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Axioms 2018, 7(4), 90; doi:10.3390/axioms7040090
Special Types of Locally Conformal Closed G2-Structures
Departamento de Álgebra, Geometría y Topología, Universidad Complutense de Madrid, Plaza de Ciencias 3, 28040 Madrid, Spain
Dipartimento di Matematica “G. Peano”, Università degli Studi di Torino, Via Carlo Alberto 10, 10123 Torino, Italy
Author to whom correspondence should be addressed.
Received: 30 October 2018 / Accepted: 23 November 2018 / Published: 28 November 2018
Motivated by known results in locally conformal symplectic geometry, we study different classes of G2-structures defined by a locally conformal closed 3-form. In particular, we provide a complete characterization of invariant exact locally conformal closed G2-structures on simply connected Lie groups, and we present examples of compact manifolds with different types of locally conformal closed G2-structures.
Keywords:locally conformal closed G2-structure; coupled SU(3)-structure
MSC:53C10; 53C15; 53C30
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 [1,2,3,4,5] and the references therein). In both cases, the condition of being locally conformal closed concerns a suitable non-degenerate 2-form and is encoded in the equation , where is a closed 1-form called the Lee form. LCK structures belong to the pure class of Gray–Hervella’s celebrated 16 classes of almost Hermitian manifolds (see ). 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 , 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 admits an LCS structure whose Lee form is (a multiple of) (see ).
In odd dimensions, 7-manifolds admitting G2-structures provide a natural setting where the locally conformal closed condition is meaningful. Recall that G2 is one of the exceptional Riemannian holonomy groups resulting from Berger’s classification , and that a G2-structure on a 7-manifold M is defined by a 3-form with a pointwise stabilizer isomorphic to G2. Such a 3-form gives rise to a Riemannian metric and to a volume form on with corresponding Hodge operator . By an h-principle argument, it is possible to show that every compact 7-manifold admitting G2-structures always admits a coclosed G2-structure, i.e., one whose defining 3-form fulfills . A G2-structure satisfying the conditionsfor some closed 1-form is locally conformal to one which is both closed and coclosed. G2-structures fulfilling Equation (1) correspond to the class in Fernández–Gray’s classification , and they are called locally conformal parallel (LCP), as being closed and coclosed for a G2-form is equivalent to being parallel with respect to the associated Levi Civita connection (see ). It was proved by Ivanov, Parton, and Piccinni in  (Theorem A) that a compact LCP G2-manifold is a mapping torus bundle over with fiber a simply connected nearly Kähler manifold of dimension six and finite structure group. This shows that LCP G2-structures are far from abundant.
Relaxing the LCP requirement by ruling out the second condition in Equation (1) leads naturally to locally conformal closed, a.k.a. locally conformal calibrated (LCC), G2-structures. Additionally, the unique closed 1-form for which is called the Lee form. LCC G2-structures have been investigated in [12,13,14]; in particular, in , the authors showed that a result similar to that of Ivanov, Parton, and Piccinni holds for compact manifolds with a suitable LCC G2-structure. Roughly speaking, they are mapping tori bundle over with fiber a 6-manifold endowed with a coupled -structure, of which nearly Kähler structures constitute a special case. We refer the reader to Theorem 1 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 [5,15]); 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. The geometry of an LCS structure of the first kind is very rich and is related to the existence of a contact structure on the leaves of the corresponding foliation (cf. [1,15]). Another way to distinguish LCS structures is according to the vanishing of the class of in the Lichnerowicz cohomology defined by the Lee form. This leads to the notions of exact and non-exact LCS structures. An LCS structure of the first kind is always exact, but the converse is not true (see, e.g.,  (Example 5.4)). The LCS structures constructed by Eliashberg and Murphy in  are exact.
The purpose of this note is to bring ideas of LCS geometry into the study of LCC G2-structures. In Section 3 and Section 4, after recalling the notion of conformal class of an LCC G2-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 G2-structure , which is everywhere transversal to the kernel of the Lee form. As for exactness, every LCC G2-structure defines a class in the Lichnerowicz cohomology associated with the Lee form ; is said to be exact if . As we shall see, LCC G2-structures of the first kind are always exact, but the opposite does not need to be true (cf. Example 3). It is an open question whether an h-principle argument can be used to prove the existence of an exact LCC G2-structure on a compact manifold admitting G2-structures.
In the literature, there exist many examples of left-invariant LCP and LCC G2-structures on solvable Lie groups (see e.g., [12,14,16]). In the LCC case, the examples exhibited in  admit a lattice and hence provide compact solvmanifolds endowed with an invariant LCC G2-structure. In Section 5, we completely characterize the left-invariant exact LCC G2-structures on simply connected Lie groups: their Lie algebra is a rank-one extension of a six-dimensional Lie algebra with a coupled -structure by a suitable derivation (see Theorem 2). Moreover, using the classification of seven-dimensional nilpotent Lie algebras carrying a closed G2-structure , we prove that no such nilpotent Lie algebra admits an LCC G2-structure (Proposition 5). Finally, in Section 6, we show that there exist solvable Lie groups admitting a left-invariant LCC G2-structure, which is not exact (see Example 1). This does not happen on nilpotent Lie groups, as every left-invariant LCC G2-structure must be exact by a result of Dixmier  on the Lichnerowicz cohomology. We also show that, unlike the LCS case, there exist exact LCC structures on unimodular Lie algebras that are not of the first kind (see Remark 6).
Let M be a seven-dimensional manifold. A G2-reduction of its frame bundle, i.e., a G2-structure, is characterized by the existence of a 3-form , which can be pointwise written aswith respect to a basis of the cotangent space Here, the notation is a shorthand for . A G2-structure gives rise to a Riemannian metric with volume form via the identityfor all vector fields . We shall denote by the corresponding Hodge operator.
When a G2-structure on M is given, the G2-action on k-forms (cf.  (Section 2)) induces the following decompositions:where
The decompositions of , for , are obtained from the previous ones via the Hodge operator.
By the above splittings, on a 7-manifold M endowed with a G2-structure there exist unique differential forms , , , and , such thatsee  (Proposition 1). Such forms are called intrinsic torsion forms of the G2-structure , as they completely determine its intrinsic torsion. In particular, is torsion-free if and only if all of these forms vanish identically, that is, if and only if is both closed () and coclosed (). When this happens, is Ricci-flat and its holonomy group is isomorphic to a subgroup of G2.
In this paper, we shall mainly deal with the G2-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 and . For the general classification of G2-structures, we refer the reader to .
Since G2 acts transitively on the 6-sphere with stabilizer , a G2-structure on a 7-manifold M induces an SU(3)-structure on every oriented hypersurface. Recall that an -structure on a 6-manifold N is the data of an almost Hermitian structure with fundamental 2-form , and a unit -form , where . By , the whole SU(3)-structure is completely determined by the 2-form and the 3-form . In particular, at each point p of N, there exists a basis of the cotangent space such that
In a similar way, as in the case of G2-structures, the intrinsic torsion of an SU(3)-structure is encoded in the exterior derivatives , , (see ). According to  (Definition 4.1), an SU(3)-structure is called half-flat if and . A half-flat SU(3)-structure is said to be coupled if , for some , while it is called symplectic half-flat if , that is, if the fundamental 2-form is symplectic. We shall refer to c as the coupling constant.
If is an oriented hypersurface of a 7-manifold M endowed with a G2-structure , and V is a unit normal vector field on N, then the SU(3)-structure on N induced by is defined by the differential forms
The reader may refer to  for more details on the relationship between G2- and SU(3)-structures in this setting.
3. Locally Conformal Closed G2-Structures
A G2-structure on a 7-manifold M is said to be locally conformal closed or locally conformal calibrated (LCC for short) iffor some . Notice that such a 1-form is unique and closed, as the mapis injective for . Moreover, it can be written in terms of as follows:(see  (Lemma 2.1)).
The unique closed 1-form θ fulfilling Equation (3) is called the Lee form of the LCC G2-structure φ.
Henceforth, we denote an LCC G2-structure with Lee form by . As the name suggests, an LCC G2-structure is locally conformal equivalent to a closed one. Indeed, since , each point of M admits an open neighborhood where , for some , and the 3-form defines a closed G2-structure on with associated metric and orientation . Moreover, an LCC G2-structure is globally conformal equivalent to a closed one when is exact, and it is closed if and only if vanishes identically.
Given an LCC G2-structure , we may consider its conformal classIt is easily seen that is also LCC, so the de Rham class is an invariant of the conformal class.
- LCC G2-structures belong to the class in Fernández–Gray classification . The subclasses and correspond to closed and locally conformal parallel G2-structures, respectively.
Simple examples of manifolds admitting an LCC G2-structure can be obtained as follows. Start with a 6-manifold N endowed with a coupled SU(3)-structure such that (various examples can be found, for instance, in [14,25,26]). The product manifold then admits an LCC G2-structure given by the 3-form , where denotes the global 1-form on . The Lee form of is .
More generally, if is coupled and is a diffeomorphism such that , then the quotient of by the infinite cyclic group of diffeomorphisms generated by is a smooth seven-dimensional manifold endowed with an LCC G2-structure (see  (Proposition 3.1)). is called the mapping torus of , and the natural projection , , is a smooth fiber bundle with fiber N. Notice that .
In , Fernández and Ugarte proved that the LCC condition can be characterized in terms of a suitable differential subcomplex of the de Rham complex. In detail,
Proposition 1 ().
A G2-structure on a 7-manifold M is LCC if and only if the exterior derivative of every 3-form in belongs to . Consequently, is LCC if and only if there exists the complexwhere denotes the restriction of the differential d to , for .
As the Lee form of an LCC G2-structure is closed, it is also possible to introduce the Lichnerowicz (or Morse–Novikov) cohomology of M relative to . This is defined as the cohomology corresponding to the complex , where
It is clear that Equation (3) is equivalent to . Thus, defines a cohomology class . If , namely if for some , then the LCC G2-structure is said to be -exact or exact. Notice that being exact is a property of the conformal class of .
More generally, if a G2-structure is -exact with respect to some closed 1-form , then it is LCC with Lee form . The converse might not be true, as we shall see in Example 1.
4. LCC G2-Structures of the First and of the Second Kind
A special class of exact LCC G2-structures can be introduced after some considerations of the infinitesimal automorphisms.
Recall that the automorphism group of a seven-dimensional manifold M endowed with a G2-structure is
Clearly, is a closed Lie subgroup of the isometry group of the Riemannian manifold . Moreover, its Lie algebra is given byand every infinitesimal automorphism is a Killing vector field for .
If is closed and , then the 2-form is easily seen to be harmonic. When M is compact, this implies that is Abelian with dimension bounded by (see ).
Let us now focus on the case when is LCC with Lee form not identically vanishing. For every infinitesimal automorphism , we havehence we see that . Consequently, is constant, and the mapis a well-defined morphism of Lie algebras. This suggests that various meaningful ideas of locally conformal symplectic geometry (e.g., [1,5,15,28]) make sense for LCC G2-structures, too. In particular, as the map is either identically zero or surjective, we give the following G2-analogue of a definition first introduced by Vaisman in .
An LCC G2-structure is of the first kind if the Lie algebra morphism is surjective, while it is of the second kind otherwise.
If there exists at least one point p of M where , then the LCC G2-structure is necessarily of the second kind. As a consequence, if is an LCC G2-structure with Lee form such that for some smooth function , then the 3-form defines an LCC G2-structure of the second kind, as the corresponding Lee form is . Hence, being of the first kind is not an invariant of the conformal class of .
Assume now that is an LCC G2-structure of the first kind. Then, its Lee form is nowhere vanishing; consequently, if M is compact. Let us consider an infinitesimal automorphism such that . The condition is equivalent towhere . Thus, an LCC G2-structure of the first kind is always exact. More precisely, it belongs to the image of the restriction of to .
- Comparing our situation to the LCS case  , we are choosing the opposite sign for the infinitesimal automorphism This is only a matter of convention and simplifies our presentation.
- As we mentioned above, if is a coupled structure on a 6-manifold N and satisfies , then the mapping torus of ν admits an LCC G2-structure . It follows from the proof of  (Proposition 3.1) that there exists an infinitesimal automorphism such that . Thus, is of the first kind.
We shall say that an exact G2-structure is of the first kind if it can be written as with .
Let be an LCC G2-structure. Then, if and only if In particular, φ is of the first kind if and only if .
The first assertion follows from the identity
The second assertion is an immediate consequence of the above definition. ☐
Some examples of LCC G2-structures of the first and of the second kind will be discussed in Section 6. In particular, we will see that there exist exact G2-structures of the form with .
In  (Theorem 6.4), the structure of compact 7-manifolds admitting an LCC G2-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 1 ().
Let M be a compact seven-dimensional manifold endowed with an LCC G2-structure of the first kind. If the -dual vector field of θ belongs to , then
- M is the total space of a fiber bundle over , and each fiber is endowed with a coupled -structure;
- M has an LCC G2-structure such that , where is a 1-form with integral periods.
Motivated by the structure results for locally conformal symplectic structures of the first kind obtained in [1,15], we state the following more general problem.
What can one say about the structure of a (compact) 7-manifold M endowed with an LCC G2-structure of the first kind?
We conclude this section by mentioning a mild issue related to the above statement. In order to prove Theorem 1, one first deforms the Lee form of the given LCC G2-structure on M to a closed 1-form with integral periods. Then, by a result of Tischler , M is the mapping torus of a compact 6-manifold N and a diffeomorphism and one shows that N is endowed with a coupled -structure . However, in general, is not preserved by . In particular, it is not clear whether admits LCC G2-structures arising from the mapping torus construction. A similar issue appears in locally conformal symplectic geometry. In , Banyaga proved that a compact manifold M endowed with an LCS structure of the first kind is the total space of a mapping torus fiber bundle of a compact contact manifold and a diffeomorphism , which need not preserve the contact form (if it does, then one can show that admits a natural LCS structure of the first kind). A different approach, which does not deform the given structure, was taken in : the authors showed that, if is a compact LCS manifold of the first kind and the codimension-one foliation given by the kernel of has a compact leaf, then M is diffeomorphic to the mapping torus of a compact contact manifold and a strict contactomorphism , and, moreover, the LCS structure on M is precisely the one given by the mapping torus construction.
5. Lie Algebras with an LCC G2-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 an LCC G2-structure, and we prove a structure result for Lie algebras with an exact LCC G2-structure. All Lie algebras considered in this section are assumed to be real.
5.1. Rank-One Extension of Lie Algebras
Let be a Lie algebra of dimension n, and denote by its Lie bracket and by the corresponding Chevalley–Eilenberg differential. The structure equations of with respect to a basis are given bywith , , and . Equivalently, if is the dual basis of , then the structure equations of can be written as follows:A Lie algebra is then described up to isomorphism by the n-tuple .
The rank-one extension of induced by a derivation is the -dimensional Lie algebra given by the vector space endowed with the Lie bracketfor all . We shall denote this Lie algebra by . Moreover, we let , and we denote by the 1-form on such that and , for all . Notice that, if is a nilpotent Lie algebra and D is a nilpotent derivation, then is nilpotent.
Let d denote the Chevalley–Eilenberg differential on . Using the Koszul formula, it is possible to check that for every k-form , the following identity holds:where the natural action of an endomorphism on is given byfor all . Moreover, it is clear that .
5.2. A Structure Result for Lie Algebras with an Exact LCC G2-Structure
Let be a six-dimensional Lie algebra. A pair defines an SU(3)-structure on if there exists a basis of such thatWe shall call an -basis for . An SU(3)-structure on is half-flat if and . A half-flat SU(3)-structure satisfying the condition for some is coupled if , while it is symplectic half-flat if .
Similarly, a 3-form on a seven-dimensional Lie algebra defines a G2-structure if there is a basis of such thatWe shall refer to as a G2-basis for . A G2-structure is closed if , while it is locally conformal closed (LCC ) if for some 1-form with .
If is the rank-one extension of a six-dimensional Lie algebra endowed with an SU(3)-structure , then it admits a G2-structure defined by the 3-formIndeed, if is an SU(3)-basis for , then with is a G2-basis for .
In the next proposition, we collect some conditions guaranteeing the existence of an LCC G2-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 seven-dimensional Lie algebras simply by d.
Let be a six-dimensional Lie algebra endowed with a coupled -structure with , and consider the rank-one extension , , endowed with the G2-structure . Then, the following holds:
- is LCC with Lee form , for some , if and only if . In particular, it is closed if and only if .
- If with , then φ is -exact. Moreover, it is of the first kind if and only if .
Using Equation (4), we see that the G2-structure is LCC with Lee form if and only ifFrom this, (i) follows.
As for (ii), we first observe that the hypothesis impliesThus, is LCC with Lee form by Point (i). Moreover,Consequently,Hence, is exact. Notice that . Therefore, according to Proposition 2, is of the first kind if and only if☐
- When the SU(3)-structure on is symplectic half-flat and satisfies , then is a closed G2-structure on by Point (i). This was already observed by Manero in  (Proposition 1.1).
- Recall that for a six-dimensional Lie algebra endowed with an SU(3)-structure , the following isomorphisms hold:
The next result is the converse of Point (ii) of Proposition 3.
Let be a seven-dimensional Lie algebra endowed with an exact LCC G2-structure , where is closed and . Assume that the non-zero vector for which satisfies . Then, splits as a -orthogonal direct sum , where and is a six-dimensional ideal endowed with a coupled -structure induced by φ. Moreover, there is a derivation such that , and .
It is clear that is a six-dimensional ideal of , as is non-zero and closed. Since , we see that the vector space decomposes into the direct sum , with . The -linear mapis well-defined, as , and it is a derivation of by the Jacobi identity. From this, it is easy to see that as a Lie algebra.
Let be the -dual vector of . By definition, . Thus, and the decomposition is -orthogonal, i.e., for all . Consequently, depending on the choice of a unit vector , with , the ideal admits an SU(3)-structure defined by the pair
Notice that , as . We claim that is coupled with coupling constant First, observe that for all , we have
Therefore, , and the claim is proved. Let us now determine the expression of , from which we will deduce the expression of . For all , we havewhere the second equality follows from Koszul formula and the condition . Since , on we haveThus,☐
Combining Propositions 3 and 4, we obtain the following analogue of  (Theorem 1.4) for exact locally conformal symplectic Lie algebras.
There is a one-to-one correspondence between seven-dimensional Lie algebras admitting an exact G2-structure of the form , with and , and six-dimensional Lie algebras endowed with a coupled -structure , with coupling constant c, and a derivation such that , for some .
Comparing Theorem 2 with Theorem 1, we see that in the former we do not have any issue with deformations. Indeed, the ideal of admitting a coupled SU(3)-structure is precisely the kernel of the Lee form θ, while the fibration considered in Theorem 1 is associated with a closed 1-form arising from a deformation of the Lee form.
According to a result of Dixmier (see  (Theorem 1)), the Lichnerowicz cohomology of a nilpotent Lie algebra with respect to any closed 1-form vanishes. Hence, every LCC G2-structure on a seven-dimensional nilpotent Lie algebra is exact. We use this observation to prove the following result.
None of the seven-dimensional non-Abelian nilpotent Lie algebras admitting closed -structures admits LCC -structures.
By the classification result of Conti-Fernández , a seven-dimensional non-Abelian nilpotent Lie algebra admitting closed -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 on a seven-dimensional Lie algebra defines a G2-structure if and only if the symmetric bilinear mapis definite (cf. ). Now, for every nilpotent Lie algebra appearing above, we consider the generic closed 1-form , with some of the real numbers possibly zero as , and the generic -exact 3-form , where . Then, we compute the map associated with such a 3-form , and we observe that in each case it cannot be definite. Indeed, it is just a matter of computation to show that for the nilpotent Lie algebras , with and that for the remaining ones.
We now use the results of the previous section to construct various examples of LCC G2-structures that clarify the interplay between the conditions discussed in Section 3 and Section 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  (Theorem 4.1)).
Theorem 3 ().
Up to isomorphism, a six-dimensional non-Abelian nilpotent Lie algebra admitting coupled -structures is isomorphic to one of the followingIn both cases, is an -basis for a certain coupled structure .
Let us consider the coupled SU(3)-structure on . Since is an SU(3)-basis, the forms and can be written as in Equation (5), and a simple computation shows that . As observed in , the inner product induced by is a nilsoliton, i.e., its Ricci operator is of the formwhere is given bybeing the basis of whose dual basis is the SU(3)-basis of . For more details on nilsolitons we refer the reader to .
We know that the rank-one extension of induced by a derivation admits a G2-structure defined by the 3-form and that the G2-basis is given by with . In what follows, we shall always write the structure equations of with respect to such a basis.
The first example we consider was discussed in . It consists of a solvable Lie algebra endowed with an LCC G2-structure inducing an Einstein inner product. As we will see, is not exact, that is, its class in the Lichnerowicz cohomology is not zero.
Let us consider the derivation appearing in Equation (6). The rank-one extension of has structure equations
Since and the coupling constant is , the G2-structure on is LCC with Lee form , by Point (i) of Proposition 3. Moreover, it induces the inner product , which is Einstein with Ricci operator by  (Lemma 2). A simple computation shows that φ cannot be equal to for any 2-form . In particular, it is of the second kind.
We conclude this example observing that the Lie algebra is solvable and not unimodular, as Thus, the corresponding simply connected solvable Lie group does not admit any compact quotient.
The next two examples were obtained in  (Section 5). In the first one, the LCC G2-structure is of the first kind, while in the second one the LCC G2-structure is exact but it is not of the first kind.
Consider the derivation defined as follows:Then, the rank-one extension has structure equationsand . Thus, by Point (ii) of Proposition 3, we have that the 3-form defines an LCC G2-structure of the first kind on with Lee form .
Consider the rank-one extension , where is given by
The structure equations of are the following:Since but , the G2-structure on is LCC with Lee form , by Point (i) of Proposition 3. We observe thatwhere does not belong to . In this case, the only infinitesimal automorphisms of φ are of the form , with . Thus, φ is of the second kind.
As shown in , the Lie algebras considered in Examples 2 and 3 are solvable and unimodular, and the corresponding simply connected solvable Lie groups admit a lattice. Thus, both examples give rise to a compact seven-dimensional solvmanifold endowed with an LCC GG2-structure.
It was proved in  (Proposition 5.5) that, on a unimodular Lie algebra, every exact locally conformal symplectic structure is of the first kind. This is not the case in the G2 setting: indeed, the LCC G2-structure of Example 3 is exact but not on the first kind, while the Lie algebra is unimodular.
The authors contributed equally to this work.
This research received no external funding.
The authors would like to thank Daniele Angella for useful conversations. They are also grateful to the anonymous referee for her/his useful suggestions. The first author was supported by a Juan de la Cierva—Incorporación Fellowship of Spanish Ministerio de Ciencia e Innovación. Both authors were partially supported by GNSAGA of INdAM—Istituto Nazionale di Alta Matematica.
Conflicts of Interest
The authors declare no conflict of interest.
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