A G2-structure on a seven-dimensional manifold M is defined by a positive 3-form (the G2 form) on M, which induces a Riemannian metric and a volume form on M such that
for any vector fields X, Y on M. If the 3-form is covariantly constant with respect to the Levi-Civita connection of the metric or, equivalently, the 3-form is closed and coclosed  , then the holonomy group of is a subgroup of the exceptional Lie group G2, and the metric is Ricci-flat. When this happens, the G2-structure is said to be torsion-free . The first compact examples of Riemannian manifolds with holonomy G2 were constructed first by Joyce  , and then by Kovalev . Recently, other examples of compact manifolds with holonomy G2 were obtained in  .
There are many different G2-structures attending to the behavior of the exterior derivative of the G2 form  . In the following, we will focus our attention on G2-structures where the 3-form is closed. In this case, the G2-structure is said to be closed (or calibrated). The first example of a compact G2-calibrated manifold, which does not admit any torsion-free G2-structure, was obtained in . This example is a compact nilmanifold, that is a compact quotient of a simply connected nilpotent Lie group by a lattice, endowed with an invariant calibrated G2-structure. In  , Conti and the first author classified the 7-dimensional compact nilmanifolds admitting a left invariant closed G2-structure. More examples were given in    .
Calibrated geometry was introduced by Harvey and Lawson in  and it concerns to a special type of minimal submanifolds of a Riemannian manifold, which are defined by a closed form (the calibration) on the manifold. Such submanifoldds are called calibrated submanifolds (see Section 5 for details). Every compact calibrated submanifold is volume-minimizing in its homology class (  Proposition 3.7.2).
In addition to compact Kähler manifolds and compact 7-manifolds with a torsion-free G2-structure, 7-manifolds with a closed G2-structure are also calibrated manifolds. In fact, if M is a 7-manifold with a closed G2-structure , then is a calibration . The 3-dimensional orientable submanifolds calibrated by the G2 form , that is, those 3-dimensional submanifolds such that restricted to Y is a volume form for Y, are called associative 3-folds of .
In this paper, we consider a parametrized family of 7-dimensional compact solvmanifolds with an invariant closed G2-structure , which is not coclosed, where k is a real number such that is an integer number different from 2. We show that is formal (Proposition 4.1) and its first Betti number . Moreover, we construct associative calibrated (so volume-minimizing) 3-tori in with respect to the closed G2 form (Proposition 5.3).
By   , a closed G2-structure on a compact manifold cannot induce an Einstein metric, unless the induced metric has holonomy contained in G2. It is still an open problem to see if the same property holds on noncompact manifolds. For the homogeneous case, a negative answer has been recently given in . Indeed, in  it is proved that if a solvable Lie algebra has a closed G2-structure then the induced inner product is Einstein if and only if it is flat.
Natural generalizations of Einstein metrics are given by Ricci solitons, which have been introduced by Hamilton in . All known examples of nontrivial homogeneous Ricci solitons are solsolitons. They are right invariant (or left invariant) metrics on simply connected solvable Lie groups, whose Ricci curvature tensor satisfies the condition
for some and some derivation D of the corresponding Lie algebra, where I is the identity map.
A natural question is thus to see if a closed G2-structure on a noncompact manifold induces a (non-Einstein) Ricci soliton metric. For the metric determined by the invariant closed G2 form on mentioned before, we show that if is the simply connected solvable (non-nilpotent) Lie group underlying to , then induces a solsoliton on (see Proposition 4.2).
The other motivation of this paper comes from the Laplacian flow on 7-manifolds admitting closed G2-structures. Let M be a 7-dimensional manifold with a closed G2-structure . The Laplacian flow on M starting from is given by
where is a closed G2 form on M and is the Hodge Laplacian operator associated with the metric induced by the 3-form . This geometric flow was introduced by Bryant in  as a tool to find torsion-free G2-structures on compact manifolds. Short-time existence and uniqueness of the solution, in the case of compact manifolds, were proved in . Properties of this flow were proved in   .
The first noncompact examples with long-time existence of the solution were obtained on seven-dimensional nilpotent Lie groups in  , but in those examples the Riemannian curvature tends to 0 as t goes to infinity. Further solutions on solvable Lie groups were described in    . Moreover, a cohomogeneity one solution converging to a torsion-free G2-structure on the 7-torus was worked out in .
In Section 6, we consider the solvable (non-nilpotent) Lie group underlying to the compact solvmanifold , and we show that the Laplacian flow of on exists for all time. In fact, in Theorem 6.2, we explicitly determine the solution for the flow of on , and we prove that it is defined on a time interval of the form , where is a real number. (This solution was previously given in  from a family of symplectic half-flat structures on a 6-dimensional ideal of the Lie algebra of .) We also show that the Ricci endomorphism of the underlying metric of is independent of the time t, and so the solution does not converge to a torsion-free G2-structure as t goes to infinity.
2. Closed G2-Structures
In this section we collect some basic facts and definitions concerning G2 forms on smooth manifolds (see           for details).
Let us consider the space of the Cayley numbers, which is a non-associative algebra over of dimension 8. Thus, we can identify with the subspace of consisting of pure imaginary Cayley numbers. Then, the product on defines on the 3-form given by
(see     for details), where is the standard basis of . Here, stands for , and so on. The group G2 is the stabilizer of (2) under the standard action of on . G2 is one of the exceptional Lie groups, and it is a compact, connected, simply connected simple Lie subgroup of of dimension 14.
A G2- structure on a 7-dimensional manifold M is a reduction of the structure group of its frame bundle from to the exceptional Lie group G2, which can actually be viewed naturally as a subgroup of . Thus, a G2-structure determines a Riemannian metric and an orientation on M. In fact, one can prove that the existence of a G2-structure is equivalent to the existence of a global differential 3-form (the G2 form) on M, which can be locally written as (2) with respect to some (local) basis of the (local) 1-forms on M. Such a 3-form was introduced by Bonan in  , and it induces a Riemannian metric and a volume form on M satisfying (1). We say that the manifold M has a closed (or calibrated) G2-structure if there is a G2-structure on M such that is closed, that is , and so defines a calibration .
Now, let G be a 7-dimensional simply connected nilpotent Lie group with Lie algebra . Then, a G2-structure on G is left invariant if and only if the corresponding 3-form is left invariant. Thus, a left invariant G2-structure on G corresponds to an element of that can be written as (2), that is,
with respect to some orthonormal coframe of the dual space . We say that a G2-structure on is calibrated if is closed, i.e.
where d denotes the Chevalley-Eilenberg differential on . If is a discrete subgroup of G, a G2-structure on induces a G2-structure on the quotient . In particular, if is solvable and is a discrete subgroup of G such that the quotient is compact, then a G2-structure on determines a G2-structure on the compact manifold , which is called a compact solvmanifold; and if has a calibrated G2-structure, the G2-structure on is also calibrated.
3. Formal Manifolds
First, we need some definitions and results about minimal models. Let be a differential algebra, that is, A is a graded commutative algebra over the real numbers, with a differential d which is a derivation, that is, , where is the degree of a.
A differential algebra is said to be minimal if it satisfies the following two conditions:
1) A is free as an algebra, that is, A is the free algebra V over a graded vector space ,
2) there exists a collection of generators , for some well-ordered index set I, such that if and each is expressed in terms of preceding ( ). This implies that does not have a linear part, that is, it lives in .
Morphisms between differential algebras are required to be degree-preserving algebra maps which commute with the differentials. Given a differential algebra , we denote by its cohomology. We say that A is connected if , and A is one-connected if, in addition, .
We will say that is a minimal model of the differential algebra if is minimal and there exists a morphism of differential graded algebras inducing an isomorphism on cohomology. Halperin  proved that any connected differential algebra has a minimal model unique up to isomorphism.
A minimal model is said to be formal if there is a morphism of differential algebras that induces the identity on cohomology. The formality of a minimal model can be distinguished as follows.
Theorem 3.1  A minimal model is formal if and only if and the space V decomposes as a direct sum with , d is injective on N and such that every closed element in the ideal generated by N in is exact.
A minimal model of a connected differentiable manifold M is a minimal model for the de Rham complex of differential forms on M. If M is a simply connected manifold, the dual of the real homotopy vector space is isomorphic to for any i. (For details see, for example,  .)
Definition 3.2 We will say that a differentiable manifold M is formal if its minimal model is formal or, equivalently, the differential algebras and have the same minimal model.
Many examples of formal manifolds are known: spheres, projective spaces, compact Lie groups, symmetric spaces, flag manifolds, and all compact Kähler manifolds .
We will also use the following property
Lemma 3.3 Let and be differentiable manifolds. Then, the product manifold is formal if and only if and are formal.
In  , the condition of formal manifold is weaken to s-formal manifold as follows.
Definition 3.4 Let be a minimal model of a differentiable manifold M. We say that is s-formal, or M is an s-formal manifold if such that for each , the space of generators of degree i decomposes as a direct sum , where the spaces and satisfy the three following conditions:
2) the differential map is injective,
3) any closed element in the ideal , generated by in , is exact in .
The relation between the formality and the s-formality for a manifold is given in the following theorem.
Theorem 3.5 Let M be a connected and orientable compact differentiable manifold of dimension 2n or . Then M is formal if and only if it is -formal.
4. The Compact Solvmanifolds M7(k)
Let be the simply connected and solvable Lie group of dimension 5 consisting of matrices of the form
where , for , and k is a real number such that is an integer number different from 2. Then a global system of coordinates for is defined by , and a standard calculation shows that a basis for the right invariant 1-forms on consists of
We notice that the Lie group may be described as a semidirect product , where acts on via the linear transformation of given by the matrix
Thus the operation on the group is given by
where and similarly for . Therefore , where is a connected abelian subgroup, and is the nilpotent commutator subgroup.
Now we show that there exists a discrete subgroup of such that the quotient space is compact. To construct it suffices to find some real number such that the matrix defining is conjugate to an element A of the special linear group with distinct real eigenvalues and . Indeed, we could then find a lattice in which is invariant under , and take . To this end, we choose the matrix given by
with double eigenvalues and . Taking , we have that the matrices and A are conjugate. In fact, put
Then a direct calculation shows that . So, if is the transpose of the vector , where , the lattice in defined by
is invariant under the subgroup . Thus is a cocompact subgroup of . So, the quotient space
is a 5-dimensional compact solvable manifold.
Alternatively, may be viewed as the total space of a T4-bundle over the circle . In fact, let be the 4-dimensional torus and the representation defined as follows: is the transformation of T4 covered by the linear transformation of given by the matrix
So acts on by
and S is the quotient . The projection is given by
Next, we consider the 7-dimensional compact manifold
where T2 is the 2-torus .
To compute the real cohomology of , we notice that is completely solvable, that is the map has only real eigenvalues for all , where denotes the Lie algebra of . Thus Hattori’s theorem  says that the de Rham cohomology ring is isomorphic to the cohomology ring of the Lie algebra of . For simplicity we denote the right invariant forms on and their projections on by the same symbols. Then, if we denote by the (right invariant) closed 1-forms on the 2-torus T2 whose cohomology classes generate the De Rham cohomology group , we have that the 1-forms on are such that
and such that at each point of , is a basis for the 1-forms on . Here stands for , and so on. Then, the real cohomology groups of are:
Thus, the Betti numbers of are
Proposition 4.1. The 5-manifold is 2-formal and so formal. Therefore, is formal.
Proof. To prove that is 2-formal, we see that its minimal model must be a differential graded algebra , where is the free algebra of the form , where the generator has degree 1, the generators have degree 2, and the differential d is given by . The morphism , inducing an isomorphism on cohomology, is defined by and .
According to Definition 3.4, we get and , thus is 1-formal. Moreover, is 2-formal since and . Hence, is 2-formal, and so formal by Theorem 3.5. Now, Lemma 3.3 implies that is formal.
We define the 3-form on given by
Clearly, is a G2 form on which is closed. Indeed, on the right-hand side of (14) all the terms are closed, and so is closed. Note that the dual form has the following expression
So, taking into account (11) and (12), we see that and are the unique nonclosed summands in . In fact, . Therefore, does not define a torsion-free G2-structure on .
Now, let be the simply connected solvable (non-nilpotent) Lie group . Then, is a basis for the right invariant 1-forms on and the structure equations of are given by (11). So, the closed G2 form defined in (14) is a right invariant closed G2 form on .
Let N be a simply connected solvable Lie group of dimension n, and denote by its Lie algebra. Recall that a right invariant metric g on N is called a Ricci solsoliton metric (or simply solsoliton metric) if its Ricci endomorphism differs from a derivation D of by a scalar multiple of the identity map , i.e. if there exists a real number such that
Not all solvable Lie groups admit solsoliton metrics, but if a solsoliton exists, then it is unique up to automorphism and scaling .
Proposition 4.2. Let be the seven dimensional Lie group , and let be the right invariant closed G2 form on defined in (14). Then the metric determined by is a solsoliton on .
Proof. Clearly, the metric induced on by is such that the basis for the 1-forms on is orthonormal, that is . Then, is a solsoliton since
is a derivation of the Lie algebra of .
5. Associative 3-Folds in M7(k)
In this section, we show associative 3-folds of the compact G2-calibrated solvmanifold defined in (10) with the closed G2 form given by (14). First, we need some definitions and results about calibrations (see   for details).
Let be a Riemannian manifold. An oriented tangent k-plane V on M is a vector subspace V of some tangent space to M, with and equipped with an orientation. If V is an oriented tangent k-plane on M, then is a Euclidean metric on V. So, combining with the orientation on V gives a natural volume form on V, which is a k-form on V.
Let a closed k-form on a Riemannian manifold . We say that is a calibration on M if for any and every oriented k-dimensional subspace V of the tangent space we have , for some (see  and  3.7). Thus, if Y is an oriented submanifold of M with dimension k then, for any , the tangent space is an oriented tangent k-plane on M. We say that Y is a calibrated submanifold if , for all .
All calibrated submanifolds are minimal submanifolds. Even more, every compact calibrated submanifold is volume-minimizing in its homology class (  Proposition~3.7.2).
Harvey and Lawson in  proved that any closed G2 form on a 7-manifold M is a calibration on M. The 3-dimensional orientable submanifolds calibrated by the G2 form , i.e. those submanifolds that satisfy , for each and for some unique orientation of Y, are called associative 3-folds.
Next, we shall produce examples of associative 3-folds in from the fixed locus of a G2-involution of the compact manifold applying the following.
Proposition 5.1 (  [Proposition 10.8.1]) Let N be a 7-manifold with a closed G2 form , and let be an involution of N satisfying and such that is not the identity map. Then the fixed point set is an embedded associative 3-fold. Furthermore, if N is compact then so is P.
Remark 5.2 Note that Proposition 10.8.1 in  is stated for the G2-structures that are closed and coclosed, but the coclosed condition is not used in the proof.
Proposition 5.3 There exist nine disjoint copies of 3-tori in , which define nine embedded, associative (calibrated by ), minimal 3-tori in .
Proof. Let be the seven dimensional Lie group defined in Proposition 4.2. We consider on the involution given by
that is is the product of the involutions with the identity map of , where is defined by
The involution is such that , and so descends to the 5-dimensional compact manifold . Hence, defines also an involution of . From now on, we denote by
the involution of induced by the involution of defined in (15). Then, taking into account (5), we have that the induced action on the 1-forms is given by
Therefore, the G2 form on defined in (14) is preserved by the involution of . In fact, by (16), each term on the right-hand side of (14) is σ-invariant.
Let P be the fixed locus of . Then, P consists of all the 3-dimensional spaces given as follows:
Consequently, P is a disjoint union of 9 copies of a 3-torus T3.
Since the G2 form on defined in (14) is preserved by the involution of , each of the 9 torus in fixed by is an associative 3-fold in by Proposition 5.1.
6. The Laplacian Flow
The purpose of this section is to prove that the Laplacian flow of on the 7-dimensional Lie group exists for all time. Moreover, we prove that the Ricci endomorphisms of the underlying metrics of the solution are independent of the time t, and so the solution does not converge to a torsion-free G2-structure as t goes to infinity.
Consider a 7-manifold M endowed with a calibrated G2-structure . The Laplacian flow starting from is the initial value problem
where denotes the Hodge Laplacian of the Riemannian metric induced by . This flow was introduced by Bryant in  to study seven-dimensional manifolds admitting calibrated G2-structures. Notice that the stationary points of the flow Equation in (17) are harmonic G2-structures, which coincide with torsion-free G2-structures on compact manifolds.
Short-time existence and uniqueness of the solution of (17) when M is compact were proved in .
Theorem 6.1 Assume that M is compact. Then, the Laplacian flow (17) has a unique solution defined for a short time , with depending on .
In the following theorem, we determine a global solution of the Laplacian flow of the closed G2 form given by (14) on the Lie group , where is the Lie group defined in Section 4.
Theorem 6.2 On the simply connected solvable (non-nilpotent) Lie group , the solution of the Laplacian flow (17) starting from the calibrated G2-structure is given by
Proof. Let be some differentiable real functions depending on a parameter such that and , for any , where I is a real open interval. For each , we consider the basis of left invariant 1-forms on defined by
Taking into account (11), the structure equations of with respect to the basis are
From now on, we write , , and so forth. Then, for any , we consider the G2-structure on given by
Note that the 3-form defined by (20) is such that and, for any t, determines the metric on such that the basis
of left invariant vector fields on dual to is orthonormal. Moreover, by (19), is closed, for any . Therefore, to solve the flow (17) of it is sufficient to determine the functions and the interval I so that , for .
Clearly since . Moreover,
So, and are the unique nonclosed summands in . Then, taking into account (19), we obtain
Thus, in terms of the forms , the expression of becomes
On the other hand,
Comparing (21) and (22) we have that if and only if the functions satisfy the following equations
The equations (23) with the initial conditions imply
Now, the equalities and imply and , respectively, and thus
Moreover, from we have
and from we have
Now, using (26) and (28), the system of differential equations formed by the Equations (24) and (25) is written as
Multiplying the first equation of (29) by , and the second one by , one can check that (29) implies that
Then, using that , we have
Thus, the system (29) is written as follows
Integrating this equation, we obtain
for some constant . But the initial condition implies , and hence
From (26), (27), (28), (30) and (31), we get
Therefore, taking into account (20), the family of closed G2 forms given by (18) is the solution of the Laplacian flow of on , and it is defined for all . □
Remark 6.3 Note that the metric , with , is a
solsoliton on . In fact, the metric with respect to the basis is given by
where is the function given by (31). Then, the Ricci endomorphism satifies
is a derivation of the Lie algebra of . Moreover, on is non-zero and independent of the time t. So, the solution does not converge to a torsion-free G2-structure as t goes to infinity.
Furthermore, taking into account the symmetry properties of the Riemannian curvature we obtain
where . Thus, the Riemannian curvature does not converge when t tends to infinity.
The authors were partially supported by MINECO-FEDER Grant MTM2014-54804-P and Gobierno Vasco Grant IT1094-16, Spain.
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