . . . , e n ), then the structure equations of h can be written as follows A Lie algebra h is then described up to isomorphism by the n-tuple (d h e 1 , . . . , d h e n ).
The rank-one extension of h induced by a derivation D ∈ Der(h) is the (n+1)-dimensional Lie algebra given by the vector space h ⊕ R endowed with the Lie bracket We shall denote this Lie algebra by h⋊ D R. Moreover, we let ξ := (0, 1), and we denote by η the 1-form on h⋊ D R such that η(ξ) = 1 and η(X) = 0, for all X ∈ h. Notice that if h is a nilpotent Lie algebra and D is a nilpotent derivation, then h⋊ D R is nilpotent.
In the next proposition, we collect some conditions guaranteeing the existence of a LCC G 2 -structure on the rank-one extension of a six-dimensional Lie algebra. For the sake of convenience, from now on we shall denote the Chevalley-Eilenberg differential on sevendimensional Lie algebras simply by d. Proof. Using (5.1), we see that the G 2 -structure ϕ = ω ∧ η + ψ is LCC with Lee form θ = aη if and only if From this, i) follows. As for ii), we first observe that the hypothesis D * ω = µω implies Thus, ϕ is LCC with Lee form θ = −(c + µ)η by point i). Moreover, Consequently, we get ω c is exact. Notice that ω c = ι ξ c ϕ ∈ Λ 2 7 ((h⋊ D R) * ). Therefore, according to Proposition 4.2, ϕ is of the first kind if and only if 3) Recall that for a six-dimensional Lie algebra h endowed with an SU(3)-structure (ω, ψ) the following isomorphisms hold: In particular, if (ω, ψ) is coupled and A * ω = 0, then A ∈ sp(6, R) ∩ sl(3, C) = su (3).
The next result is the converse of point ii) of Proposition 5.1.
Proposition 5.3. Let g be a seven-dimensional Lie algebra endowed with an exact LCC Assume that the non-zero vector X ∈ g for which σ = ι X ϕ satisfies θ(X) = 0. Then, g splits as a g ϕ -orthogonal direct sum g = h ⊕ R, where R = X and h := ker(θ) is a six-dimensional ideal endowed with a coupled SU(3)-structure (ω, ψ) induced by ϕ. Moreover, there is a derivation D ∈ Der(h) such that D * ω = − (1 + θ(X)) ω, and g ∼ = h⋊ D R.
Proof. It is clear that h := ker(θ) is a six-dimensional ideal of g, as θ ∈ Λ 1 (g * ) is non-zero and closed. Since θ(X) = 0, we see that the vector space g decomposes into the direct sum is well-defined, as dθ = 0, and it is a derivation of h by the Jacobi identity. From this it is easy to see that g ∼ = h⋊ D R as a Lie algebra.
According to a result of Dixmier (see [11,Théorème 1]), the Lichnerowicz cohomology of a nilpotent Lie algebra with respect to any closed 1-form vanishes. Hence, every LCC G 2structure on a seven-dimensional nilpotent Lie algebra is exact. We can use this observation to show that among the seven-dimensional non-Abelian nilpotent Lie algebras admitting closed G 2 -structures (see [10] for the classification) there is no one where LCC G 2 -structures exist.
Proposition 5.5. None of the seven-dimensional non-Abelian nilpotent Lie algebras admitting closed G 2 -structures admits LCC G 2 -structures.
Proof. By [10], a seven-dimensional non-Abelian nilpotent Lie algebra admitting closed G 2 -structures is isomorphic to one of the following: To show the proposition, we will use Dixmier's result together with the following fact: a 3-form φ on a seven-dimensional Lie algebra g defines a G 2 -structure if and only if the symmetric bilinear map is definite (cf. [21]). Now, for every nilpotent Lie algebra n i appearing above, we consider the generic closed 1-form θ = 7 k=1 θ k e k ∈ Λ 1 (n * i ), with some of the real numbers θ k possibly zero as dθ = 0, and the generic d θ -exact 3-form φ = dσ−θ∧σ, where σ = 1≤j<k≤7 σ jk e jk ∈ Λ 2 (n * i ). Then, we compute the map b φ associated with such a 3-form φ, and we observe that in each case it cannot be definite. Indeed, it is just a matter of computation to show that b φ (e 6 , e 6 ) = 0 for the nilpotent Lie algebras n i , with i = 1, 2, 3, 4, 5, 6, and that b φ (e 7 , e 7 ) = 0 for the remaining ones.

Examples
We now use the results of the previous section to construct various examples of LCC G 2 -structures that clarify the interplay between the conditions discussed in sections 3 and 4.
First of all, we need to start with a six-dimensional Lie algebra admitting coupled SU(3)structures. In the nilpotent case, the following classification is known (see [17,Thm. 4.1]). In both cases, (e 1 , . . . , e 6 ) is an SU(3)-basis for a certain coupled structure (ω, ψ).
Let us consider the coupled SU(3)-structure (ω, ψ) on h 1 . Since (e 1 , . . . , e 6 ) is an SU(3)basis, the forms ω and ψ can be written as in (5.2), and a simple computation shows that d h 1 ω = −ψ. As observed in [17], the inner product g = 6 i=1 (e i ) 2 induced by (ω, ψ) is a nilsoliton, i.e., its Ricci operator is of the form (e 1 , . . . , e 6 ) being the basis of h 1 whose dual basis is the SU(3)-basis of (h 1 , ω, ψ). For more details on nilsolitons we refer the reader to [23]. We know that the rank-one extension h 1 ⋊ D R of h 1 induced by a derivation D ∈ Der(h 1 ) admits a G 2 -structure defined by the 3-form ϕ = ω ∧η +ψ, and that the G 2 -basis is given by e 1 , . . . , e 6 , e 7 with e 7 := η. In what follows, we shall always write the structure equations of h 1 ⋊ D R with respect to such a basis.
The first example we consider was discussed in [17]. It consists of a solvable Lie algebra endowed with a LCC G 2 -structure ϕ inducing an Einstein inner product. As we will see, ϕ is not exact, that is, its class [ϕ] θ in the Lichnerowicz cohomology is not zero. Since D * 1 ψ = 2ψ and the coupling constant is c = −1, the G 2 -structure ϕ = ω ∧ η + ψ on h 1 ⋊ D 1 R is LCC with Lee form θ = −η, by point i) of Proposition 5.1. Moreover, it induces the inner product g ϕ = g + η 2 , which is Einstein with Ricci operator Ric(g ϕ ) = −3 Id by [24,Lemma 2]. A simple computation shows that ϕ cannot be equal to d θ σ for any 2-form σ ∈ Λ 2 ((h 1 ⋊ D 1 R) * ). In particular, it is of the second kind.
We conclude this example observing that the Lie algebra h 1 ⋊ D 1 R is solvable and not unimodular, as tr(ad e 7 ) = tr(D 1 ) = 4. Thus, the corresponding simply connected solvable Lie group does not admit any compact quotient.