# Remarks on Exact G2-Structures on Compact Manifolds

Aaron Kennon  Department of Physics UCSB<EMAIL_ADDRESS>
###### Abstract.

An important open question in G2 geometry concerns whether or not a compact
seven-manifold can support an exact G2-Structure. Given the significance of
this question we initiate a study of exact G2-Structures on compact manifolds.
We focus on exact G2-Structures subject to no additional constraints but we
also consider the relationship between the exact condition and other
conditions for closed G2-Structures such as the Extremally Ricci-Pinched and
Laplacian Soliton conditions.

The research of the author is partially suported by Simons Foundation Award
#488629 (Morrison)

###### Contents

  1. 1 Introduction
  2. 2 Background on G2-Structures
  3. 3 Constraints on the Two-Form Underlying an Exact G2-Structure
  4. 4 Exact G2-Structures Subjected to Additional Constraints
  5. 5 The Curvature Tensor of Exact G2-Structures

##  1\. Introduction

Manifolds with G2-holonomy metrics are quite distinguished geometric objects.
From the perspective of mathematics they are significant because the Lie group
G2 is the only possible special holonomy group for a simply-connected,
irreducible, non-symmetric Riemannian seven-manifold [1]. Such manifolds are
also Ricci-flat and admit a parallel spinor [2] which makes them very
interesting as backgrounds for M-Theory compactifications to four-dimensional
Minkowski space. [3, 4, 5]

In order for a manifold to have G2 holonomy it needs to admit a torsion-free
G2-Structure. Such a structure is equivalent to the existence of a
distinguished harmonic three-form $\varphi$ pointwise identified with a model
form on $\mathbb{R}^{7}$. Provided that a compact manifold admits a torsion-
free G2-Structure the manifold has holonomy precisely equal to G2 if and only
if the fundamental group is finite.

All known constructions of G2-holonomy metrics on compact manifolds start from
closed G2-Structures which are weaker structures than torsion-free
G2-Structures where the three-form $\varphi$ is only required to be closed and
not necessarily co-closed. Not much is known about these closed G2-Structures
and understanding them better is an important step toward determining when a
manifold admits torsion-free G2-Structures.

In particular, it is not known if a compact manifold can admit an exact
G2-Structure. Of course, on any manifold with vanishing third Betti number any
closed G2-Structure will have to be exact. We can refine this question by
asking if compact manifolds subject to additional constraints can admit exact
G2-Structures. If exact G2-Structures cannot exist on manifolds with vanishing
first Betti number, then that would mean that a manifold which admits a
G2-Holonomy metric could not also admit an exact G2-Structure. Similarly, if
an exact G2-Structure could not exist on a manifold with nonvanishing third
Betti number, then a manifold which admits a torsion-free G2-Structure of any
holonomy could not admit an exact G2-Structure.

The case of closed G2-Structures should be contrasted with that of co-closed
G2-Structures. In that case we have a parametric h-principle which says that
any compact, orientable, spinnable seven-manifold admits a co-closed
G2-Structure C0 close to any given G2-Structure [11]. As a consequence, co-
closed G2-Structures are very plentiful even on compact manifolds. It is known
that pointwise the closed condition is stronger than the co-closed condition,
but it is unclear precisely how restrictive the closed conditon is on compact
manifolds. If it turns out that exact G2-Structures on compact manifolds do
not exist, then compact manifolds with vanishing third Betti number cannot
admit closed G2-Structures at all, which would imply it is more difficult to
satisfy the closed condition on compact manifolds than the co-closed
condition. Such a result would be the first nonexistence result for closed
G2-Structures subject to no additional constraints on compact manifolds that
otherwise admit G2-Structures.

We begin by introducing background on G2-Structures in §2. We then prove
pointwise identities for exact G2-Structures and use them to prove results on
the two-form underlying an exact G2-Structure on a compact manifold in §3. We
finish by considering additional constraints on exact G2-Structures on compact
manifolds in §4 and results on the Riemann curvature in §5.

##  2\. Background on G2-Structures

Here we include a general introduction to G2-Structures on seven-manifolds.
The material here may be found in the review by Bryant [13], the review by
Karigiannis [16], or the monograph by Joyce [6].

A G2-Structure is a principle subbundle of the frame bundle with structure
group G2. A seven-manifold admits G2-Structures if and only if it is
orientable and spin. We may also characterize the existence of a G2-Structure
in terms of a three-form pointwise identified with the three-form
$\varphi_{0}$ on $\mathbb{R}^{7}$.

$\varphi_{0}=dx^{123}-dx^{167}-dx^{527}-dx^{563}+dx^{415}+dx^{426}+dx^{437}$
(2.1)

Such a three-form is called positive and we also refer to such a form as the
G2 three-form. A positive three-form defines a Riemannian metric and also
determines a four-form by taking the Hodge star of the three-form with respect
to this metric. Since both the bundle of positive three-forms and that of
three-forms on a seven-manifold have rank 35 positivity is an open condition.
In other words, if we have a G2-Structure and perturb it by a three-form with
sufficiently small C0 norm, then the resulting form is also a G2-Structure.

On a manifold with a positive three-form, we may decompose the spaces of
differential k-forms into irreducible G2 representations. The equations
defining the various representations are then all given in terms of the G2
three-form $\varphi$ and G2 four-form $\psi$ [18].

$\displaystyle\Lambda^{1}(T^{*}M)=\Lambda^{1}_{8}(T^{*}M)$ (2.2)
$\displaystyle\Lambda^{2}(T^{*}M)=\Lambda^{2}_{7}(T^{*}M)\oplus\Lambda^{2}_{14}(T^{*}M)$
(2.3)
$\displaystyle\Lambda^{3}(T^{*}M)=\Lambda^{3}_{1}(T^{*}M)\oplus\Lambda^{3}_{7}(T^{*}M)\oplus\Lambda^{3}_{27}(T^{*}M)$
(2.4)

$\displaystyle\Lambda^{2}_{7}(T^{*}M)=\\{\beta\in\Lambda^{2}(T^{*}M):*(\psi\wedge\beta)=2\beta\\}$
(2.5)
$\displaystyle\Lambda^{2}_{14}(T^{*}M)=\\{\beta\in\Lambda^{2}(T^{*}M):*(\psi\wedge\beta)=-\beta\\}$
(2.6) $\displaystyle\Lambda^{3}_{1}(T^{*}M)=\\{f\varphi\hskip
5.69054pt\textnormal{for}\hskip 2.84526ptf\in C^{\infty}(M)\\}$ (2.7)
$\displaystyle\Lambda^{3}_{7}(T^{*}M)=\\{X\lrcorner\psi\hskip
5.69054pt\textnormal{for}\hskip 2.84526ptX\in\Gamma(M)\\}$ (2.8)
$\displaystyle\Lambda^{3}_{27}(T^{*}M)=\\{\eta\in\Lambda^{3}(T^{*}M):\eta\wedge\varphi=\eta\wedge\psi=0\\}$
(2.9)

We may also write out the representation equations in local coordinates. For
instance, for two-forms the equations defining the representations may be
expressed easily in terms of contractions between the relevant tensors.

$\displaystyle\beta\in\Lambda^{2}_{7}(T^{*}M)\hskip
5.69054pt\textnormal{iff}\hskip 5.69054pt\beta_{ij}\psi_{abij}=4\beta_{ab}$
(2.10) $\displaystyle\beta\in\Lambda^{2}_{14}(T^{*}M)\hskip
5.69054pt\textnormal{iff}\hskip 5.69054pt\beta_{ij}\varphi_{ijk}=0$ (2.11)

There is an important isomorphism of G2 representations $\i_{\varphi}$ which
maps the space of symmetric, traceless two-tensors S${}_{0}^{2}(M)$ onto
$\Lambda_{27}^{3}(T^{\ast}M)$. The mapping may be defined explicitly on
decomposable, symmetric, traceless two tensors $\alpha\circ\beta$ [13].

$\i_{\varphi}(\alpha\circ\beta)=\alpha\wedge\ast(\beta\wedge\ast\varphi)+\beta\wedge\ast(\alpha\wedge\ast\varphi)$
(2.12)

As any symmetric traceless two-tensor is built out of decomposable elements we
can define this map for an arbitrary such tensor by extending by linearity. We
also may explicitly invert the isomorphism $i_{\varphi}$ to get an isomorphism
$\j_{\varphi}$ which maps an element $\gamma\in\Lambda_{27}^{3}(T^{\ast}M)$ to
a symmeteric traceless two-tensor with vector arguments $v$ and $w$.

$\j_{\varphi}(\gamma)(v,w)=\ast\bigl{(}(v\lrcorner\varphi)\wedge(w\lrcorner\varphi)\wedge\gamma\bigr{)}$
(2.13)

In order to ensure a holonomy reduction, we need to find a G2-Structure that
is torsion-free. The G2-Structure being torsion-free is equivalent to the
positive three-form being closed and co-closed with respect to the Riemannian
metric determined by the three-form. On a compact manifold, a differential
form being both closed and co-closed is equivalent to it being harmonic.

$\displaystyle d\varphi=0$ (2.14) $\displaystyle d\ast\varphi=0$ (2.15)

Provided we have a torsion-free G2-Structure, the holonomy of the
corresponding Levi-Civita connection is known to be contained in G2 but it may
be a proper subgroup. The possibilities are the trivial group, SU(2), and
SU(3). If the holonomy is trivial, then the corresponding connection is flat
as a corollary of the Ambrose-Singer theorem [6]. Given a torsion-free
G2-Structure, we have straightforward means to ensure the holonomy is
precisely G2. On a compact manifold we can do so by ensuring the fundamental
group is finite [6] and on a simply-connected complete manifold we may ensure
that there are no parallel one-forms [14].

We may generalize the torsion-free condition by studying positive three-forms
that are not necessarily harmonic. For an arbitrary G2-Structure we may
decompose the exterior derivatives of the G2 three-form and four-form into
irreducible G2 representations.

$\displaystyle d\varphi=\tau_{0}\psi+3\tau_{1}\wedge\phi+\ast\tau_{3}$ (2.16)
$\displaystyle d\psi=4\tau_{1}\wedge\psi+\ast\tau_{2}$ (2.17)

This decomposition identifies special functions and differential forms
$\tau_{0}\in C^{\infty}(M)$, $\tau_{1}\in\Gamma(M)$,
$\tau_{2}\in\Lambda^{2}_{14}(T^{*}M)$, and
$\tau_{3}\in\Lambda^{3}_{27}(T^{*}M)$. There are then sixteen possibilities
for types of G2-Structures depending on which of these torsion forms vanish.
We can amalgamate all of these torsion forms into a tensor called the torsion
tensor. We may express this tensor in terms of the covariant derivative of the
three-form and the four-form in coordinates.

$T_{lm}=1/24(\nabla_{l}\varphi_{abc})\psi_{mabc}$ (2.18)

We may also express the torsion tensor in terms of the torsion forms
$\tau_{i}$.

$T_{lm}=\frac{\tau_{0}}{4}g_{lm}-(\tau_{3})_{lm}-(\tau_{1})_{lm}-\frac{1}{2}(\tau_{2})_{lm}$
(2.19)

Here $\tau_{1}$, which properly speaking is a one-form, is identified with a
two-form in the $\Lambda^{2}_{7}(T^{*}M)$ representation of G2 on two-forms.
The curvature of the metric associated to the given G2-Structure is completely
determined by the torsion tensor. An expression for the Riemann curvature in
terms of the torsion tensor is given in the work of Karigiannis [17], and
expressions for the Ricci tensor and scalar curvature of arbitrary
G2-Structures in terms of the torsion forms are given by Bryant [13].

One particularly interesting class of G2-Structures corresponds to the case
when the positive three-form is closed but not necessarily co-closed. Such
G2-Structures are called closed G2-Structures and are sometimes also called
calibrated G2-Structures. Looking at the expression for the exterior
derivative of $\varphi$ for a closed G2-Structure, it is apparent that every
torsion component vanishes except for $\tau_{2}$. As we will be discussing
closed G2-Structures exclusively, we will abbreviate $\tau_{2}$ by $\tau$.

Closed G2-Structures are significant as they are used in constructions for
manifolds with holonomy G2. The idea is to construct a closed G2-Structure
whose torsion is sufficiently small and then use perturbative analysis to
deform the solution to a torison-free G2-Structure. The precise criterion
needed to perturb these structures are encapsulated in a theorem of Joyce [7]
and a lot of effort in G2-Geometry is devoted to the question of when a closed
G2-Structure can be deformed into a torsion-free G2-Structure. For instance,
providing a refinement of our understanding of this issue is a primary
motivation of studying the Laplacian flow of closed G2-Structures [12].

The expressions for the torsion tensor and the various cuvature tensors
simplify considerably when the G2-Structure is closed. The torsion tensor is
then directly proportional to the torsion two-form $\tau$ which implies that
the torsion tensor is totally antisymmetric for a closed-G2-Structure. The
Ricci curvature and scalar curvature expressions also simplify considerably.

$\displaystyle\textnormal{Ric}(g_{\varphi})$
$\displaystyle=\biggl{(}\frac{1}{4}\,|\tau|^{2}\biggl{)}\,g_{\varphi}-\j_{\varphi}\Bigl{(}\frac{1}{4}\,d\tau-\frac{1}{8}\,\ast(\tau\wedge\tau\Bigr{)}$
(2.20) $\displaystyle\textnormal{Scal}(g_{\varphi})$
$\displaystyle=-\frac{1}{2}|\tau|^{2}$ (2.21)

In particular, the scalar curvature is non-positive and vanishes identically
if and only if the G2-Structure is torsion-free. In this case the metric is
not only scalar-flat but also Ricci-flat.

Besides these results concerning the torsion and curvature relatively little
is known about these structures. There are now several methods of constructing
closed G2-Structures in the literature. The original approach utilized
symmetries to simplify the PDE system $d\varphi=0\hskip 5.69054pt$[19, 20,
21]. The examples have the form $M=\Gamma/G$ where $G$ is a seven-dimensional
simply-connected nilpotent or more general solvable Lie group and $\Gamma$ is
a co-compact discrete subgroup of $G$. The closed G2-Structures on these
spaces come from left-invariant ones on $G$. An advantage of these examples is
that the G2-Structure is determined by the three-form on the Lie algebra of G
and the closed condition reduces to an algebraic equation.

A newer method to constuct closed G2-Structures on compact manifolds is based
on the known constructions of G2 holonomy manifolds. A preliminary step for
constructing these G2 holonomy metrics is to get a closed G2-Structure on the
manifold with small torsion so these methods can be thought of as
constructions for closed G2-Structures with more stringent requirements. Of
course we may disregard these requirements if the goal is just to get closed
G2-Structures.

One G2 holonomy construction which has been considered in the context of
closed G2-Structures is Joyce’s orbifold construction [7, 8]. In the original
version Joyce starts with the standard torsion-free G2-Structure on $T^{7}$
(with trivial holonomy) and acts on this space with a finite group which
preserves the G2-Structure. Then he glues in spaces which resolve these
orbifold singularities producing a smooth manifold. On this manifold he uses
the original G2-Structures to build a new G2-Structure but this G2 three-form
only closed as opposed to harmonic. These structures have arbitrarily small
torsion depending on parameters involved in the gluing. For the closed
G2-Structure construction the starting manifold is a compact nilmanifold
$M=\Gamma/N$ for a particular nilpotent Lie group N [22]. Then for a
particular finite group the resulting orbifold may be desingularized and shown
to carry a closed G2-Structure. The resulting manifold is interesting because
it has nonvanishing first Betti number so it cannot admit a G2 holonomy
metric.

To date there are no known examples of compact manifolds which admit
G2-Structures but do not admit closed G2-Structures. However, there are a
number of geometric constraints that closed G2-Structures are known not to be
able to satisfy. Several of these have to do with the curvature of the metrics
determined by these structures. For instance, the metric corresponding to a
closed G2-Structure on a compact manifold cannot be Einstein and also cannot
be too Ricci-pinched [13, 23]. The automorphism group of these structures is
also constrained by the topology of the underlying manifold [25]. We will see
that many of these results which are known to hold for closed G2-Structures
generally may be strengthened when the G2-Structure is exact.

There are also several classes of exact G2-Structures on non-compact manifolds
that have recently been produced. Examples on unimodular Lie algebras with
vanishing third Betti number are considered by Fernandez, Fino, and Raffero
[26], and examples which are closed G2-Eigenforms are considered by Friebert
and Salamon [27]. It is worthwhile to note that there are no compact quotients
of the unimodular Lie algebra examples, and it is known that a compact
manifold cannot admit closed G2-eigenforms [12].

##  3\. Constraints on the Two-Form Underlying an Exact G2-Structure

We start by working out a number of projection identities for differential
forms onto irreducible G2 components that hold generally when the G2-Structure
is closed. For this work as we are mostly interested in exact G2-Structures,
which are defined by a two-form, the two-form projection identities are
particularly significant.

######  Lemma 3.1 (Projection identities for derivatives of two-forms).

If (M, $\varphi$) is a manifold with closed G2-Structure $\varphi$ , X is a
vector field, and $\alpha\in\Lambda^{2}_{14}(T^{*}M)$ then the type
decompositions of $d(X\lrcorner\varphi)$ and $d\alpha$ may be given in terms
of X, $\alpha$, and $\gamma_{1},\gamma_{2}\in\Lambda^{3}_{27}(T^{*}M)$.

$\displaystyle
d(X\lrcorner\varphi)=-3/7(d^{\ast}X)\varphi-1/2(d^{\ast}(X\lrcorner\varphi))\lrcorner\psi+\gamma_{1}$
(3.1) $\displaystyle
d\alpha_{14}=1/7\langle\alpha_{14},\tau\rangle\varphi+(1/4d^{\ast}\alpha_{14})\lrcorner\psi+\gamma_{2}$
(3.2)

Proof. $d(X\lrcorner\varphi)$ and $d\beta_{14}$ are both three-forms so they
will take the form $f\varphi+Y\lrcorner\psi+\gamma$ for some function $f$,
vector field $Y$ and three-form $\gamma\in\Lambda^{3}_{27}(T^{*}M)$. In both
cases we can isolate the one dimensional component by wedging the relevant
three-form with $\psi$. For $d(X\lrcorner\varphi)$ doing so gives a factor of
$d^{\ast}X$.

$d(X\lrcorner\varphi)\wedge\psi=7f\textnormal{vol}$ (3.3)

We can then use the fact that the exterior derivative is a derivation to
rearrange this expression.

$d(X\lrcorner\varphi\wedge\psi)-X\lrcorner\varphi\wedge\tau\wedge\varphi=7f\textnormal{vol}$
(3.4)

However, $X\lrcorner\varphi$ and $\tau$ are pointwise orthogonal so the second
term vanishes identically. The argument in the differential also simplifies
considerably [18].

$X\lrcorner\varphi\wedge\psi=3\ast X$ (3.5)

We may then take the Hodge star to get the desired expression for $f$. For
$d\alpha_{14}$ wedging with $\psi$ gives the pointwise inner product of
$\alpha_{14}$ with the torsion two-form $\tau$.

$d\alpha_{14}\wedge\psi=7f\textnormal{vol}$ (3.6)

We may then rearrange this expression and use the fact that
$\alpha_{14}\wedge\psi=0$ to eliminate the exact term.

$-\alpha_{14}\wedge\tau\wedge\varphi=7f\textnormal{vol}$ (3.7)

But the left-hand side is just an expression for the pointwise interior
product of $\alpha_{14}$ and $\tau$. The result follows.

To isolate the seven-dimensional components of these forms we need to wedge
the relevant forms by $\varphi$. For $d(X\lrcorner\varphi)$ we start by using
G2 identities to get an expression for the Hodge dual of Y [16].

$d(X\lrcorner\varphi)\wedge\varphi=-4\ast Y$ (3.8)

Then we can use the Leibnitz rule to get the answer solely in terms of the
vector field X.

$d(X\lrcorner\varphi\wedge\varphi)=-1/2d\ast(X\lrcorner\varphi)=-4\ast Y$
(3.9)

We can then take the Hodge star to isolate the vector field Y.

$-1/2d^{\ast}(X\lrcorner\varphi)=Y$ (3.10)

For $\alpha_{14}$ we can follow a similar procedure.

$d\alpha_{14}\wedge\varphi=-4\ast Y$ (3.11)

We can use the equation that defines the 14-dimensional representation to
simplify the last equation and get the answer in terms of the co-exterior
derivative of $\alpha_{14}$.

$1/4(d^{\ast}\alpha_{14})=Y$ (3.12)

All that is left is the twenty-seven dimensional components of these forms
which may be thought of as what is left after subtracting off the components
corresponding to the one and seven-dimensional representations. ∎

Now we apply some of these general projection identities to infer information
about the underlying two-form and related differential forms associated to an
exact G2-Structure. Many of these results require compactness, so we are
careful to emphasize when we are utilizing this assumption.

Let $\varphi=d\beta$ be an exact G2-Structure. An important point to note is
that for such a structure we may add a closed two-form $\alpha$ to $\beta$ and
the resulting G2-Structure $d\beta$ will be unchanged. In particular,
everything determined by the G2-Structure such as the torsion or decomposition
of forms into irreducible representations of G2 will be unchanged under such a
transformation. Keeping this gauge freedom in mind, we can fix a particular
two-form $\beta$ corresponding to exact G2-Structure $d\beta$ and we can
decompose $\beta$ into irreducible representations of G2.

$\beta=\beta_{7}+\beta_{14}$ (3.13)

Then if we differentiate we get an expression for the G2-Structure in terms of
derivatives of the two-form components.

$d\beta=d\beta_{7}+d\beta_{14}$ (3.14)

We can then decompose $d\beta_{7}$ and $d\beta_{14}$ into irreducible G2
representations and we get some pointwise constraints on the sums of these
components since d$\beta$ is the G2-Structure.

$\displaystyle\pi_{1}(d\beta_{7})+\pi_{1}(d\beta_{14})=1$ (3.15)
$\displaystyle\pi_{7}(d\beta_{7})+\pi_{7}(d\beta_{14})=0$ (3.16)
$\displaystyle\pi_{27}(d\beta_{7})+\pi_{27}(d\beta_{14})=0$ (3.17)

We can now substitute the equations we found for each of these components into
these constraints. Doing so for the one-dimensional component, we get a
relation between the pointwise inner product of $\beta_{14}$ and $\tau$ and
the co-exterior derivative of $X$.

$\langle\beta_{14},\tau\rangle-3d^{\ast}X=7$ (3.18)

If we integrate this pointwise statement over a compact manifold, the
divergence term vanishes and we get an equation for the total volume.

$\langle\beta_{14},\tau\rangle_{L^{2}}=7\textnormal{Vol}$ (3.19)

We could arrive at the same conclusion by considering the $L^{2}$ norm of the
G2-Structure $d\beta$. As the G2-Structure has pointwise norm equal to seven,
the $L^{2}$ norm gives an expression for the total volume.

$7\textnormal{Vol}=|d\beta|^{2}_{L^{2}}$ (3.20)

Assuming compactness, we can then integrate the $L^{2}$ norm of $d\beta$ by
parts and utilize the fact that $\tau$ is in the fourteen dimensional
representation of G2 on two-forms to arrive at the result.

$|d\beta|^{2}_{L^{2}}=\langle\beta,\tau\rangle_{L^{2}}=\langle\beta_{14},\tau\rangle_{L^{2}}$
(3.21)

As a consequence of this result it is clear that $\beta_{14}$ cannot vanish
identically. Later in this section we will prove that $\beta_{7}$ also cannot
vanish and we will be able to relate the $L^{2}$ norms of these two-forms.

We now turn to the constraint from $\pi_{7}(d\beta)=0$. From this identity we
see that we get a canonical co-closed two-form built out of the components of
the two-form $\beta$ underlying an exact G2-Structure.

$d^{\ast}(2\beta_{7}-\beta_{14})=0$ (3.22)

Note that the co-closed form $2\beta_{7}-\beta_{14}$ can’t vanish unless its
components vanish individually, which would imply that $\beta$ itself
vanishes.

Now we assume compactness and we want to see if $\beta$ even needs to have
both components. It turns out that it does, and moreover, we are able to get a
constraint relating the $L^{2}$ norms of these components.

######  Theorem 3.2 (L2 norms of $\beta$ components).

The L2 norms of the irreducible G2-components of the two-form $\beta$
underlying an exact G2-Structure on a compact manifold are related up to a
factor of two.

$2|\beta_{7}|^{2}_{L^{2}}=|\beta_{14}|^{2}_{L^{2}}$ (3.23)

Proof. We consider the exterior derivative of $\beta$ wedged against itself
three times. We may expand this expression using the product rule.

$1/3\hskip 5.69054ptd(\beta\wedge\beta\wedge\beta)=\beta\wedge\beta\wedge
d\beta$ (3.24)

If we assume compactness then we may integrate the left-hand side of this
equation and get zero by citing Stokes’ theorem. However, on the right-hand
side d$\beta$ is the G2-Structure, and we get an identity for a two-form
wedged twice against a G2-Structure using the representation equations for
two-forms and orthogonality of the representations.

$\displaystyle\beta\wedge\varphi=2\ast\beta_{7}-\ast\beta_{14}$ (3.25)
$\displaystyle\beta\wedge\beta\wedge\varphi=2|\beta_{7}|^{2}-|\beta_{14}|^{2}$
(3.26)

But we know that if we integrate this, then that is equal to zero so the
result follows. ∎

The main implication of this result is that neither irreducible G2 component
of the two-form $\beta$ can vanish identically if $\beta$ defines an exact
G2-Structure. Moreover, since $\beta_{7}$ is determined by a one-form and
doesn’t vanish identically we also have a canonical vector field X associated
to an exact G2-Structure which cannot vanish. We can then take the derivative
of the corresponding one-form to get a two-form dX which we can relate to
$\beta$ and $\tau$.

It will be useful to elucidate the relationship between $\beta_{7}$ and dX.
Note that we may rewrite the interior product in terms of the Hodge star
operator.

$\beta_{7}=X\lrcorner\varphi=\ast(X\wedge\ast\varphi)$ (3.27)

We may take the Hodge star and then the exterior derivative to get an
expression for the co-closure which involves dX.

$d\ast\beta_{7}=dX\wedge\ast\varphi-X\wedge\tau\wedge\varphi$ (3.28)

Wedging dX into $\ast\varphi$ isolates $\pi_{7}(dX)$, and we may also simplify
the second term into a contraction of X into the torsion two-form.

$d\ast\beta_{7}=\pi_{7}(dX)\wedge\ast\varphi-\ast(X\lrcorner\tau)$ (3.29)

We know even without assuming compactness that the two-form 2$\beta_{7}$ \-
$\beta_{14}$ is co-closed. Therefore, if $\beta_{7}$ is also co-closed, then
$\beta_{14}$ and $\beta$ itself must be co-closed. This constraint shows how
dX and $X\lrcorner\tau$ must be related for $\beta_{7}$ to be co-closed.
Assuming compactness, we can see that if $\pi_{7}(dX)$ vanishes then dX must
vanish identically. This result follows from an expression for the components
of two-forms and an application of Stokes’ theorem.

$dX\wedge
dX\wedge\varphi=(2|\pi_{7}(dX)|^{2}-|\pi_{14}(dX)|^{2})\textnormal{vol}$
(3.30)

If dX equals zero, then the constraint shows how the co-closure of $\beta_{7}$
and $X\lrcorner\tau$ must be related.

##  4\. Exact G2-Structures Subjected to Additional Constraints

An important subclass of closed G2-Structures are the Extremally Ricci-Pinched
(ERP) G2-Structures. These G2-Structures satisfy a condition on the derivative
of their torsion.

$d\tau=1/6(|\tau|^{2}\varphi+\ast(\tau\wedge\tau))$ (4.1)

Bryant showed that all closed G2-Structures which are not torsion-free obey a
pinching inequality relating the squared norm of the traceless Ricci tensor to
the squared scalar curvature.

$\int_{M}|\textnormal{Ric}_{0}|^{2}\textnormal{vol}\geq
4/21\int_{M}R^{2}\textnormal{vol}$ (4.2)

The ERP G2-Structures are those for which this inequality is actually an
equality. In this sense ERP G2-Structures are special because they are as
Ricci-pinched as a G2-Structure can be without being torsion-free.

ERP G2-Structures actually belong to a larger class of G2-Structures that are
called $\lambda$-quadratic. The derivative of torsion of these closed
G2-Structures depends quadratically on the torsion tensor and features a
constant $\lambda$.

$d\tau=\frac{1}{7}|\tau|^{2}\varphi+\lambda(\frac{1}{7}\varphi+\ast(\tau\wedge\tau))$
(4.3)

The case $\lambda=1/6$ corresponds to the ERP G2-Structures, but this class of
structures also includes familiar closed G2-Structures such as Einstein
structures as special cases [13]. Recently, Ball has shown that many more
values for $\lambda$ than were previously understood may be found on non-
compact manifolds [28]. However, only the ERP condition can occur on a compact
manifold so we will focus on that case [13].

For an exact G2-Structure on a compact manifold, we know it cannot be torsion-
free because by Hodge theory, if this was the case then it would vanish
identically. It still stands that it could be ERP; however, perhaps not
suprisingly, we can prove that this cannot happen.

######  Theorem 4.1 (ERP and Exact G2-Structures).

An Exact G2-Structure on a compact manifold cannot be Extremally Ricci
Pinched.

Proof. The ERP condition may be given in terms of the exterior derivatve of
the torsion two-form.

$d\tau=1/6(|\tau|^{2}\varphi+\ast(\tau\wedge\tau))$ (4.4)

Here $\varphi=d\beta$ as the G2-Structure is assumed to be exact. An important
property of ERP G2-Structures on compact manifolds is that the norm of the
torsion two-form is a constant [13]. As such we may commute this norm with the
exterior derivative in the ERP equation.

$d(\tau-1/6|\tau|^{2}\beta)=1/6\ast(\tau\wedge\tau)$ (4.5)

We may then write out an expression for the pointwise norm of
$\tau\wedge\tau$.

$\tau\wedge\tau\wedge\ast(\tau\wedge\tau)=|\tau\wedge\tau|^{2}\textnormal{vol}$
(4.6)

We may then substitute our expression for $\ast(\tau\wedge\tau)$ into this
expression.

$6(\tau\wedge\tau\wedge
d(\tau-1/6|\tau|^{2}\beta))=|\tau\wedge\tau|^{2}\textnormal{vol}$ (4.7)

We may then note that $\tau\wedge\tau$ is automatically closed for an ERP
G2-Structure [13], so we may again use Stokes’ theorem to show that
$\tau\wedge\tau$ vanishes. We then use a general result for G2-Structures that
relates the pointwise norm of a differential two form
$\eta\in\Lambda^{2}_{14}(T^{*}M)$ to the norm of $\eta\wedge\eta$ [13].

$|\eta\wedge\eta|^{2}=|\eta|^{4}$ (4.8)

As a result we may conclude that $\tau$ vanishes, so the G2-Structure is
torsion-free. But this would mean that the G2 three-form is harmonic, and we
already know it is exact. Then by Hodge theory the three-form must vanish,
which would contradict positivity. ∎

We can apply several of these results on exact G2-Structures to the study of
certain closed Laplacian solitons. A Laplacian soliton is a triple ($\varphi$,
$\bar{X}$, $\lambda$) consisting of a G2-Structure $\varphi$ whose Hodge
Laplacian is equal to a constant $\lambda$ times $\varphi$ plus the Lie
derivative of $\varphi$ with respect to a vector field $\bar{X}$ .

$\Delta\varphi=\lambda\varphi+L_{\bar{X}}\varphi$ (4.9)

If ($\varphi_{0},\bar{X},\lambda$) satisfies this equation, then we can use
this initial data to write down a solution of the Laplacian flow that
essentially scales the initial G2 three-form up to diffeomorphism. The precise
expression for $\varphi(t)$ depends on a function $\rho(t)$, which is linear
in t, and the action of the group of diffeomorphisms on the initial
G2-Structure $\varphi_{0}$ which are generated by the vector field $\bar{X}$.

$\varphi(t)=\rho(t)\phi_{t}^{*}\varphi_{0}$ (4.10)

If the constant $\lambda$ is positive, negative, or zero, the Laplacian
soliton is called expanding, shrinking, or steady, resectively. Lin proves
that a Laplacian soliton cannot be shrinking on a compact manifold, and that
if it is steady then the G2-Structure is torsion-free [29]. If the
G2-Structure is closed, then a Laplacian soliton is called a closed Laplacian
Soliton. For a closed Laplacian soliton, by results of Lotay and Wei it is
known that $\bar{X}$ cannot vanish unless the soliton is steady and the
G2-Structure is torsion-free [12]. They also provide a simple proof that a
closed Laplacian soliton cannot be shrinking on a compact manifold and that if
it is steady the G2-Structure is torsion-free.

For a compact manifold, then, we will focus on the case $\lambda>0$. To date,
no examples of such expanding closed Laplacian solitons are known to exist on
compact manifolds.

Manipulating the Laplacian soliton equation a bit, it follows that expanding
(or shrinking) closed Laplacian solitons are automatically exact
G2-Structures. To see this we can express the Hodge Laplacian of the G2-three-
form in terms of the derivative of the torsion form.

$\Delta\varphi=dd^{\ast}\varphi+d^{\ast}d\varphi=dd^{\ast}\varphi=d\tau$
(4.11)

$L_{\bar{X}}\varphi$ also turns out to be an exact form as a consequence of
Cartan’s formula.

$L_{\bar{X}}\varphi=d(\bar{X}\lrcorner\varphi)+\bar{X}\lrcorner
d\varphi=d(\bar{X}\lrcorner\varphi)$ (4.12)

We can then rearrange the Laplacian soliton equation to get an expression for
$\lambda\varphi$. Note that by assumption the constant $\lambda$ doesn’t
vanish.

$\lambda\varphi=d(\tau-\bar{X}\lrcorner\varphi)$ (4.13)

As a consequence, all expanding closed Laplacian solitons are actually exact,
so if exact G2-Structures generally can’t exist on compact manifolds, then a
closed Laplacian soliton would have to have $\lambda=0$ and the G2-Structure
would have to be torsion-free.

Not only does the Laplacian soliton equation imply that $\varphi$ is exact,
but it also gives information about the two-form. Fixing a choice of gauge, we
can take the two-form to be exactly
$\lambda\beta=\tau-\bar{X}\lrcorner\varphi$. Given this choice, the
decomposition of $\beta$ into irreducible G2 representations is immediate
because each of these terms are in particular G2 representations.

$\displaystyle\lambda\beta_{7}=\lambda
X\lrcorner\varphi=-\bar{X}\lrcorner\varphi$ (4.14)
$\displaystyle\lambda\beta_{14}=\tau$ (4.15)

We can now apply some of the results we found for general exact G2-Structures
to expanding closed Laplacian solitons. It turns out that several of the key
results found by Lotay and Wei are immediate consequences of the fact that the
G2-Structure underlying (non-steady) closed Laplacian solitons is exact.

######  Theorem 4.2 (Projection Identities for Shrinking and Expanding Closed
Laplacian Solitons).

Given a Closed Laplacian Soliton $(\varphi,\bar{X},\lambda)$ with $\lambda\neq
0$, the norm of the torsion, the constant $\lambda$, and $d^{\ast}\bar{X}$
satisfy a linear relation. Moreover, the form $\bar{X}\lrcorner\varphi$ is co-
closed.

$\displaystyle\frac{1}{3}|\tau|^{2}+d^{\ast}\bar{X}=7/3\lambda$ (4.16)
$\displaystyle d^{\ast}(\bar{X}\lrcorner\varphi)=0$ (4.17)

Proof. The first relation is just the statement
$\pi_{1}(d\beta_{7})+\pi_{1}(d\beta_{14})=7$ applied to the case of a
Laplacian soliton. For such a structure, the components of the exact G2 two-
form are given in terms of $\tau$ and $\bar{X}\lrcorner\varphi$. We saw that
we may simplify this expression into an expression relating the co-exterior
derivative of the vector field X and the pointwise inner product of
$\beta_{14}$ and $\tau$.

$\langle\beta_{14},\tau\rangle+d^{\ast}X=1$ (4.18)

For the Laplacian soliton, $\beta_{7}$ is given in terms of the vector field
$\bar{X}$ and $\beta_{14}$ is given in terms of the torsion two-form.

$\displaystyle\lambda\beta_{7}=\lambda
X\lrcorner\varphi=-\bar{X}\lrcorner\varphi$ (4.19)
$\displaystyle\lambda\beta_{14}=\tau$ (4.20)

We can then plug these expressions for the components into the general
projection expression for an exact G2-Structure to provide the desired
relation.

We can now look at the relation $\pi_{7}(d\beta_{7})+\pi_{7}(d\beta_{14})=0$
for a Laplacian soliton. We saw as a consequence of this relation that there
was a canonical co-closed two-form built out of the components of the exact
G2-Structure.

$d^{\ast}(2\beta_{7}-\beta_{4})=0$ (4.21)

For the Laplacian soliton, not only is this form co-closed, but $\beta_{14}$
is also co-closed since it is directly proportional to $\tau$. As a
consequence $\beta_{7}$ also has to be co-closed, which proves the desired
relation. ∎

As a consequence, the two-form (with this choice of gauge) underlying the
G2-Structure of a Laplacian soliton is co-closed.

We can also apply our result relating the L2 norms of $\beta_{7}$ and
$\beta_{14}$ to the case of these closed Laplacian solitons.

######  Theorem 4.3.

For an expanding closed Laplacian soliton ($\varphi,\bar{X},\lambda$) on a
compact manifold the L2 norms of the torsion $\tau$ and the vector field
$\bar{X}$ are proportional up to a factor of six.

$|\tau|^{2}=6|\bar{X}|^{2}$ (4.22)

Proof. We plug in the components $\lambda\beta_{7}=-\bar{X}\lrcorner\varphi$
and $\lambda\beta_{14}=\tau$ into the result from Theorem 4.1.

$2|\bar{X}\lrcorner\varphi|_{L^{2}}^{2}=|\tau|_{L^{2}}^{2}$ (4.23)

We can then relate the norm of $\bar{X}\lrcorner\varphi$ to that of $\bar{X}$.
It is generally true for vector fields that the norm of $\bar{X}$ is three
times the norm of $\bar{X}\lrcorner\varphi$ [16]. The result follows. ∎

This result provides an alternate proof of the fact that $\bar{X}$ cannot
vanish for a closed Laplacian soliton on a compact manifold unless the
G2-Structure is torsion-free and the soliton is steady. This precise
relationship between the norms of $\tau$ and $\bar{X}$, however, appears to be
new. Note that $\lambda$ does not enter into the norm expression; however, we
have assumed that it does not vanish because it is only in this case that the
G2-Structure is exact and the method of proof applies.

We have also seen that an exact G2-Structure cannot by ERP on a compact
manifold. As an ERP G2-Structure is not torsion-free and thus cannot
correspond to a steady soliton, if an ERP G2-Structure did correspond to a
soliton the G2-Structure would have to be exact. We may then infer that an ERP
G2-Structure may never be a closed Laplacian soliton on a compact manifold.

We may also clarify a constraint on $\bar{X}\lrcorner\tau$ found by Lotay and
Wei for gradient Laplacian solitons. For these solitons the dual one-form of
the vector field $\bar{X}$ is the gradient of a function $f$. We saw the
constraint relating the co-closure of $\beta_{7}$ to $dX$ and
$X\lrcorner\tau$, which holds for a general exact G2-Structure. We may
multiply through by $\lambda$ to rephrase this constraint for a closed
Laplacian soliton.

$d\ast(\bar{X}\lrcorner\varphi)=\pi_{7}(d\bar{X})\wedge\ast\varphi-\ast(\bar{X}\lrcorner\tau)$
(4.24)

For a closed Laplacian soliton, $\beta_{7}$ is co-closed, so the left-hand
side vanishes. For a gradient soliton $d\bar{X}$ also has to vanish, so these
together imply that $\bar{X}\lrcorner\tau$ has to vanish, which is the result
found by Lotay and Wei. From this result we may also establish a converse
statement which provides a means to ensure that a soliton is a gradient
soliton.

######  Theorem 4.4 (Gradient Solitons).

A necessary and sufficient criterion for a Laplacian soliton
$(\varphi,\bar{X},\lambda)$ to be a gradient soliton on a simply-connected
compact manifold is for $\bar{X}\lrcorner\tau=0$.

Proof. Lotay and Wei prove that gradient solitons, without an assumption on
compactness, satisfy $\bar{X}\lrcorner\tau=0$. For the converse we may utilize
the same relation.

$d\ast(\bar{X}\lrcorner\varphi)=\pi_{7}(d\bar{X})\wedge\ast\varphi-\ast(\bar{X}\lrcorner\tau)$
(4.25)

If $\bar{X}\lrcorner\tau$ vanishes then this implies that $\pi_{7}(d\bar{X})$
has to vanish pointwise because $\beta_{7}$ always is co-closed for a
Laplacian soliton. However, on a compact manifold the $L^{2}$ norms of
$\pi_{7}(d\bar{X})$ and $\pi_{14}(d\bar{X})$ are related up to a factor of
two.

$2|\pi_{7}(d\bar{X})|_{L^{2}}^{2}=|\pi_{14}(d\bar{X})|_{L^{2}}^{2}$ (4.26)

Therefore, $\pi_{14}(d\bar{X})$ also has to vanish so $d\bar{X}$ has to
vanish. If the underlying manifold is simply-connected or more generally has
vanishing first Betti number, then $\bar{X}=df$ for some function $f$ and the
soliton is a gradient soliton. ∎

The significance of gradient solitons is derived by analogy to the Ricci flow.
In that context Perelman showed that every Ricci soliton is a gradient soliton
using variational methods [31]. If the Laplacian flow behaves analogously to
the Ricci flow then we may also expect gradient solitons to be generic.
However, the precise degree of analogy between these geometric flows is rather
unclear. In particular, it is unclear if gradient solitons exist at all on
compact manifolds. However, there are examples of complete steady gradient
solitons recently constructed by Ball [28].

##  5\. The Curvature Tensor of Exact G2-Structures

In addition to several results on the Ricci curvature for closed
G2-Structures, there are also several results on the Riemann curvature which
were proven by Cleyton and Ivanov [23, 24]. In this section we show how to
strengthen these results on the Riemann curvature when the G2-Structure is
exact.

Recall that for a general n-dimensional Riemannian manifold, we may decompose
its Riemann curvature as a (0,4) tensor orthogonally into components [32].

$R_{ijkl}=S_{ijkl}+E_{ijkl}+W_{ijkl}$ (5.1)

These components may be expressed in terms of the metric, scalar curvature,
and traceless Ricci tensor Ric0.

$\displaystyle S_{ijkl}=\frac{R}{n(n-1)}(g_{il}g_{jk}-g_{ik}g_{jl})$ (5.2)
$\displaystyle
E_{ijkl}=\frac{1}{n-2}((\textnormal{Ric}_{0})_{il}g_{jk}-(\textnormal{Ric}_{0})_{jl}g_{ik}-(\textnormal{Ric}_{0})_{ik}g_{jl}+(\textnormal{Ric}_{0})_{jk}g_{il})$
(5.3) $\displaystyle W_{ijkl}=R_{ijkl}-S_{ijkl}-E_{ijkl}$ (5.4)

These tensors have the same algebraic symmetries as the Riemann tensor. The
Weyl tensor W is also traceless and is known to measure the deviation of the
manifold from being locally conformally flat [32].

The previous decomposition may be thought of as a decomposition of the space
of algebraic curvature tensors into O(n) representations. In the G2 context we
can decompose the space of algebraic curvature tensor further into irreducible
G2 representations. This decomposition was carried out by Cleyton and Ivanov,
who found that the Weyl tensor from the SO(7) decomposition further splits
into three pieces.

$W=W_{27}+W_{64}+W_{77}$ (5.5)

This full decomposition (with the terms S and E as well) shows how to
decompose the space of algebraic curvature tensors into irreducible
representations of G2. The Weyl tensor spaces found by Cleyton and Ivanov are
isomorphic to familiar G2 representations.

$\displaystyle W_{77}\cong V_{77}^{(2,0)}$ (5.6) $\displaystyle W_{64}\cong
V_{64}^{(1,1)}$ (5.7) $\displaystyle W_{27}\cong\Lambda^{3}_{27}$ (5.8)

These spaces may be defined explicitly in terms of the Ricci curvature and
scalar curvature of the G2-Structure, as well in terms of as a tensor that
Cleyton and Ivanov refer to as the $\phi$-Ricci Tensor [23]. We will not need
the precise definitions of these spaces for our purposes but it is useful to
keep in mind the representations of G2 to which they are isomorphic.

In order to prove results on the curvature, we study the first Pontryagin
class of the underlying manifold. Given an affine connection $\nabla$ we can
build a representative of this class using the curvature of the connection.

$p_{1}(\nabla)=-\frac{1}{8\pi^{2}}\textnormal{tr}(R_{\nabla}^{2})$ (5.9)

In their proof that a closed G2-Structure on a compact manifold cannot be
Einstein unless it is torsion-free, Cleyton and Ivanov consider the integral
of the closed G2-Structure wedged against the difference of the Pontryagin
forms corresponding to the Levi-Civita connection and the canonical G2
connection.

$\int_{M}(p_{1}(\nabla_{LC})-p_{1}(\nabla_{G_{2}}))\wedge\varphi$ (5.10)

As the class is topological and in particular doesn’t depend on the choice of
connection, the difference of the representatives of the Pontryagin classes of
the two connections is an exact four-form. Since $\varphi$ is closed, we can
use Stokes’ theorem to assert that this integral vanishes. If we assume the
stronger condition that the G2-Structure is exact then we do not need to take
the difference of two Pontryagin class representatives to utilize Stokes’
theorem. Since the Pontryagin form for a given connection is a closed form, we
may immediately say that the integral of such a form wedged against the G2
three-form vanishes.

$\int_{M}p_{1}(\nabla)\wedge\varphi=0\\\ $ (5.11)

We may then rewrite these identities in terms of the curvatures of the
connections to get constraints on the curvature of an exact G2-Structure. The
Levi-Civita constraint is particularly easy to work with due to the symmetries
of the Riemann curvature, and immediately provides a result on the G2
decomposition of the Riemann cuvature viewed as a Lie algebra-valued two-form.

######  Theorem 5.1 (Integral Identity for two-form components of Riemann
curvature of Exact G2-Structure).

If Rm is the Riemann curvature of the metric determined by an exact
G2-Structure on a compact manifold then the L2 norms of the irreducible G2
components of the Riemann curvature are related up to a factor of two.

$2|\pi_{7}(\textnormal{Rm})|_{L^{2}}^{2}=|\pi_{14}(\textnormal{Rm})|_{L^{2}}^{2}$
(5.12)

Proof. We start with the Pontryagin class identity for the Levi-Civita
connection, substituting in the expression for the representative of the class
in terms of the curvature.

$\int_{M}\textnormal{tr}(\textnormal{Rm}\wedge\textnormal{Rm})\wedge\varphi=0$
(5.13)

This expression is reminiscent of an expression for the components of an
ordinary two-form $\alpha$ that we have utilized in several preceding
theorems.

$\alpha\wedge\alpha\wedge\varphi=2|\alpha_{7}|^{2}-|\alpha_{14}|^{2}$ (5.14)

Viewing the curvature as a Lie algebra-valued two-form and using the
symmetries of the Riemann curvature, we get an analogous identity. The result
then follows. ∎

We may compare this result to what Joyce finds for a torsion-free
G2-Structure. For such a structure, the curvature only has a component in the
14 dimensional representation of G2 on two-forms. As a result, for a torsion-
free G2-Structure, wedging the G2 three-form against the Levi-Civita
representative of the first Pontryagin class and integrating gives the L2 norm
of the curvature.

$\int_{M}p_{1}(\nabla_{LC})\wedge\varphi=-\int_{M}|R|^{2}\textnormal{vol}$
(5.15)

From this expression we see that the first Pontryagin class being trivial
would imply that the connection is flat; in other words, that it would have
trivial holonomy. However, for an exact G2-Structure the integral always
vanishes. This is consistent with a vanishing first Pontryagin class but not
necessary. So, at least by this reasoning, the first Pontryagin class
vanishing is not an obstruction for a compact manifold to admit an exact
G2-Structure.

We may also point out that it is known for an arbitrary G2-Structure that
$\pi_{7}(\textnormal{Rm})$ cannot vanish unless the G2-Structure is torsion-
free [17]. Moreover, if $\pi_{14}(\textnormal{Rm})$ vanishes then the
G2-Structure is not only torsion-free but flat. So we could have anticipated
that neither of these curvature components could vanish for an exact
G2-Structure, but it is certainly not true for an arbitrary G2-Structure that
these components would be related precisely as in this theorem.

We may also express the wedge product of the Pontryagin form against the
G2-Structure in terms of the decomposition of the Riemann tensor due to
Cleyton and Ivanov. They find a relationship between the G2 components of the
Riemann tensor and the first Pontryagin class for an arbitrary closed
G2-Structure on a compact manifold [23].

$\langle
p_{1}(M)\cup\varphi,[M]\rangle\geq-\frac{1}{8\pi^{2}}\int_{M}|W_{77}|^{2}-1/2|W_{64}|^{2}-3/16S^{2}$
(5.16)

Here equality holds iff the closed G2-Structure is ERP. We can now consider
the implications of this identity when the G2-Structure is exact.

######  Theorem 5.2 (Integral Identity for components of Riemann curvature of
Exact G2-Structure).

For an exact G2-Structure on a compact manifold M, the L2 norms of the
components of the Riemann tensor are related by an inequality.

$\int_{M}|W_{77}|^{2}>1/2\int_{M}|W_{64}|^{2}+3/16\int_{M}S^{2}$ (5.17)

Proof. We start with the identity of Cleyton and Ivanov and note that for an
exact G2-Structure, the integral of the first Pontryagin class wedged against
the G2 three-form vanishes.

$0\geq-\int_{M}|W_{77}|^{2}+1/2\int_{M}|W_{64}|^{2}+3/16\int_{M}S^{2}$ (5.18)

Rearranging, we basically get the required result, but we can also make the
inequality a proper inequality by noting that the case of equality only
happens when the closed G2-Structure is ERP. We proved in theorem 4.2 that an
exact G2-Structure cannot be ERP, so this statement cannot be an equality. ∎

As a corollary we may see that W77 may not vanish for an exact G2-Structure.
This result complements the finding by Cleyton and Ivanov that W27 also cannot
vanish for a closed G2-Structure unless that structure is torsion-free. Of
course, S2 also may not vanish since that would also mean that the
G2-Structure was torsion-free.

###  5.1. Acknowledgements

I would like to thank Gavin Ball, Robert Bryant, Jason Lotay, and my advisor
Dave Morrison for helpful comments related to my work on exact G2-Structures.
I’m also very appreciative of the support I have recieved more generally from
the Simons Collaboration on Special Holonomy in Geometry, Analysis, and
Physics. I’d also like to thank my friends and family for support and Jessica
Li in particular for being very critical of my grammar which resulted in
noticeable improvements for this paper.

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# ViraPart: A Text Refinement Framework for Automatic Speech Recognition and
Natural Language Processing Tasks in Persian

Narges Farokhshad  Milad Molazadeh  Saman Jamalabbasi  Hamed Babaei Giglou
Saeed Bibak Authors are with Part AI Research Center, NLP Department, Tehran,
Iran (e-mails:{narges.farrokhshad, milad.molazadeh, saman.abbasi,
hamed.babaei<EMAIL_ADDRESS>
###### Abstract

The Persian language is an inflectional subject-object-verb language. This
fact makes Persian a more uncertain language. However, using techniques such
as Zero-Width Non-Joiner (ZWNJ) recognition, punctuation restoration, and
Persian Ezafe construction will lead us to a more understandable and precise
language. In most of the works in Persian, these techniques are addressed
individually. Despite that, we believe that for text refinement in Persian,
all of these tasks are necessary. In this work, we proposed a ViraPart
framework that uses embedded ParsBERT in its core for text clarifications.
First, used the BERT variant for Persian following by a classifier layer for
classification procedures. Next, we combined models outputs to output
cleartext. In the end, the proposed model for ZWNJ recognition, punctuation
restoration, and Persian Ezafe construction performs the averaged F1 macro
scores of 96.90%, 92.13%, and 98.50%, respectively. Experimental results show
that our proposed approach is very effective in text refinement for the
Persian language.

Persian Ezafe, Punctuation, ZWNJ, NLP, ParsBERT, Transformers

##  1 Introduction

Most of the spoken languages around the world such as Persian are rhythmic and
this fact will make the words of sentences more meaningful while speaking and
one can change the meaning of a sentence by changing the purpose and
significance of a particular word. By making sure that marks are used
accurately, we can avoid probable uncertainty. Otherwise, the sentence will be
understood differently; hence, not paying attention to the accurate use of
punctuation leads to confusion and misunderstandings [2].

Automated Speech Recognition (ASR) is a technology that allows users to speak
entries rather than punching numbers or letters on a keypad. However, in a
typical ASR system, punctuation and capitalization of words are removed
because they do not affect the pronunciation of words. The output contains
purely a sequence of words or alphabets characters. This output is sufficient
for many applications that usually use a short and independent segment of
speeches, but it is difficult to be used in applications that decipher long
speech segments. Furthermore, ASR results are fed into natural language
processing (NLP) models that punctuations and word capitalizations are
important pieces of information that can help to boost the NLP model’s
performances [1].

To achieve a readable system in Persian there are different tasks. We
considered punctuation restoration, zero-width non-joiner (ZWNJ) recognition,
and Ezafe construction tasks to process texts for NLP tasks. Punctuation
restoration is the task of identifying punctuation marks that increases the
readability of texts. Punctuation restoration and correcting word casing is
one requirement to offer a reliable post-processing system for texts before
applying complex NLP algorithms. For English, we obtained comparable state-of-
the-art results, while for Persian, there are a limited number of works in the
field. In the English language, hyphens are used to join words and to separate
syllables of a single word. This process is called hyphenation and this type
of word is called multi-part word. However, the Persian language consists of
multi-part words as well. In Persian morphology, the ZWNJ character is used to
separate parts of multi-part words [3]. Persian ezafe is an unstressed
morpheme syllable that appears at the end of the words. It is pronounced as -e
after consonants and as -ye after vowels. This syntactic phenomenon links a
head noun, head pronoun, head adjective, head preposition, or head adverb to
their modifiers in a constituent called ezafe construction [4].

In most studies, all of these three concepts are investigated separately
(Punctuation restoration, ZWNJ recognition, and ezafe construction) one reason
could be the avaliable dataset that are for different purposes. Another reason
is the complexity of the tasks is varied from one task to another. However, in
this work, we analogy that for an enriched text we must pay attention to all
three aspects, so in this study, we proposed a framework that is a combination
of three models for text refinement. In practice, we used publically avaliable
datasets to create models that make the readable text for numerous NLP tasks.
In general our research contributions are two folds:

  * •

We proposed a framework for text refinements in Persian for various Nlp tasks.
We used three major text clarifications models in one framework.

  * •

We used ParsBERT (a BERT variant for the Persian language) for sequence
labeling.

After reviewing the related works of three tasks in Section II, we introduce
our proposed method in Section III. We then discuss punctuation restorations,
ZWNJ recognition, and ezafe recognition tasks and their results in Sections IV
and at the end, we made a conclusion of this paper in Section V.

##  2 Related Works

We first review methods for English, and then methods for the Persian
language. Since our work relies on multiple tasks, we summarize related
literature on these topics as well.

###  2.1 English

There are lots of works in the English language, we divided the related work
in the field into 4 categories namely prosodic features, n-gram language
model, RNN based approaches, and transformers. Prosodic features are features
that appear when we put sounds together in connected speech. The [5] used a
statistical finite-state model that combines prosodic, linguistic, and
punctuation class features for the punctuation restoration task. In a similar
work, the [6] proposed a combined punctuation generation and speech
recognition system that uses prosody features to boost their performance. The
n-gram language models (LM) [7] have a long history in NLP tasks that are
useful for punctuation annotation systems. The [8] proposed a metadata
detection system based on n-gram LM that combines information from different
types of textual knowledge sources with information from a prosodic classifier
to automatic detection of sentence boundaries and disfluencies. In [9] authors
used n-gram LM and larger training datasets consistently, and they conclude
that using large data will improve performance while increasing n-gram order
does not help. With progress in deep learning models, RNN based models
achieved higher attention in the text-domain. Subsequent research revealed
different approaches to this task. For example, authors of the [10] proposed
Deep Neural Network (DNN) sequence labeling model that uses Bidirectional Long
Short-Term Memory (BiLSTM) and a Convolutional Neural Network (CNN) to predict
the punctuation. The word and character-level embedding with CNN-RNN model for
punctuation restoration excavated in [17]. LSTM approach has been proposed by
[18] authors for the punctuation prediction task.
The transformer in NLP is a novel architecture that aims to solve sequence-to-
sequence tasks while handling long-range dependencies with ease [11].
Transformers are groundbreaking models in various NLP tasks such as sequence
labeling. In works of [12], [13], [14], [15], and [16] authors used
transformers models for punctuation restoration task and achieved a promising
results.

###  2.2 Persian

In this section, we divided the related works into three sections, to explore
similarly.

####  2.2.1 ZWNJ Recognition

ZWNJ has been investigated in Persian and there are a few related works for
this task. To the best of our knowledge, the oldest work on the shelf belongs
to [20], in which they proposed a model based on a statistical machine
translation paradigm for space corrections. In machine translation, the text
in the source language is translated into a text in the destination language,
in the proposed model the output is space corrected text. The BLEU score
statistical machine-translation method reaches 0.91. Parsivar [19] is a
toolkit that performs different kinds of activities composed of normalization,
space correction, tokenization, stemming, parts of speech tagging, and shallow
parsing. They proposed a Naïve Bayes model for space correction with a BIO
scheme to find words with multiple parts separated by spaces with an F1 score
of 89.50%. In a similar manner authors of [21] investigated various nlp
toolkits for the Persian language and they found out that except Parsivar none
of them is capable of handling spaces desirably than Parsivar. They believe
that Hazm[35] is not properly handling the spaces in Persian, so they trained
a model that uses Hazm tokenization as input for the space correction model.
First, they combined various datasets and achieved 1,750,607 words. Next, they
trained the n-gram language model using the KenLM [23] toolkit for space
correction. Their method outperforms the Hazm model for space correction with
an F1 score of 81.94%. The BERT-based approach proposed by [22] for word
segmentation correction and ZWNJ recognition. They approach the problem
jointly as a sequence labeling problem and proposed a BERT-CRF model which
consists of a BERT model followed by a CRF layer for sequence labeling task
for ZWNJ recognition. They specified three variant classes namely none, space,
and ZWNJ classes. Next, They proposed an algorithm that extends the existing
dataset by adding noses. This allowed boosting their model performance from an
F1 score of 96.67% to 98.14%. However, they collected 500 difficult sentences
for further evaluation and reported an F1 score of 92.40%.

####  2.2.2 Punctuation Restoration

Punctuation prediction/restoration is a crucial task in ASR. The sequence of
words with no punctuation needed to be processed to make sense for humans and
even other areas of NLP. The [24] introduced the first-ever corpus for
automatic punctuation prediction in Persian texts. They have made many
revisions to their corpus such as word replacements, normalizations, word type
corrections, and numerous corrections to the punctuation marks. They trained
the CRF model and achieved an F1 score of 69.00% in their preliminary
experiments.

####  2.2.3 Ezafe Construction

Ezafe construction is a peculiar aspect of the Persian language which is able
to make a good indicator of important information in the text. The [26]
mentioned that adding information about ezafe can boost performances of
dependency parsing and shallow parsing by 4.6% and 9% respectively. Most of
the work in the field used rule-based and hand-crafted features to tackle this
challenge. The [27] proposed the head-driven phrase structure grammar to
formalize Persian syntax and determine phrase boundaries and [28] used a rule-
based method to create a Persian morphological analyzer. The [29] employed the
CART tree classification approach for ezafe constructions. The [30] used a
probabilistic approach namely CRF to create an ezafe recognition model in
Persian and achieved accuracy 98.04%. Another work is [31] that uses both a
rule-based method and a genetic algorithm for ezafe recognition and reported
accuracy of 95.26%. The newest work in ezafe recognition is the work of [25]
that uses transformers and ezafe recognition role in part-of-speech tagging.
They employed transformer-based methods, BERT and XLMRoBERT, and achieved an
F1 score of 98.10%.

##  3 Proposed Method

In this section, the methodology of the proposed system is described
elaborately. The proposed system consists of four layers including the input
processing layer, model layers which are divided into two modules (prediction
and mapping layers), and the output construction layer. The architecture of
the proposed method is presented in figure 1. First, in the processing layer
documents will be cleaned, normalized, and tokenized for the next layer. Next,
the model layer that consists of prediction and mapping layers uses processed
documents for punctuation construction, ZWNJ construction, and ezafe
recognition. In the last layer, outputs will be combined for the final result
of the system. The model layers consist of three sections, each for a
different task that works in parallel form. The prediction layer in model
layers tries to train models in different sections; in the punctuation section
it tries to identify the possibility of specified punctuation marks. The ZWNJ
section at the prediction layer predicts whatever ZWNJ should occur or not.
Similarly, it works in the ezafe section in this layer as well. The next layer
in model layers is the mapping layer which tries to map the prediction results
on text. Mapping starts with the punctuation section by constructing
punctuations. Then the produced output of the punctuation section feeds into
the ZWNJ section for ZWNJ construction at the mapping layer. Likewise, the
punctuation and ZWNJ constructed output from the previous section will feed
into the ezafe section to determine which token needs ezafe construction. In
the following sections, we will describe each layer more elaborately.

Figure 1: Proposed System Architecture

###  3.1 Problem Formulations

In this paper, our training data consists of $N$ sentences that each sentence
consists of $n$ tokens. Regarding this; we can define $T_{X}$ and $T_{Y}$ as
input and output representation of input sequences, respectively. The
definitions are presented as follows:

$T_{X}=\sum_{i=1}^{n}W_{i}\;\;\&\;\;T_{Y}=\sum_{i=1}^{n}Y_{i}$

Where $T_{y}$ changes based on models. In this paper, we defined the $T_{Y}$
for each model at the model layer as followings:

$T_{Y_{punct}}=\sum_{i=1}^{n}Y_{i}\;\;\;where\;\;Y_{i}\in
C(".",":",",","?","unk")$
$T_{Y_{zwnj}}=\sum_{i=1}^{n}Y_{i}\;\;\;where\;\;\;Y_{i}\in C(1,0)$
$T_{Y_{ezafe}}=\sum_{i=1}^{n}Y_{i}\;\;\;where\;\;\;Y_{i}\in C(1,0)$

Where $C$ is the label set for tokens in different tasks. For example, for
punctuation tasks, we specified only four types of marks for restoration and
the last unk label is for a token that doesn’t take any punctuation mark, in
our model it is a five-class sequence labeling model. For other tasks, it is a
binary class classification that 1 means changes required else changes are not
required. Changes for binary classification are construction and specification
(recognition) for ZWNJ and Ezafe tasks respectively. Overall, we are dealing
with a classification task. Considering definitions, we can define the
training, validation, and test sets as a following for each problem.

$T_{train}=\\{(T_{X_{1}},T_{Y_{1}}),(T_{X_{2}},T_{Y_{2}}),...,(T_{X_{n}},T_{Y_{n}})\\}$
$T_{val}=\\{(T_{X_{1}},T_{Y_{1}}),(T_{X_{2}},T_{Y_{2}}),...,(T_{X_{k}},T_{Y_{k}})\\}$
$T_{test}=\\{(T_{X_{1}},T_{Y_{1}}),(T_{X_{2}},T_{Y_{2}}),...,(T_{X_{m}},T_{Y_{m}})\\}$

Where $n$, $k$, and $m$ are sizes of the train, validation, and test sets
respectively. The goal is to create system $\gamma$, $\beta$, and $\alpha$
models that $F_{mapper}$ uses for punctuation, ZWNJ, and ezafe constructions.

$F_{mapper}(T_{X})=\alpha(\beta(\gamma(T_{X})))$

Where $\gamma$ is the punctuation construction model that predicts and
constructs the specified punctuation marks in input $T_{X}$, and the resulting
output $\hat{T_{Y}}$ will be used in the $\beta$ model for the joining of
ZWNJ’s. In the end, the output of the previous model will enter into $\alpha$
model for ezafe construction.

###  3.2 Processing Layer

Raw documents first enter into the processing layer, the processing layer aims
to create $(T_{X},T_{Y})$ pairs from input raw documents. Generally speaking,
raw documents can be sentences or paragraphs as well and they need to be
processed into the form of input pairs for models. Nevertheless, in this
layer, we removed every punctuations mark from inputs. This technique will
allow us to use the proposed framework for punctuation restoration tasks. It
presents everything similar to the output of ASR systems. However, it is
usable for punctuation compilation as well.

First, a document is converted into $W_{i}$ tokens using a personalized
tokenizer that considers spaces, half-spaces, and punctuations for splitting
raw documents into texts. Next, the special character has been erased from
$W_{i}$ tokens. Next, English numbers have been converted into Persian
numerics. In the end, modifications on $Y_{i}$ applied to output required
labels for $W_{i}$ sequences.

###  3.3 Model Layers

Model layers aim to make a prediction and mappings for refinements. In these
layers, a model will be trained in the prediction layer to predict new unseen
input sequences and the mapping layer will use the trained model for
refinements. In the following, we will describe each layer carefully.

Figure 2: Proposed Model Architecture

####  3.3.1 Prediction

ParsBERT [33] is a monolingual version of the BERT language model [32] for the
Persian language that uses the base configuration of the BERT model such as 12
hidden layers, hidden size of 768 with 12 attention heads. ParsBERT is a
transformer-based language model with an encoder-only structure. The input
sequence ${W_{1},W_{2},...,W_{n}}$ is mapped to a contextualized encoded
sequence ${W^{{}^{\prime}}_{1},W^{{}^{\prime}}_{2},...,W^{{}^{\prime}}_{n}}$
by going through a series of bi-directional self-attention blocks with two
feed-forward layers in each block. Then we took the output sequences and
mapped them to a binary and multi-classification with sigmoid and softmax
activation functions for each task, respectively. First, we used ParsBERT
tokenizer to create attention mask and word mappers, then we fine-tuned
ParsBERT for tasks. For the training procedure, we used CrossEntropy and
CategoricalCrossEntropy loss functions for binary and multi-class
classification models, respectively. The rest of the parameters are the same
for all of the models. We set, learning rate to 2e5, epoch to 3, and dropout
to 0.1 to train models.

####  3.3.2 Mapping

The mapping goal is to use the trained model in the prediction layer for
making changes into tokens. The $F_{mapper}$ takes raw $T_{X}$ as input and
then first uses the punctuation prediction model to identify which punctuation
mark is needed to be applied for tokens, after punctuation construction
($\gamma$), the refined $T_{X}$ enters into the ZWNJ construction ($\beta$)
for adding changes. Similarly, the output of $\beta$ will enter into ezafe
recognition ($\alpha$) to sign ezafe in predicted tokens.

###  3.4 Output Layer

The output layer at top of the proposed framework combines the previous layer
predictions from the mapper to construct the sentences from output sequences.

The figure 3 represents sample input and outputs from the proposed framework.
In this figure, the blue-colored text presents the correct ZWNJ constructed in
input sentences and the model do not need to make any changes on them.
However, the green-colored and red-colored texts are wrong ZWNJ construction
in input sequences that our models recognized and corrected in the outputs. In
general, there are two type of modifications for ZWNJ; the first type is
occurring in green-colored texts, where ZWNJ is required in input sentences,
the second type is occurring in the red-colored text, where ZWNJ is not
required in input sentences and our model removes them in outputs. Punctuation
mark restorations in texts happened with background-colored punctuations where
yellow color means that it was incorrectly has been utilized in input
sentences and blue color means that input sentences missing those
punctuations. More experimental evaluations will be presented in the next
sections.

Figure 3: Output layer result for raw input sentences.

##  4 Results

In this section, we will explore the dataset and results of the proposed
framework.

###  4.1 Dataset

For training and evaluation of the proposed framework, we used the Bijankhan
corpus [34]. The Bijankhan is a well-known dataset for Persian natural
language processing tasks. The dataset consists of news articles and common
texts consisting of in total 4300 different topics. This dataset consists of
2,597,939 words that are annotated with morpho-syntactic and partly semantic
features. The corpus has been tagged by 550 tags in hierarchical order with
more fine-grained POS tags like noun-plural-subj. Nearly 23% of words in this
corpus are tagged with Ezafe. Punctuation marks are specified with the DELM
tag in this corpus with a total number of 246,023. Each word in this corpus is
specified as a single row and sentences divided by #. Multi-words are also
specified in a single row, so in this way, we made modifications to the
dataset using its structure to obtain a dataset for the ZWNJ model. We split
multi-words into separate words and labeled them for ZWNJ recognition. We
found almost 30% of the corpus appropriate for the ZWNJ recognition task. We
split the dataset for each task in 80% 10% 10% for train, validation, and test
sets, respectively. The table 1 presents the details about train, test, and
validations sets based on tasks.

Table 1: The number of sentences and tokens in train, validation, and test sets. Set | # of Sentences | # of Tokens
---|---|---
Punctuation |  |
Train | 72780 | 2,040,213
Validation | 9098 | 256,767
Test | 9097 | 255,197
Ezafe |  |
Train | 78741 | 2,068,316
Validation | 9843 | 256,282
Test | 9843 | 261,362
ZWNJ |  |
Train | 75562 | 2,211,741
Validation | 9445 | 279,284
Test | 9445 | 276,685

###  4.2 Metrics

TThe model’s evaluation metrics are precision, recall, F1, and accuracy
measures. Because the dataset is skewed for tasks, Macro $F1$ measurement
provided a better view of the proposed method performance over all classes.
For each class $C$, we calculate its precision ($p$), recall ($r$), and $F1$
score as follow:

$p=\frac{\\#\;of\;tokens\;predicted\;as\;C\;accurately}{\\#\;of\;tokens\;predicted\;as\;C}$
$r=\frac{\\#\;of\;tokens\;predicted\;as\;C\;accurately}{\\#\;of\;tokens\;with\;grand\;truth\;label\;of\;C}$

$F1=2*\frac{p*r}{p+r}$

Considering all the classes together, the averaged F1 macro score is:

$Macro\;F1=\frac{1}{n_{c}}\sum^{i=1}_{n_{c}}F1_{i}$

Where $n_{c}$ is the number of classes, and $F1_{i}$ is the averaged F1 macro
score for class $i$. The overall accuracy for all tokens are calculated as
follows:

$accuracy=\frac{\\#\;of\;tokens\;predicted\;accurately}{\\#\;of\;tokens}$

###  4.3 Evaluation

The table 2 presents the achieved results on both validation and test sets.
For both punctuation and ezafe models, our proposed method achieved a higher
F1 score regarding the related works in the fields. However, our proposed
method performed second place for ZWNJ’s. In the following, we will discuss
each model separately.

####  4.3.1 Punctuation

The proposed method achieved an F1 score of 92.13% in the test set. It is the
first result in the punctuation restoration task using the Bijankhan dataset.
However, we have found the work of [24] with an F1 score of 69.00% on a
different dataset.

####  4.3.2 Ezafe

The proposed approach for ezafe recognition achieved the first F1 score of
98.50% on the test set. Regarding obtained performance, the second best
performing approach is [25] that used XLMRoBERT and obtained an F1 score of
98.10% with 0.40% margin within our proposed model. In conclusion, the
proposed model for ezafe recognitions is the best performer model.

####  4.3.3 ZWNJ

The work of [22] used BERT-CRF with some additions to the training datasets.
It boosted their performance from an F1 score of 96.67% to 98.14%.
Nevertheless, in normal conditions, our proposed method performed 0.23%
higher. However, in the end, our model performed 2.24% less than the highest
approach in the task. However, they created a BIO structure while training
their model but we used only a binary classification scheme and achieved F1
score of 96.90%.

Table 2: Results Model | Accuracy | Recall | Precision | F1
---|---|---|---|---
Validation |  |  |  |
Punctuation | 98.59 | 91.10 | 93.28 | 92.09
Ezafe | 98.89 | 98.55 | 98.30 | 98.42
ZWNJ | 99.29 | 98.86 | 99.29 | 96.92
Test |  |  |  |
Punctuation | 98.60 | 90.13 | 94.39 | 92.13
Ezafe | 98.94 | 98.60 | 98.40 | 98.50
ZWNJ | 99.29 | 96.71 | 97.09 | 96.90

##  5 Conclusion

In this paper, we presented ViraPart framework for automatic text refinements
for various NLP tasks in the Persian language such as part-of-speech tagging,
named entity recognition, ASR, grapheme to phoneme, spell checker, text
summarization, and machine translation, etc. The ViraPart consists of 3
models, namely punctuation restoration, ZWNJ recognition, and ezafe
recognition. We fine-tuned ParsBERT as a sequence labeling model for tasks and
achieved promising results. We obtained averaged F1 macro score of 92.13%,
98.50%, and 96.90% for punctuation restoration, ezafe recognition, and ZWNJ
recognition, respectively. The presented results show the effectiveness of the
proposed framework on text refinement.

## Acknowledgment

The authors would like to give big thanks to Part AI Research Center (the
biggest AI company in Iran) for supporting and funding this research.

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