A G₂-structure on a seven-dimensional manifold M is defined by a positive 3-form φ (the G₂ form) on M, which induces a Riemannian metric g φ and a volume form d V φ on M such that

g φ ( X , Y ) d V φ = 1 6 ι X φ ∧ ι Y φ ∧ φ , (1)

for any vector fields X, Y on M. If the 3-form φ is covariantly constant with respect to the Levi-Civita connection of the metric g φ or, equivalently, the 3-form φ is closed and coclosed [1] , then the holonomy group of g φ is a subgroup of the exceptional Lie group G₂, and the metric g φ is Ricci-flat. When this happens, the G₂-structure is said to be torsion-free [2]. The first compact examples of Riemannian manifolds with holonomy G₂ were constructed first by Joyce [3] , and then by Kovalev [4]. Recently, other examples of compact manifolds with holonomy G₂ were obtained in [5] [6].

There are many different G₂-structures attending to the behavior of the exterior derivative of the G₂ form [1] [7]. In the following, we will focus our attention on G₂-structures where the 3-form φ 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 [8]. 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 [9] , Conti and the first author classified the 7-dimensional compact nilmanifolds admitting a left invariant closed G₂-structure. More examples were given in [10] [11] [12] [13].

Calibrated geometry was introduced by Harvey and Lawson in [14] 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 ( [15] 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 φ , then φ is a calibration [14]. The 3-dimensional orientable submanifolds Y ⊂ M calibrated by the G₂ form φ , that is, those 3-dimensional submanifolds Y ⊂ M such that φ restricted to Y is a volume form for Y, are called associative 3-folds of ( M , φ ) .

In this paper, we consider a parametrized family of 7-dimensional compact solvmanifolds M 7 ( k ) with an invariant closed G₂-structure φ k , which is not coclosed, where k is a real number such that e k + e − k is an integer number different from 2. We show that M 7 ( k ) is formal (Proposition 4.1) and its first Betti number b 1 ( M 7 ( k ) ) = 3 . Moreover, we construct associative calibrated (so volume-minimizing) 3-tori in M 7 ( k ) with respect to the closed G₂ form φ k (Proposition 5.3).

By [16] [17] , 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 [18]. Indeed, in [18] 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 [19]. 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

R i c ( g ) = λ I + D ,

for some λ ∈ ℝ 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 φ k on M 7 ( k ) mentioned before, we show that if H ( k ) is the simply connected solvable (non-nilpotent) Lie group underlying to M 7 ( k ) , then φ k induces a solsoliton on H ( k ) (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 φ . The Laplacian flow on M starting from φ is given by

{ ∂ ∂ t φ ( t ) = Δ t   φ ( t ) , d   φ ( t ) = 0 , φ ( 0 ) = φ ,

where φ ( t ) is a closed G₂ form on M and Δ t = d   d * + d * d is the Hodge Laplacian operator associated with the metric g φ ( t ) induced by the 3-form φ ( t ) . This geometric flow was introduced by Bryant in [16] 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 [20]. Properties of this flow were proved in [21] [22] [23].

The first noncompact examples with long-time existence of the solution were obtained on seven-dimensional nilpotent Lie groups in [24] , but in those examples the Riemannian curvature tends to 0 as t goes to infinity. Further solutions on solvable Lie groups were described in [25] [26] [27] [28]. Moreover, a cohomogeneity one solution converging to a torsion-free G₂-structure on the 7-torus was worked out in [29].

In Section 6, we consider the solvable (non-nilpotent) Lie group H ( k ) underlying to the compact solvmanifold M 7 ( k ) , and we show that the Laplacian flow of φ k on H ( k ) exists for all time. In fact, in Theorem 6.2, we explicitly determine the solution φ k ( t ) for the flow of φ k on H ( k ) , and we prove that it is defined on a time interval of the form ( T , ∞ ) , where T < 0 is a real number. (This solution was previously given in [25] from a family of symplectic half-flat structures on a 6-dimensional ideal of the Lie algebra h ( k ) of H ( k ) .) We also show that the Ricci endomorphism R i c ( g k ( t ) ) of the underlying metric g k ( t ) of φ k ( t ) is independent of the time t, and so the solution φ k ( t ) does not converge to a torsion-free G₂-structure as t goes to infinity.

In this section we collect some basic facts and definitions concerning G₂ forms on smooth manifolds (see [1] [2] [7] [14] [15] [16] [30] [31] [32] [33] for details).

Let us consider the space O of the Cayley numbers, which is a non-associative algebra over ℝ of dimension 8. Thus, we can identify ℝ 7 with the subspace of O consisting of pure imaginary Cayley numbers. Then, the product on O defines on ℝ 7 the 3-form given by

e 127 + e 347 + e 567 + e 135 − e 236 − e 146 − e 245 (2)

(see [1] [32] [33] [34] for details), where { e 1 , ⋯ , e 7 } is the standard basis of ( ℝ 7 ) * . Here, e 127 stands for e 1 ∧ e 2 ∧ e 7 , and so on. The group G₂ is the stabilizer of (2) under the standard action of GL ( 7, ℝ ) on Λ 3 ( ℝ 7 ) * . G₂ is one of the exceptional Lie groups, and it is a compact, connected, simply connected simple Lie subgroup of SO ( 7 ) of dimension 14.

A G₂- structure on a 7-dimensional manifold M is a reduction of the structure group of its frame bundle from GL ( 7, ℝ ) to the exceptional Lie group G₂, which can actually be viewed naturally as a subgroup of SO ( 7 ) . 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 φ (the G₂ form) on M, which can be locally written as (2) with respect to some (local) basis { e 1 , ⋯ , e 7 } of the (local) 1-forms on M. Such a 3-form φ was introduced by Bonan in [35] , and it induces a Riemannian metric g φ and a volume form d V φ on M satisfying (1). We say that the manifold M has a closed (or calibrated) G₂-structure if there is a G₂-structure φ on M such that φ is closed, that is d φ = 0 , and so φ defines a calibration [14].

Now, let G be a 7-dimensional simply connected nilpotent Lie group with Lie algebra g . Then, a G₂-structure on G is left invariant if and only if the corresponding 3-form φ is left invariant. Thus, a left invariant G₂-structure on G corresponds to an element φ of Λ 3 ( g * ) that can be written as (2), that is,

φ = e 127 + e 347 + e 567 + e 135 − e 146 − e 236 − e 245 , (3)

with respect to some orthonormal coframe { e 1 , ⋯ , e 7 } of the dual space g ∗ . We say that a G₂-structure on g is calibrated if φ is closed, i.e.

d φ = 0 ,

where d denotes the Chevalley-Eilenberg differential on g ∗ . If Γ is a discrete subgroup of G, a G₂-structure on g induces a G₂-structure on the quotient Γ \ G . In particular, if g is solvable and Γ is a discrete subgroup of G such that the quotient Γ \ G is compact, then a G₂-structure on g determines a G₂-structure on the compact manifold Γ \ G , which is called a compact solvmanifold; and if g has a calibrated G₂-structure, the G₂-structure on Γ \ G is also calibrated.

First, we need some definitions and results about minimal models. Let ( A , d ) 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, d ( a ⋅ b ) = ( d a ) ⋅ b + ( − 1 ) d e g ( a ) a ⋅ ( d b ) , where d e g ( a ) is the degree of a.

A differential algebra ( A , d ) 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 V = ⊕ V i ,

2) there exists a collection of generators { a τ , τ ∈ I } , for some well-ordered index set I, such that d e g ( a μ ) ≤ d e g ( a τ ) if μ < τ and each d a τ is expressed in terms of preceding a μ ( μ < τ ). This implies that d a τ does not have a linear part, that is, it lives in Λ V > 0 ⋅ Λ V > 0 ⊂ V .

Morphisms between differential algebras are required to be degree-preserving algebra maps which commute with the differentials. Given a differential algebra ( A , d ) , we denote by H * ( A ) its cohomology. We say that A is connected if H 0 ( A ) = ℝ , and A is one-connected if, in addition, H 1 ( A ) = 0 .

We will say that ( M , d ) is a minimal model of the differential algebra ( A , d ) if ( M , d ) is minimal and there exists a morphism of differential graded algebras ρ : ( M , d ) → ( A , d ) inducing an isomorphism ρ * : H * ( M ) → H * ( A ) on cohomology. Halperin [36] proved that any connected differential algebra ( A , d ) has a minimal model unique up to isomorphism.

A minimal model ( M , d ) is said to be formal if there is a morphism of differential algebras Ψ : ( M , d ) → ( H * ( M ) , d = 0 ) that induces the identity on cohomology. The formality of a minimal model can be distinguished as follows.

Theorem 3.1 [37] A minimal model ( M , d ) is formal if and only if M = Λ V and the space V decomposes as a direct sum V = C ⊕ N with d ( C ) = 0 , d is injective on N and such that every closed element in the ideal I ( N ) generated by N in Λ V is exact.

A minimal model of a connected differentiable manifold M is a minimal model ( Λ V , d ) for the de Rham complex ( Ω * M , d ) of differential forms on M. If M is a simply connected manifold, the dual of the real homotopy vector space π i ( M ) ⊗ ℝ is isomorphic to V i for any i. (For details see, for example, [37] [38].)

Definition 3.2 We will say that a differentiable manifold M is formal if its minimal model is formal or, equivalently, the differential algebras ( Ω * M , d ) and ( H * ( M ) , d = 0 ) 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 [37].

We will also use the following property

Lemma 3.3 Let M 1 and M 2 be differentiable manifolds. Then, the product manifold M = M 1 × M 2 is formal if and only if M 1 and M 2 are formal.

In [39] , the condition of formal manifold is weaken to s-formal manifold as follows.

Definition 3.4 Let ( M , d ) be a minimal model of a differentiable manifold M. We say that ( M , d ) is s-formal, or M is an s-formal manifold ( s ≥ 0 ) if M = Λ V such that for each i ≤ s , the space V i of generators of degree i decomposes as a direct sum V i = C i ⊕ N i , where the spaces C i and N i satisfy the three following conditions:

1) d ( C i ) = 0 ,

2) the differential map d : N i → V is injective,

3) any closed element in the ideal I s = I s ( ⊕ i ≤ s N i ) , generated by ⊕ i ≤ s N i in Λ (   ⊕ i ≤ s V i ) , is exact in Λ V .

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 ( 2 n − 1 ) . Then M is formal if and only if it is ( n − 1 ) -formal.

Let G ( k ) be the simply connected and solvable Lie group of dimension 5 consisting of matrices of the form

a = ( e k x 5 0 0 0 0 x 1 0 e − k x 5 0 0 0 x 2 0 0 e k x 5 0 0 x 3 0 0 0 e − k x 5 0 x 4 0 0 0 0 1 x 5 0 0 0 0 0 1 ) , (4)

where x i ∈ ℝ , for 1 ≤ i ≤ 5 , and k is a real number such that e k + e − k is an integer number different from 2. Then a global system of coordinates { x i ,1 ≤ i ≤ 5 } for G ( k ) is defined by x i ( a ) = x i , and a standard calculation shows that a basis for the right invariant 1-forms on G ( k ) consists of

e 1 = d x 1 − k x 1 d x 5 ,   e 2 = d x 2 + k x 2 d x 5 , e 3 = d x 3 − k x 3 d x 5 ,   e 4 = d x 4 + k x 4 d x 5 , e 5 = d x 5 . (5)

We notice that the Lie group G ( k ) may be described as a semidirect product G ( k ) = ℝ ⋉ ρ k ℝ 4 , where ℝ acts on ℝ 4 via the linear transformation ρ k ( t ) of ℝ 4 given by the matrix

ρ k ( t ) = ( e k t 0 0 0 0 e − k t 0 0 0 0 e k t 0 0 0 0 e − k t ) .

Thus the operation on the group G ( k ) is given by

x ⋅ a = ( x 1 + a 1 e k x 5 , x 2 + a 2 e − k x 5 , x 3 + a 3 e k x 5 , x 4 + a 4 e − k x 5 , x 5 + a 5 ) ,

where a = ( a 1 , ⋯ , a 5 ) and similarly for x . Therefore G ( k ) = ℝ ⋉ ρ k ℝ 4 , where ℝ is a connected abelian subgroup, and ℝ 4 is the nilpotent commutator subgroup.

Now we show that there exists a discrete subgroup Γ ( k ) of G ( k ) such that the quotient space G ( k ) / Γ ( k ) is compact. To construct Γ ( k ) it suffices to find some real number t 0 such that the matrix defining ρ k ( t 0 ) is conjugate to an element A of the special linear group SL ( 4, ℤ ) with distinct real eigenvalues λ and λ − 1 . Indeed, we could then find a lattice Γ 0 in ℝ 4 which is invariant under ρ k ( t 0 ) , and take Γ ( k ) = ( t 0 ℤ ) ⋉ ρ k Γ 0 . To this end, we choose the matrix A ∈ SL ( 4, ℤ ) given by

A = ( 2 1 0 0 1 1 0 0 0 0 2 1 0 0 1 1 ) , (6)

with double eigenvalues 3 + 5 2 and 3 − 5 2 . Taking t 0 = 1 k log ( 3 + 5 2 ) , we have that the matrices ρ k ( t 0 ) and A are conjugate. In fact, put

P = ( 1 1 2 ( 5 − 1 ) 0 0 1 − 1 2 ( 5 + 1 ) 0 0 0 0 1 1 2 ( 5 − 1 ) 0 0 1 − 1 2 ( 5 + 1 ) ) . (7)

Then a direct calculation shows that P A = ρ k ( t 0 ) P . So, if ( m 1 , m 2 , m 3 , m 4 ) t is the transpose of the vector ( m 1 , m 2 , m 3 , m 4 ) , where m 1 , m 2 , m 3 , m 4 ∈ ℤ , the lattice Γ 0 in ℝ 4 defined by

Γ 0 = P ( m 1 , m 2 , m 3 , m 4 ) t , (8)

is invariant under the subgroup ℤ . Thus Γ ( k ) = ( t 0 ℤ ) ⋉ ρ k Γ 0 is a cocompact subgroup of G ( k ) . So, the quotient space

S ( k ) = G ( k ) / Γ ( k ) (9)

is a 5-dimensional compact solvable manifold.

Alternatively, S ( k ) may be viewed as the total space of a T⁴-bundle over the circle S 1 . In fact, let T 4 = ℝ 4 / Γ 0 be the 4-dimensional torus and ν : ℤ → Diff ( T 4 ) the representation defined as follows: ν ( m ) is the transformation of T⁴ covered by the linear transformation of ℝ 4 given by the matrix

ρ k ( m t 0 ) = ( e k m t 0 0 0 0 0 e − k m t 0 0 0 0 0 e k m t 0 0 0 0 0 e − k m t 0 ) .

So ℤ acts on T 4 × ℝ by

( ( x 1 , x 2 , x 3 , x 4 ) , x 5 ) ↦ ( ρ k ( m t 0 ) ⋅ ( x 1 , x 2 , x 3 , x 4 ) t , x 5 + m ) ,

and S is the quotient ( T 4 × ℝ ) / ℤ . The projection π is given by

π [ ( x 1 , x 2 , x 3 , x 4 ) , x 5 ] = [ x 5 ] .

Next, we consider the 7-dimensional compact manifold

M 7 ( k ) = S ( k ) × T 2 , (10)

where T² is the 2-torus T 2 = ℝ 2 / ℤ 2 .

To compute the real cohomology of M 7 ( k ) , we notice that S ( k ) is completely solvable, that is the map a d X : g ( k ) → g ( k ) has only real eigenvalues for all X ∈ g ( k ) , where g ( k ) denotes the Lie algebra of G ( k ) . Thus Hattori’s theorem [40] says that the de Rham cohomology ring H * ( S ( k ) ) is isomorphic to the cohomology ring H * ( g ( k ) * ) of the Lie algebra g ( k ) of G ( k ) . For simplicity we denote the right invariant forms { e i } ( i = 1 , ⋯ , 5 ) on G ( k ) and their projections on S ( k ) by the same symbols. Then, if we denote by e 6 , e 7 the (right invariant) closed 1-forms on the 2-torus T² whose cohomology classes generate the De Rham cohomology group H 1 ( T 2 , ℝ ) , we have that the 1-forms e i ( 1 ≤ i ≤ 7 ) on M 7 ( k ) are such that

d e 1 = − k e 15 ,   d e 2 = k e 25 ,   d e 3 = − k e 35 ,   d e 4 = k e 45 ,   d e i = 0 ,   i = 5 , 6 , 7 , (11)

and such that at each point of M 7 ( k ) , { e 1 , e 2 , e 3 , e 4 , e 5 , e 6 , e 7 } is a basis for the 1-forms on M 7 ( k ) . Here e 15 stands for e 1 ∧ e 5 , and so on. Then, the real cohomology groups of M 7 ( k ) are:

H 0 ( M 7 ( k ) ) = 〈 1 〉 , H 1 ( M 7 ( k ) ) = 〈 [ e 5 ] , [ e 6 ] , [ e 7 ] 〉 , H 2 ( M 7 ( k ) ) = 〈 [ e 12 ] , [ e 14 ] , [ e 23 ] , [ e 34 ] , [ e 56 ] , [ e 57 ] , [ e 67 ] 〉 , H 3 ( M 7 ( k ) ) = 〈 [ e 125 ] , [ e 126 ] , [ e 127 ] , [ e 145 ] , [ e 146 ] , [ e 147 ] , [ e 235 ] ,                                             [ e 236 ] , [ e 237 ] , [ e 345 ] , [ e 346 ] , [ e 347 ] , [ e 567 ] 〉 ,

H 4 ( M 7 ( k ) ) = 〈 [ e 1234 ] , [ e 1256 ] , [ e 1257 ] , [ e 1267 ] , [ e 1456 ] , [ e 1457 ] , [ e 1467 ] ,                                             [ e 2356 ] , [ e 2357 ] , [ e 2367 ] , [ e 3456 ] , [ e 3457 ] , [ e 3467 ] 〉 , H 5 ( M 7 ( k ) ) = 〈 [ e 12345 ] , [ e 12346 ] , [ e 12347 ] , [ e 12567 ] , [ e 14567 ] , [ e 23567 ] , [ e 34567 ] 〉 , H 6 ( M 7 ( k ) ) = 〈 [ e 123456 ] , [ e 123457 ] , [ e 123467 ] 〉 , H 7 ( M 7 ( k ) ) = 〈 [ e 1234567 ] 〉 . (12)

Thus, the Betti numbers of M 7 ( k ) are

b 0 ( M 7 ( k ) ) = b 7 ( M 7 ( k ) ) = 1 , b 1 ( M 7 ( k ) ) = b 6 ( M 6 ( k ) ) = 3 , b 2 ( M 7 ( k ) ) = b 5 ( M 7 ( k ) ) = 7 , b 3 ( M 7 ( k ) ) = b 4 ( M 7 ( k ) ) = 13. (13)

Proposition 4.1. The 5-manifold S ( k ) is 2-formal and so formal. Therefore, M 7 ( k ) = S ( k ) × T 2 is formal.

Proof. To prove that S ( k ) is 2-formal, we see that its minimal model must be a differential graded algebra ( M , d ) , where M is the free algebra of the form M = Λ ( a 1 ) ⊗ Λ ( a 2 , b 2 , c 2 , e 2 ) ⊗ Λ V ≥ 3 , where the generator a 1 has degree 1, the generators a 2 , b 2 , c 2 , e 2 have degree 2, and the differential d is given by d a 1 = d a 2 = d b 2 = d c 2 = d e 2 = 0 . The morphism ρ : M → Ω ( S ( k ) ) , inducing an isomorphism on cohomology, is defined by ρ ( a 1 ) = e 5 , ρ ( a 2 ) = e 12 , ρ ( b 2 ) = e 14 , ρ ( c 2 ) = e 23 and ρ ( e 2 ) = e 34 .

According to Definition 3.4, we get C 1 = 〈 a 1 〉 and N 1 = 0 , thus S ( k ) is 1-formal. Moreover, S ( k ) is 2-formal since C 2 = 〈 a 2 , b 2 , c 2 , e 2 〉 and N 2 = 0 . Hence, S ( k ) is 2-formal, and so formal by Theorem 3.5. Now, Lemma 3.3 implies that M 7 ( k ) = S ( k ) × T 2 is formal.

We define the 3-form φ on M 7 ( k ) given by

φ k = e 127 + e 347 + e 567 + e 135 − e 146 − e 236 − e 245 . (14)

Clearly, φ k is a G₂ form on M 7 ( k ) which is closed. Indeed, on the right-hand side of (14) all the terms are closed, and so φ k is closed. Note that the dual form ⋆ φ k φ k has the following expression

⋆ φ k φ k = e 1234 + e 1256 + e 1367 + e 1457 + e 2357 − e 2467 + e 3456 .

So, taking into account (11) and (12), we see that e 1367 and e 2467 are the unique nonclosed summands in ⋆ φ k φ k . In fact, d ⋆ φ k φ k = 2 k ( e 13567 − e 24567 ) ≠ 0 . Therefore, φ k does not define a torsion-free G₂-structure on M 7 ( k ) .

Now, let H ( k ) be the simply connected solvable (non-nilpotent) Lie group H ( k ) = G ( k ) × ℝ 2 . Then, { e 1 , e 2 , e 3 , e 4 , e 5 , e 6 , e 7 } is a basis for the right invariant 1-forms on H ( k ) and the structure equations of H ( k ) are given by (11). So, the closed G₂ form φ k defined in (14) is a right invariant closed G₂ form on H ( k ) .

Let N be a simply connected solvable Lie group of dimension n, and denote by n 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 R i c ( g ) differs from a derivation D of n by a scalar multiple of the identity map I n , i.e. if there exists a real number λ such that

R i c ( g ) = λ I n + D .

Not all solvable Lie groups admit solsoliton metrics, but if a solsoliton exists, then it is unique up to automorphism and scaling [41].

Proposition 4.2. Let H ( k ) be the seven dimensional Lie group H ( k ) = G ( k ) × ℝ 2 , and let φ k be the right invariant closed G₂ form on H ( k ) defined in (14). Then the metric g k determined by φ k is a solsoliton on H ( k ) .

Proof. Clearly, the metric g k induced on H ( k ) by φ k is such that the basis { e 1 , e 2 , e 3 , e 4 , e 5 , e 6 , e 7 } for the 1-forms on H ( k ) is orthonormal, that is g k = ∑ i = 1 7 ( e i ) 2 . Then, g k is a solsoliton since

R i c ( g k ) = diag ( 0,0,0,0, − 4 k 2 ,0,0 ) = − 4 k 2 I 7 + D ,

where

D = diag ( 4 k 2 ,4 k 2 ,4 k 2 ,4 k 2 ,0,4 k 2 ,4 k 2 ) ,

is a derivation of the Lie algebra h ( k ) of H ( k ) .

In this section, we show associative 3-folds of the compact G₂-calibrated solvmanifold M 7 ( k ) defined in (10) with the closed G₂ form φ k given by (14). First, we need some definitions and results about calibrations (see [14] [15] for details).

Let ( M , g ) be a Riemannian manifold. An oriented tangent k-plane V on M is a vector subspace V of some tangent space T p M to M, with dim V = k and equipped with an orientation. If V is an oriented tangent k-plane on M, then g | V is a Euclidean metric on V. So, combining g | V with the orientation on V gives a natural volume form v o l V on V, which is a k-form on V.

Let θ a closed k-form on a Riemannian manifold ( M , g ) . We say that θ is a calibration on M if for any p ∈ M and every oriented k-dimensional subspace V of the tangent space T p M we have θ | V = λ v o l V , for some λ ≤ 1 (see [14] and [15] 3.7). Thus, if Y is an oriented submanifold of M with dimension k then, for any p ∈ Y , the tangent space T p Y is an oriented tangent k-plane on M. We say that Y is a calibrated submanifold if θ ( p ) | T p Y = v o l T p Y , for all p ∈ Y .

All calibrated submanifolds are minimal submanifolds. Even more, every compact calibrated submanifold is volume-minimizing in its homology class ( [15] Proposition~3.7.2).

Harvey and Lawson in [14] proved that any closed G₂ form φ on a 7-manifold M is a calibration on M. The 3-dimensional orientable submanifolds Y ⊂ M calibrated by the G₂ form φ , i.e. those submanifolds Y ⊂ M that satisfy φ ( p ) | T p Y = v o l Y ( p ) , for each p ∈ Y and for some unique orientation of Y, are called associative 3-folds.

Next, we shall produce examples of associative 3-folds in M 7 ( k ) from the fixed locus of a G₂-involution of the compact manifold M 7 ( k ) applying the following.

Proposition 5.1 ( [15] [Proposition 10.8.1]) Let N be a 7-manifold with a closed G₂ form ϕ , and let σ : N → N be an involution of N satisfying σ * ϕ = ϕ and such that σ is not the identity map. Then the fixed point set P = { p ∈ N | σ ( p ) = p } 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 [15] 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 M 7 ( k ) , which define nine embedded, associative (calibrated by φ k ), minimal 3-tori in M 7 ( k ) .

Proof. Let H ( k ) be the seven dimensional Lie group H ( k ) = G ( k ) × ℝ 2 defined in Proposition 4.2. We consider on H ( k ) the involution given by

σ : ( x 1 , x 2 , x 3 , x 4 , x 5 , x 6 , x 7 ) ↦ ( − x 1 , − x 2 , − x 3 , − x 4 , x 5 , x 6 , x 7 ) , (15)

that is σ is the product of the involutions σ 1 : G ( k ) → G ( k ) with the identity map of ℝ 2 , where σ 1 is defined by

σ 1 : ( x 1 , x 2 , x 3 , x 4 , x 5 ) ↦ ( − x 1 , − x 2 , − x 3 , − x 4 , x 5 ) .

The involution σ 1 is such that σ 1 ( Γ ( k ) ) = Γ ( k ) , and so σ 1 descends to the 5-dimensional compact manifold S ( k ) = G ( k ) / Γ ( k ) . Hence, σ defines also an involution of M 7 ( k ) . From now on, we denote by

σ : M 7 ( k ) → M 7 (k)

the involution of M 7 ( k ) induced by the involution σ of H ( k ) defined in (15). Then, taking into account (5), we have that the induced action on the 1-forms e i is given by

σ * e i = − e i ,     i = 1 , 2 , 3 , 4 ,   ρ * e j = e j ,     j = 5 , 6 , 7. (16)

Therefore, the G₂ form φ k on M 7 ( k ) defined in (14) is preserved by the involution σ of M 7 ( k ) . In fact, by (16), each term on the right-hand side of (14) is σ-invariant.

Let P be the fixed locus of σ . Then, P consists of all the 3-dimensional spaces P a given as follows:

P a = { ( a 1 , a 2 , a 3 , a 4 , x 5 , x 6 , x 7 ) | ( x 5 , x 6 , x 7 ) ∈ T 3 } ⊂ M 7 ( k ) ,

where a = ( a 1 , a 2 , a 3 , a 4 ) with

( a 1 , a 2 ) , ( a 3 , a 4 ) ∈ { ( 0,0 ) , ( 1 2 , 1 2 ) , 1 4 ( 5 − 1 ) , − 1 4 ( 5 + 1 ) } .

Consequently, P is a disjoint union of 9 copies of a 3-torus T³.

Since the G₂ form φ k on M 7 ( k ) defined in (14) is preserved by the involution σ of M 7 ( k ) , each of the 9 torus P a in M 7 ( k ) fixed by σ : M 7 ( k ) → M 7 ( k ) is an associative 3-fold in ( M 7 ( k ) , φ k ) by Proposition 5.1.

The purpose of this section is to prove that the Laplacian flow of φ k on the 7-dimensional Lie group H ( k ) exists for all time. Moreover, we prove that the Ricci endomorphisms R i c ( g k ( t ) ) of the underlying metrics g k ( t ) of the solution φ k ( t ) are independent of the time t, and so the solution φ k ( t ) 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 φ 0 . The Laplacian flow starting from φ 0 is the initial value problem

{ d d t φ ( t ) = Δ t φ ( t ) , d φ ( t ) = 0 , φ ( 0 ) = φ 0 . (17)

where Δ t denotes the Hodge Laplacian of the Riemannian metric g ( t ) induced by φ ( t ) . This flow was introduced by Bryant in [16] 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 [20].

Theorem 6.1 Assume that M is compact. Then, the Laplacian flow (17) has a unique solution defined for a short time t ∈ [ 0, ε ) , with ε depending on φ 0 .

In the following theorem, we determine a global solution of the Laplacian flow of the closed G₂ form φ k given by (14) on the Lie group H ( k ) = G ( k ) × ℝ 2 , where G ( k ) is the Lie group defined in Section 4.

Theorem 6.2 On the simply connected solvable (non-nilpotent) Lie group H ( k ) = G ( k ) × ℝ 2 , the solution of the Laplacian flow (17) starting from the calibrated G₂-structure φ k is given by

φ k ( t ) = e 127 + e 347 + e 567 + ( 16 3 k t + 1 ) 3 / 4 e 135 − e 236 − e 146 − ( 16 3 k t + 1 ) 3 / 4 e 245 , (18)

where t ∈ ( − 3 16   k 2 , + ∞ ) .

Proof. Let f i = f i ( t ) ( i = 1 , ⋯ , 7 ) be some differentiable real functions depending on a parameter t ∈ I ⊂ ℝ such that f i ( 0 ) = 1 and f i ( t ) ≠ 0 , for any t ∈ I , where I is a real open interval. For each t ∈ I , we consider the basis { x 1 , ⋯ , x 7 } of left invariant 1-forms on H ( k ) defined by

x i = x i ( t ) = f i ( t ) e i ,   1 ≤ i ≤ 7.

Taking into account (11), the structure equations of H ( k ) with respect to the basis { x 1 , ⋯ , x 7 } are

d x 1 = − k 1 f 5 x 15 ,   d x 2 = k 1 f 5 x 25 , d x 3 = − k 1 f 5 x 35 ,   d x 4 = k 1 f 5 x 45 , d x 5 = d x 6 = d x 7 = 0. (19)

From now on, we write f i j = f i j ( t ) = f i ( t ) f j ( t ) , f i j k = f i j k ( t ) = f i ( t ) f j ( t ) f k ( t ) , and so forth. Then, for any t ∈ I , we consider the G₂-structure φ k ( t ) on H ( k ) given by

φ k ( t ) = x 127 + x 347 + x 567 + x 135 − x 146 − x 236 − x 245 = f 127 e 127 + f 347 e 347 + f 567 e 567 + f 135 e 135 − f 146 e 146 − f 236 e 236 − f 245 e 245 . (20)

Note that the 3-form φ k ( t ) defined by (20) is such that φ k ( 0 ) = φ k and, for any t, φ k ( t ) determines the metric g k ( t ) on H ( k ) such that the basis

{ x i = 1 f i e i ; i = 1 , ⋯ , 7 } of left invariant vector fields on H ( k ) dual to { x 1 , ⋯ , x 7 } is orthonormal. Moreover, by (19), φ k ( t ) is closed, for any t ∈ I . Therefore, to solve the flow (17) of φ k it is sufficient to determine the functions f i and the interval I so that d d t φ k ( t ) = Δ t φ k ( t ) , for t ∈ I .

Clearly Δ t φ k ( t ) = − d ⋆ t d ⋆ t φ k ( t ) since d φ k ( t ) = 0 . Moreover,

⋆ t φ k ( t ) = x 1234 + x 1256 + x 1367 + x 1457 + x 2357 − x 2467 + x 3456 .

So, x 1367 and x 2467 are the unique nonclosed summands in ⋆ t φ k ( t ) . Then, taking into account (19), we obtain

Δ t φ k ( t ) = 4 k 2 f 5 2 ( x 135 − x 245 ) .

Thus, in terms of the forms e i j k , the expression of Δ t φ k ( t ) becomes

Δ t φ k ( t ) = 4 k 2 f 5 ( f 13 e 135 − f 24 e 245 ) . (21)

On the other hand,

d d t φ k ( t ) = ( f 127 ) ′ e 127 + ( f 347 ) ′ e 347 + ( f 567 ) ′ e 567 + ( f 135 ) ′ e 135     − ( f 146 ) ′ e 146 − ( f 236 ) ′ e 236 − ( f 245 ) ′ e 245 . (22)

Comparing (21) and (22) we have that d d t φ k ( t ) = Δ t φ k ( t ) if and only if the functions f i satisfy the following equations

( f 127 ) ′ = ( f 347 ) ′ = ( f 567 ) ′ = ( f 236 ) ′ = ( f 146 ) ′ = 0 , (23)

( f 135 ) ′ = 4 k 2 f 13 f 5 , (24)

( f 245 ) ′ = 4 k 2 f 24 f 5 . (25)

The equations (23) with the initial conditions f i ( 0 ) = 1 ( i = 1 , ⋯ , 7 ) imply

f 127 = f 347 = f 567 = f 236 = f 146 = 1.

Now, the equalities f 127 = f 347 and f 236 = f 146 imply f 12 = f 34 and f 23 = f 14 , respectively, and thus

f 1 = f 3         and         f 2 = f 4 . (26)

Moreover, from f 127 = f 236 = 1 we have

f 6 = f 7 = 1 / f 12 , (27)

and from f 567 = 1 we have

f 5 = f 12 2 . (28)

Now, using (26) and (28), the system of differential equations formed by the Equations (24) and (25) is written as

{ ( f 1 2 ( f 1 f 2 ) 2 ) ′ = 4 k 2 1 f 2 2 , ( f 2 2 ( f 1 f 2 ) 2 ) ′ = 4 k 2 1 f 1 2 . (29)

Multiplying the first equation of (29) by f 2 2 , and the second one by f 1 2 , one can check that (29) implies that

f 2 2 ( f 1 4 f 2 2 ) ′ = f 1 2 ( f 1 2 f 2 4 ) ′ ,

that is,

f 2 ( f 1 ) ′ = f 1 ( f 2 ) ′ .

Then, using that f 1 ( 0 ) = f 2 ( 0 ) = 1 , we have

f 1 = f 2 . (30)

Thus, the system (29) is written as follows

( f 1 6 ) ′ = 4 k 2 1 f 1 2 .

Integrating this equation, we obtain

3 4 f 1 8 = 4 k 2 t + C ,

for some constant C ∈ ℝ . But the initial condition f 1 ( 0 ) = 1 implies C = 3 4 , and hence

f 1 ( t ) = 16 3 k 2 t + 1 8 . (31)

From (26), (27), (28), (30) and (31), we get

f 1 ( t ) = f 2 ( t ) = f 3 ( t ) = f 4 ( t ) = 16 3 k 2 t + 1 8 , f 5 ( t ) = 16 3 k 2 t + 1 ,     f 6 ( t ) = f 7 ( t ) = 1 16 3 k 2 t + 1 4 .

Therefore, taking into account (20), the family of closed G₂ forms φ k ( t ) given by (18) is the solution of the Laplacian flow of φ k on H ( k ) , and it is defined for all t ∈ ( − 3 16   k 2 , + ∞ ) . □

Remark 6.3 Note that the metric g k ( t ) , with t ∈ ( − 3 16   k 2 , + ∞ ) , is a

solsoliton on H ( k ) . In fact, the metric g k ( t ) with respect to the basis { e 1 , ⋯ , e 7 } is given by

g k ( t ) = diag ( f 1 2 , f 1 2 , f 1 2 , f 1 2 , f 1 8 , 1 f 1 4 , 1 f 1 4 ) ,

where f 1 = f 1 ( t ) is the function given by (31). Then, the Ricci endomorphism R i c ( g k ( t ) ) satifies

R i c ( g k ( t ) ) = diag ( 0,0,0,0, − 4 k 2 ,0,0 ) = − 4 k 2 I 7 + D ,

where

D = diag ( 4 k 2 ,4 k 2 ,4 k 2 ,4 k 2 ,0,4 k 2 ,4 k 2 ) ,

is a derivation of the Lie algebra h ( k ) of H ( k ) . Moreover, R i c ( g k ( t ) ) on H ( k ) is non-zero and independent of the time t. So, the solution φ k ( t ) 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 R ( g k ( t ) ) we obtain

R 1212 = R 1414 = R 2323 = R 3434 = k 2 f 1 4 , R 1313 = R 2424 = − k 2 f 1 4 , R 1515 = R 2525 = R 3535 = R 4545 = − k 2 f 1 2 , R i j k l = 0             otherwise,

where R i j k l = R ( g k ( t ) ) ( e i , e j , e k , e l ) . Thus, the Riemannian curvature R ( g k ( t ) ) 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.

The authors declare no conflicts of interest regarding the publication of this paper.

Fernández, M., Sanchez, J. and Santisteban, J.A. (2018) Compact Solvmanifolds with a Closed G₂-Structure. Advances in Pure Mathematics, 8, 813-829. https://doi.org/10.4236/apm.2018.811050