1. Introduction
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
(1)
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 [1] , 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 [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
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
, then
is a calibration [14]. 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 [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
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 [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
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 [25] 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 [1] [2] [7] [14] [15] [16] [30] [31] [32] [33] 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
(2)
(see [1] [32] [33] [34] 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 [35] , 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 [14].
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,
(3)
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 [36] 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 [37] 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, [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
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 [37].
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 [39] , 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:
1)
,
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
(4)
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
(5)
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
(6)
with double eigenvalues
and
. Taking
, we have that the matrices
and A are conjugate. In fact, put
(7)
Then a direct calculation shows that
. So, if
is the transpose of the vector
, where
, the lattice
in
defined by
(8)
is invariant under the subgroup
. Thus
is a cocompact subgroup of
. So, the quotient space
(9)
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
(10)
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 [40] 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
(11)
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:
(12)
Thus, the Betti numbers of
are
(13)
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
(14)
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 [41].
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
where
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 [14] [15] 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 [14] and [15] 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 ( [15] Proposition~3.7.2).
Harvey and Lawson in [14] 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 ( [15] [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 [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
, 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
(15)
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
(16)
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:
where
with
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
(17)
where
denotes the Hodge Laplacian of the Riemannian metric
induced by
. 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
, 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
(18)
where
.
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
(19)
From now on, we write
,
, and so forth. Then, for any
, we consider the G2-structure
on
given by
(20)
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
(21)
On the other hand,
(22)
Comparing (21) and (22) we have that
if and only if the functions
satisfy the following equations
(23)
(24)
(25)
The equations (23) with the initial conditions
imply
Now, the equalities
and
imply
and
, respectively, and thus
(26)
Moreover, from
we have
(27)
and from
we have
(28)
Now, using (26) and (28), the system of differential equations formed by the Equations (24) and (25) is written as
(29)
Multiplying the first equation of (29) by
, and the second one by
, one can check that (29) implies that
that is,
Then, using that
, we have
(30)
Thus, the system (29) is written as follows
Integrating this equation, we obtain
for some constant
. But the initial condition
implies
, and hence
(31)
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
where
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.
Acknowledgements
The authors were partially supported by MINECO-FEDER Grant MTM2014-54804-P and Gobierno Vasco Grant IT1094-16, Spain.