Nearly Parallel $\mathrm{G}_{2}$-Structures with Torus Symmetry

TL;DR

This paper investigates nearly parallel G2-structures with torus symmetry, using multi-moment map techniques to relate the geometry of these structures to three-dimensional data and construct examples with four-torus symmetry.

Contribution

It introduces a method to analyze nearly parallel G2-structures with torus symmetry and constructs new examples from three-dimensional geometric data.

Findings

Torus actions produce multi-moment maps on G2-manifolds.

The geometry of base spaces is determined by triples of closed two-forms.

Inverse construction yields invariant G2-structures from 3D data.

Abstract

We study nearly parallel $\mathrm{G}_{2}$-structures with a three-torus symmetry via multi-moment map techniques. An effective three-torus action on a nearly parallel $\mathrm{G}_{2}$-manifold yields a multi-moment map. The torus acts freely on its regular level sets, so they are torus bundles over smooth three-dimensional manifolds. We show that the geometry of the base spaces is specified by two triples of closed two-forms related by a Riemannian metric. We then describe an inverse construction producing invariant nearly parallel $\mathrm{G}_{2}$-structures from three-dimensional data. We observe that locally this may produce examples with four-torus symmetry.

Equations

$$6g(X,Y)\mathrm{vol}:=(X┘\phi )\wedge (Y┘\phi )\wedge \phi .$$

$$\phi$$

$$\star \phi$$

$$g(X\times Y,Z):=\phi (X,Y,Z).$$

$$d\phi =4\star \phi ,$$

$$\nu :=\phi (U_1,U_2,U_3)$$

$$d\nu =-4\star \phi (U_1,U_2,U_3,\cdot ).$$

$$\left(\begin{array}{c}{U}_1\\ U_2\\ U_3\end{array}\right)=V\left(\begin{array}{c}{E}_1\\ E_2\\ E_3\\ E_4\end{array}\right),\text{where}V=\left(\begin{array}{cccc}p& 0& 0& 0\\ q_1& q_2& 0& 0\\ r_1& r_2& r_3& r_4\end{array}\right)$$

$$\nu =p{q}_2{r}_3\text{and}d\nu =4p{q}_2{r}_4{e}^7.$$

$$h:=\det H=p^2{q}_2^2{\tilde{r}}^2,$$

$${\tilde{r}}^2=r_3^2+r_4^2.$$

$$\rho :=h-{\nu }^2=\frac{1}{16}|d\nu {|}^2.$$

$$H\theta =U^{♭}.$$

$${\alpha }_i:=\nu {\theta }_i-\phi (U_j,U_k,\cdot )$$

$$\theta =X\left(\begin{array}{c}{e}^1\\ e^2\\ e^3\\ e^4\end{array}\right),\alpha =p{q}_2{r}_{4}X\left(\begin{array}{c}{e}^6\\ -e^5\\ e^4\\ -e^3\end{array}\right),$$

$$X=\frac{1}{p{q}_2{\tilde{r}}^2}\left(\begin{array}{ccc}{q}_2& -q_1& q_1{r}_2-q_2{r}_1\\ 0& p& -p{r}_2\\ 0& 0& p{q}_2\end{array}\right)\left(\begin{array}{cccc}{\tilde{r}}^2& 0& 0& 0\\ 0& {\tilde{r}}^2& 0& 0\\ 0& 0& r_3& r_4\end{array}\right).$$

$$\star d\nu =\frac{4h}{\rho }{\theta }_{123}\wedge {\alpha }_{123}.$$

$$\phi :={\iota }^{\ast }(E┘\Psi ),\psi :={\iota }^{\ast }\Psi ,$$

$$V_i:=-x^{2i+1}{\partial }_{2i}+x^{2i}{\partial }_{2i+1},\text{for i=0,1,2,3.}$$

$$\Psi =d{x}^{0123}-{\gamma }_1\wedge {\beta }_5-{\gamma }_2\wedge {\beta }_6-{\gamma }_3\wedge {\beta }_7+d{x}^{4567},$$

$${\mathcal{L}}_{V_i}\Psi =-{\mathcal{L}}_{V_i}({\gamma }_2\wedge {\beta }_6+{\gamma }_3\wedge {\beta }_7)={\epsilon }_i({\gamma }_3\wedge {\beta }_6-{\gamma }_2\wedge {\beta }_7),i=0,1,2,3,$$

$$U_1:=V_0+V_3,U_2:=V_1+V_3,U_3:=V_2-V_3$$

$$\nu$$

$$={\iota }^{\ast }(\Psi (E,V_0+V_3,V_1,V_2)-\Psi (E,V_0,V_1+V_2,V_3))$$

$$=4((x^0{x}^2-x^1{x}^3)(x^4{x}^7+x^5{x}^6)+(x^1{x}^2+x^0{x}^3)(x^5{x}^7-x^4{x}^6)).$$

$$\nu =4\mathrm{Im}(z^1{z}^2{z}^3{z}^4).$$

$$\begin{array}{rl}d\nu & =4(x^2{x}^4{x}^{7}d{x}^0+x^0{x}^4{x}^{7}d{x}^2+x^0{x}^2{x}^{7}d{x}^4\\ & +x^0{x}^2{x}^{4}d{x}^7+x^0{x}^2{x}^{6}d{x}^5-x^2{x}^4{x}^{6}d{x}^1-x^0{x}^4{x}^{6}d{x}^3).\end{array}$$

$$\begin{array}{c}{x}^2{x}^4{x}^6=0,x^0{x}^4{x}^6=0,x^0{x}^2{x}^6=0,0=\lambda x^6,\\ x^2{x}^4{x}^7=\lambda x^0,x^0{x}^4{x}^7=\lambda x^2,x^0{x}^2{x}^7=\lambda x^4,x^0{x}^2{x}^4=\lambda x^7.\end{array}$$

$${\mathrm{Hess}}_q=ϵ\left(\begin{array}{cc}{H}_1& 0\\ 0& H_2\end{array}\right),\text{with}{H}_1=\frac{1}{2}(1_{4\times 4}-3{\mathrm{Id}}_4),H_2=-1_{4\times 4},$$

$$\phi =-{\eta }_I\wedge {\omega }_I+{\eta }_J\wedge {\omega }_J+{\eta }_K\wedge {\omega }_K,\star \phi =\frac{1}{4}d\phi =\frac{1}{2}(-{\omega }_I^2+{\omega }_J^2+{\omega }_K^2).$$

$$\tilde{\phi }=a{a}^2{\eta }_{IJK}+a{b}^2({\eta }_I\wedge {\beta }_I+{\eta }_J\wedge {\beta }_J+{\eta }_K\wedge {\beta }_K)$$

$$\phi =\eta \wedge \omega +\mathrm{Re}\Omega .$$

Full text

Introduction

A $\mathrm{G}_{2}$-structure on a seven-dimensional manifold $M$ is a reduction of the principal frame bundle to the compact, connected, simply connected, simple Lie group $\mathrm{G}_{2}$. This is equivalent to having a global three-form $\varphi$ on $M$ that is pointwise linearly equivalent to a standard model on $\mathbb{R}^{7}$ with the cross product defined by multiplication of imaginary octonions [8]. In particular the three-form $\varphi$ induces a Riemannian metric $g$ and an orientation on $M$, and hence, a Hodge star operator $\star$.

In this paper, we are interested in nearly parallel $\mathrm{G}_{2}$-structures, which are the ones for which the three-form $\varphi$ satisfies the condition $d\varphi=\lambda\,{\star\varphi}$, for a positive constant $\lambda$ [19]. For such structures the induced Riemannian metric is Einstein with positive scalar curvature [5, 9], so complete examples are necessarily compact, and the geometries are equivalently described as seven-manifolds with a non-parallel real Killing spinor [4, 17]. By [2], the holonomy of the Riemannian eight-dimensional cone over a nearly parallel $\mathrm{G}_{2}$ manifold may be (a) the full group $\mathrm{Spin}(7)$, or one of the subgroups (b) $\mathrm{SU}(4)$, (c) $\mathrm{Sp}(2)$, (d) ${1}$, according to the dimension of the space of Killing spinors. As will be discussed below, examples of nearly parallel $\mathrm{G}_{2}$-structures exist in abundance. However, case (d) only occurs when the metric is locally isometric to the round seven-dimensional sphere.

In this paper we focus on nearly parallel $\mathrm{G}_{2}$-structures with torus symmetry. Several examples of compact nearly parallel $\mathrm{G}_{2}$-manifolds with large symmetry groups have been classified. The homogeneous examples are to be found in [17], which includes tables indicating the full isometry group. In general, this is larger than the group of automorphisms of the $\mathrm{G}_{2}$-structure, as there can be isometries that do not preserve the three-form. However, Reidegeld [30, Theorem 1, part 2, and Table 2] summarises the automorphism groups, and one sees that the rank of the symmetry group in the homogeneous examples is at most $3$. These include the remarkable family of Aloff-Wallach spaces $\mathrm{SU}(3)/T^{1}_{k,\ell}$, described as simply-connected spaces of the form $\mathrm{U}(3)/T^{2}_{a,b}$, with $T^{2}_{a,b}\cap\mathrm{SU}(3)$ one-dimensional and $T^{2}_{a,b}\cap\mathbb{C}\mathrm{Id}_{3}$ zero-dimensional. We therefore focus on geometries with $T^{3}$-symmetry.

The general Aloff-Wallach spaces are of class (a) above, and this family contains infinitely many distinct homotopy types [24] and they are all topologically formal [15]. Geometries in class (b) are Sasaki-Einstein, their cone is Calabi-Yau. Infinitely many compact examples of these Sasaki-Einstein structures with $T^{3}$-symmetry may be constructed starting from ordinary toric geometry [11, 18], see also [14]. Geometries from class (c) are also $3$-Sasaki manifolds [7]. Here for each second Betti number $b_{2}>0$ there are infinitely many distinct compact homotopy types where the $\mathrm{G}_{2}$-structure has $T^{3}$-symmetry [7, 22]. The examples also give structures in class (a) via a canonical variation [17], and these new structures still have $T^{3}$-symmetry, and in fact at least $T^{2}\times\mathrm{SU}(2)$-symmetry. The general theories of Sasaki-Einstein and $3$-Sasaki manifolds, ensure that the rank of the symmetry group of the $\mathrm{G}_{2}$-structure cannot be higher than three in cases (b) and (c), which again justifies the choice of focussing on $T^{3}$-symmetry.

For a general nearly parallel $\mathrm{G}_{2}$-manifold $M$ with $T^{3}$-symmetry, we will show how to construct a multi-moment $\nu\colon M\to\mathbb{R}$ in the sense of [26]. This leads us to investigate the geometry of the quotients $Q=\nu^{-1}(s)/T^{3}$ of the level sets of $\nu$, when $s$ is a non-zero, regular value. These $Q$ are smooth three-manifolds which carry a surprising amount of geometric data. As $\nu^{-1}(s)$ is a $T^{3}$-bundle over $Q$, there is, by the usual Chern-Weil theory, a cohomology class in the integral lattice of $H^{2}(Q,\mathbb{Z}^{3})\otimes\mathbb{R}=H^{2}(Q,\mathbb{R}^{3})$ describing this bundle via its curvature. However, in our situation when $Q$ comes from a $\mathrm{G}_{2}$-structure as above, we show that the class has a representative $\tau/s$, $\tau=(\tau_{1},\tau_{2},\tau_{3})^{T}\in\Omega^{2}(Q,\mathbb{R}^{3})$, for which the three-components are pointwise linearly independent closed two-forms, even when $b_{2}(Q)=0$. Additionally, $Q$ has a Riemannian metric represented by $H\in C^{\infty}(Q,M_{3}(\mathbb{R}))$, such that $\sigma\coloneqq H\tau$ is also a closed $\mathbb{R}^{3}$-valued two-form. At a given $s$, we show that the nearly parallel $\mathrm{G}_{2}$-structure with $T^{3}$-symmetry can be recovered from the data $\tau$ and $H$, or equivalently $\tau$ and $\sigma$, under a particular geometric flow. As a consequence of this inverse construction, we are able to assert the local existence of nearly parallel $\mathrm{G}_{2}$-structures with $T^{4}$-symmetry, rather than just $T^{3}$.

Nearly parallel $\mathrm{G}_{2}$-structures share some features with nearly Kähler manifolds in dimension six [20, 19]. Indeed, the two geometries are the only exceptional cases in the classification [13, Theorem 5.14] of geometries with metric connection whose torsion is parallel and whose holonomy representation is irreducible. Both geometries are Einstein of positive scalar curvature, and their cones have special Riemannian holonomy [20, 32, 2, 8]. A significant difference is that in the nearly Kähler case, there are currently very few complete examples [10, 16]. In [31, 33] we studied nearly Kähler examples with $T^{2}$-symmetry. Again there is a multi-moment map and quotients of regular level sets are three-manifolds with geometric structure, but it was not at all clear how that structure would change when passing to the case of nearly parallel $\mathrm{G}_{2}$-structures with $T^{3}$-symmetry. The $\mathrm{G}_{2}$-case also turns to have a technical difficulty, in that it is not immediately apparent that the flow equations preserve the symmetry of the matrix $H$ determining the Riemannian structure on $Q$.

The nearly Kähler case admits examples of larger symmetry rank, namely the three-symmetric metric on $S^{3}\times S^{3}=\mathrm{SU}(2)^{3}/\mathrm{SU}(2)_{\Delta}$ has $T^{3}$-symmetry. Nearly Kähler six-manifolds with $T^{3}$-symmetry have been described locally by Mororianu & Nagy [28], but at this time no further complete examples have been found. Similarly, in the nearly parallel $\mathrm{G}_{2}$-case, no complete examples with the larger $T^{4}$-symmetry are currently known.

The structure of the paper is the following. In Section 1 we construct the multi-moment map $\nu$, prove that $\nu$ always has regular values and that the three-torus $T^{3}$ acts freely on the corresponding level sets of $\nu$. We see that if $p$ is a critical point with $\nu(p)$ non-zero, then the $T^{3}$-orbit through $p$ is an associative and minimal submanifold. As an example of the set-up, we consider structures on the round seven-sphere, showing that the quotient $Q=\nu^{-1}(s)/T^{3}$, for $s$ any regular value, is smoothly the three-sphere $S^{3}$. We also note that results of [23] for the squashed metric on the seven-sphere lead to the same conclusion. We also give a brief description of the relations to Sasaki-Einstein and $3$-Sasaki geometry in the cases (b) and (c) above in Section 1.2. For a regular value $s$ of $\nu$, we determine the induced geometry on the three-manifold $Q=\nu^{-1}(s)/T^{3}$ in Section 2. Section 3 is devoted to an inverse construction. Finally in Section 4, we discuss several local constructions of nearly parallel $\mathrm{G}_{2}$-structures, a couple of particular examples and the assertion that examples with $T^{4}$-symmetry exist locally. Indeed we construct an incomplete example with a cohomogeneity-one action of $T^{3}\times\mathrm{SU}(2)$. Such examples have not been seen in previous discussion of cohomogeneity-one examples [12, 29].

Acknowledgements. The first named author is partially supported by INdAM-GNSAGA, and by the PRIN 2022 Project (2022K53E57) – PE1 – “Optimal Transport: new challenges across analysis and geometry”. We thank Thomas Bruun Madsen for useful conversations on topics related to this paper.

1. Torus symmetry and multi-moment map

Let $M$ be a connected manifold of dimension seven. A $\mathrm{G}_{2}$-structure on $M$ is a three-form $\varphi$ that induces a Riemannian metric $g$ and volume form on $M$ via the formula

[TABLE]

We let $\star$ be the Hodge star operator induced by $g$ and the volume form. Pointwise, $\varphi$ and $\star\varphi$ can be written in terms of a coframe $e^{1},\dots,e^{7}$ as

[TABLE]

where the shorthand $e^{ijk}$ stands for $e^{i}\wedge e^{j}\wedge e^{k}$, etc. We will use the same notational convention on different forms. Notice that at each point $\mathrm{vol}=e^{12\dots 7}$, $\varphi\wedge{\star\varphi}=7\mathrm{vol}$, and $g=\sum_{k=1}^{7}(e^{k})^{2}$. We recall that $\varphi$ and $g$ define a cross-product via the identity

[TABLE]

A three-plane in $T_{x}M$ is associative if it is closed under the cross-product, see [21, IV.1.A. Definition 1.3].

We will denote by $E_{1},\dots,E_{7}$ the dual vectors of $e^{1},\dots,e^{7}$. The condition that $(M,\varphi)$ is a nearly parallel $\mathrm{G}_{2}$-manifold is that $d\varphi$ is proportional to $\star\varphi$. Up to a homothetic scaling, we can arrange that the nearly parallel condition reads

[TABLE]

cf. [8, p. 567].

Let $T^{3}$ be a three-dimensional torus acting on $(M,\varphi)$ effectively preserving $\varphi$. Let $U_{1}$, $U_{2}$, $U_{3}$ be the vector fields generated by the torus-action on $M$. The map

[TABLE]

is a multi-moment map, cf. Madsen-Swann [26]. Indeed, $\nu$ is $T^{3}$-invariant and satisfies

[TABLE]

The latter identity follows by Cartan’s formula, the fact that the Lie derivatives $\mathcal{L}_{U_{i}}\varphi$ vanish for each $i=1,2,3$, and $[U_{i},U_{j}]=0$ for all $i,j$.

Recall that $\mathrm{G}_{2}$ acts transitively on the Stiefel manifold of two-frames in $\mathbb{R}^{7}$ [8], and acts linearly on each tangent space of $M$ preserving $\varphi$. Note that $E_{3}=E_{1}\times E_{2}$, and that the subgroup of $\mathrm{G}_{2}$ preserving $E_{1},E_{2}$, and hence $E_{3}$, is isomorphic to $\mathrm{SU}(2)$ that acts transitively on the unit vectors orthogonal to $E_{1},E_{2},E_{3}$. Thus locally, up to the action of $\mathrm{G}_{2}$, we can then take $E_{1},\dots,E_{7}$ such that

[TABLE]

for some functions $p$, $q_{i}$, $r_{j}$. For such a basis, we have

[TABLE]

Proposition 1.1**.**

Critical points of $\nu$ may be characterised as follows:

1. (1)

$x\in\nu^{-1}(0)$* is a critical point for $\nu$ if and only if $U_{1},U_{2},U_{3}$ are linearly dependent at $x$.* 2. (2)

$x\in M\setminus\nu^{-1}(0)$* is a critical point for $\nu$ if and only if $\mathrm{Span}_{x}\{U_{1},U_{2},U_{3}\}$ is an associative three-plane.*

Thus $T^{3}$-orbits of critical points $x$ with $\nu(x)\neq 0$ are associative, hence minimal, submanifolds.

Proof.

In the following all expressions are at a fixed $x\in M$, and we use the local representation of $U_{i}$ above.

For (1), suppose $\nu(x)=0$ and $d\nu_{x}=0$. Then $pq_{2}=0$ or $r_{3}^{2}+r_{4}^{2}=0$. It is then clear by the above pointwise expressions that $U_{1},U_{2},U_{3}$ are linearly dependent. The converse is obvious by the global expressions for $\nu$ and $d\nu$.

Next, for (2), assume $x$ is a critical point for $\nu$ but $\nu(x)\neq 0$. We have that $pq_{2}r_{3}\neq 0$, $r_{4}=0$, and the vectors $U_{1},U_{2},U_{3}$ are linearly independent, by part (1). Thus the span of $U_{1},U_{2},U_{3}$ is that of $E_{1},E_{2},E_{3}$, which is associative.

Conversely, if the span is three-dimensional and associative, then the local form gives that $U_{3}$ is a non-zero linear combination $U_{3}=aU_{1}+bU_{2}+c\,U_{1}\times U_{2}$, with $c\neq 0$. Thus $r_{4}=0$, so $d\nu_{x}=0$ and $x$ is critical. Also, $\nu(x)=c\varphi_{x}(U_{1},U_{2},U_{1}\times U_{2})=c\lVert U_{1}\times U_{2}\rVert^{2}\neq 0$, as $U_{1}$ and $U_{2}$ are linearly independent.

Minimality of associative submanifolds of nearly parallel $\mathrm{G}_{2}$-manifolds was noted in [25, Corollary 3.6] and [3], for example: a submanifold $N\subset M$ is associative if and only if the cone $C(N)=\mathbb{R}_{>0}\times N\subset(C(M),\overline{g}=dt^{2}+t^{2}g)$ is calibrated for the cone $\mathrm{Spin}(7)$-structure given by $\Psi=dt\wedge\varphi+{\star\varphi}$. Thus $C(N)$ is minimal in the cone by [21]. But $C(N)\subset C(M)$ and $N\subset M$ have identical mean curvature vectors on $N=\{1\}\times N$, since $\overline{\nabla}_{\partial_{t}}\partial_{t}=0$ and $(\overline{\nabla}_{X}X)^{\perp C(M)}=(\nabla_{X}X+(1/t)g(X,X)\partial_{t})^{\perp C(M)}_{t=1}=(\nabla_{X}X)^{\perp M}$ for $X\in N$ extended by $X_{(t,p)}=X_{p}$. ∎

Let us write $H_{ij}=g(U_{i},U_{j})$ and $H=(H_{ij})_{i,j=1,2,3}$. So $H=VV^{T}$, and

[TABLE]

for

[TABLE]

From (1.4), we get $\det H=\nu^{2}+\frac{1}{16}\lvert d\nu\rvert^{2}$, or equivalently

[TABLE]

This implies that $\nu$ cannot be identically zero. If it were, then $h=\det H\equiv 0$ and the vectors $U_{1}$, $U_{2}$, $U_{3}$ would be linearly dependent at each point; but this contradicts the effectiveness of the $T^{3}$-action. We show later that $\nu$ cannot be constant in general.

Let $\theta_{i}$ be the dual one-form to $U_{i}$, and write $\theta=(\theta_{1},\theta_{2},\theta_{3})^{T}$, $U^{\flat}=(U_{1}^{\flat},U_{2}^{\flat},U_{3}^{\flat})^{T}$. Then

[TABLE]

We define $\alpha=(\alpha_{1},\alpha_{2},\alpha_{3})^{T}$, a vector of basic $T^{3}$-invariant one-forms, by

[TABLE]

for $(ijk)=(123)$ as permutations. A calculation yields the following pointwise expressions

[TABLE]

where

[TABLE]

As $X$ has rank $3$, we see that away from the critical points of $\nu$, the seven one-forms $d\nu$, $\theta_{i}$, $\alpha_{j}$, $i,j=1,2,3$, constitute a basis of the cotangent space of $M$.

Proposition 1.2**.**

The image of the multi-moment map $\nu\colon M\to\mathbb{R}$ is an interval with non-empty interior. If $M$ is complete, then $\nu(M)=[a,b]$ is a compact interval with $a<b$.

Proof.

For the first point, since $M$ is connected, it is enough to show that $\nu$ is not constant on $M$. A computation using the pointwise expressions of the forms gives

[TABLE]

In particular, if $\nu$ were constant, then $h=\det H$ should vanish identically, and hence $U_{1}$, $U_{2}$, $U_{3}$ would be linearly dependent at each point of $M$. This contradicts the effectiveness of the torus-action. When $M$ is complete it is compact, by Myers’ Theorem, so $\nu(M)$ is also compact. ∎

Proposition 1.3**.**

The multi-moment map $\nu$ has regular values.

For any regular value $s$, the $T^{3}$-action on $\nu^{-1}(s)$ is free, so $\nu^{-1}(s)\to\nu^{-1}(s)/T^{3}$ is a principal torus-bundle over a three-dimensional smooth manifold.

Proof.

By Sard’s Theorem, the set of critical values has zero Lebesgue measure in $\mathbb{R}$, so there are infinitely many regular values for $\nu$ in $(a,b)\neq\varnothing$.

Let us fix a regular value $s$, and let $x$ be any point in the level set $\nu^{-1}(s)$. Then $d\nu_{x}\neq 0$, so $U_{1}$, $U_{2}$, $U_{3}$ are linearly independent at $x$, and the torus-action is locally free.

We show that the action is free. If $K$ is the stabiliser in $T^{3}$ of a point $x\in\nu^{-1}(s)$, then it preserves $d\nu$, $\theta_{i}$, and $\alpha_{j}$ for all $i,j=1,2,3$, so all of $T_{x}M$. By the equivariant tubular neighbourhood theorem, the set $A\coloneqq\{y\in M:ky=y,dk_{y}=\mathrm{Id}_{T_{y}M}\text{ for all }k\in K\}$ is open and closed in $M$. Since $M$ is connected and $A$ is non-empty, $A=M$. But then $K$ must be trivial, otherwise the torus-action on $M$ would not be effective. ∎

1.1. The seven sphere

We consider first the round seven-sphere $S^{7}\subset V=\mathbb{R}^{8}$ as a nearly parallel $\mathrm{G}_{2}$-structure, cf. [1, 25]. Let $(x^{0},\dots,x^{7})$ be the standard Euclidean coordinates, $\partial_{i}$ the corresponding vector fields, and consider $\mathbb{R}^{7}\subset\mathbb{R}^{8}$ spanned by $x^{1},\dots,x^{7}$. On $\mathbb{R}^{7}$, we have the constant coefficient $\mathrm{G}_{2}$-form $\varphi_{0}$ and its dual ${\star}_{\mathbb{R}^{7}}\varphi_{0}$ given by (1.2) and (1.3) with $e^{i}=dx^{i}$. Defining $\Psi\coloneqq dx^{0}\wedge\varphi_{0}+{\star}_{\mathbb{R}^{7}}\varphi_{0}$, we get a four-form on $\mathbb{R}^{8}$, whose stabiliser under the action of $\mathrm{GL}(8,\mathbb{R})$ is $\mathrm{Spin}(7)\subset\mathrm{SO}(8)$. We get $S^{7}=\mathrm{Spin}(7)/\mathrm{G}_{2}$.

Note that the Euler vector field $E=\sum_{k=0}^{7}x^{k}\partial_{k}$ on $\mathbb{R}^{8}$ restricts to a unit normal $N$ of $S^{7}$ and $\mathcal{L}_{E}x^{i}=x^{i}$ for each $i$. As $\Psi$ has constant coefficients, we get $\mathcal{L}_{E}\Psi=4\Psi$. Putting

[TABLE]

we get from $d\Psi=0$ that $d\varphi=\iota^{*}(d(E\mathbin{\lrcorner}\Psi)+E\mathbin{\lrcorner}d\Psi)=\iota^{*}(\mathcal{L}_{E}\Psi)=\iota^{*}(4\Psi)=4\psi$. But $\mathrm{Spin}(7)$ acts by isometries on $S^{7}$, and at $(1,0,\dots,0)$, $\psi={\star}_{\mathbb{R}^{7}}\varphi_{0}={\star}\varphi$. Thus $\psi$ is the Hodge dual of $\varphi$, and so $\varphi$ equips $S^{7}$ with a nearly parallel $\mathrm{G}_{2}$-structure for which the induced metric is the round one.

On $\mathbb{R}^{8}$ we have four commuting circle actions generated by

[TABLE]

Write

[TABLE]

where $\gamma_{1}=dx^{01}+dx^{23}$, $\gamma_{2}=dx^{02}+dx^{31}$, $\gamma_{3}=dx^{03}+dx^{12}$, and $\beta_{5}=dx^{45}+dx^{67}$, $\beta_{6}=dx^{46}+dx^{75}$, $\beta_{7}=dx^{47}+dx^{56}$. Then, we have

[TABLE]

where $\varepsilon_{0}=-1=\varepsilon_{1}$, $\varepsilon_{2}=+1=\varepsilon_{3}$. So the three vector fields

[TABLE]

preserve $\Psi$. They generate the maximal torus $T^{3}$ of $\mathrm{Spin}(7)$. The action then preserves the nearly parallel $\mathrm{G}_{2}$-structure $\varphi$ on $S^{7}$, and we get a multi-moment map $\nu$. A direct computation gives

[TABLE]

Note that writing $z^{1}=x^{0}-ix^{1}$, $z^{2}=x^{2}-ix^{3}$, $z^{3}=x^{4}+ix^{5}$, $z^{4}=x^{6}+ix^{7}$, we have

[TABLE]

Proposition 1.4**.**

For the above nearly parallel $\mathrm{G}_{2}$-structure on the round seven-sphere, the image of the multi-moment map (1.7) is $\nu(S^{7})=[-1/4,1/4]$.

The only non-zero critical values are the maximum and minimum $\pm 1/4$, and the corresponding critical sets are diffeomorphic to $T^{3}$.

The $T^{3}$-quotient of the critical points in $\nu^{-1}(0)$ form a trivalent graph with four vertices and six edges connecting each pair of distinct vertices.

Each $s\in(-1/4,0)\cup(0,1/4)$ is a regular value of $\nu$, and $\nu^{-1}(s)/T^{3}$ is diffeomorphic to a three-sphere.

Proof.

Using the action of $T^{3}$, we move any given point to one with $x^{1}=0=x^{3}=x^{5}$ and $x^{0}\geq 0$, $x^{2}\geq 0$, $x^{4}\geq 0$. At points with $x^{1}=0=x^{3}=x^{5}$, we have in $\mathbb{R}^{8}$ that

[TABLE]

Critical points for $\nu$ on $S^{7}$ in the slice $x^{1}=0=x^{3}=x^{5}$, have $d\nu$ proportional to $\sum_{i=0}^{7}x^{i}dx^{i}=\sum_{i=0,2,4,6,7}x^{i}dx^{i}$. So

[TABLE]

If $\nu\neq 0$, then $x^{j}\neq 0$ for $j=0,2,4,7$. It follows that $\lambda\neq 0$ and thus $x^{6}=0$. Also, we have $(x^{0})^{2}=x^{0}x^{2}x^{4}x^{7}/\lambda=(x^{2})^{2}=(x^{4})^{2}=(x^{7})^{2}$. But $\sum_{j=0,2,4,7}(x^{j})^{2}=1$, so each $(x^{j})^{2}$ is $1/4$ and $\nu=\pm 1/4$. Thus these are the maximum and minimum values of $\nu$ and the image is as claimed. As we can rotate to $x^{k}\geq 0$, $k=0,2,4$, we get $x^{k}=1/2$ for these $k$, and $x^{7}=\pm 1/2$. Thus the $T^{3}$-orbit has a unique representative, $T^{3}$ acts freely at these points and these two critical sets are copies of $T^{3}$.

For $\nu=0$, the equations (1.8) give that in each of the following sets $\{x^{0},x^{2},x^{4},x^{6}\}$ and $\{x^{0},x^{2},x^{4},x^{7}\}$ two variables are zero. Thus in $S^{7}\subset\mathbb{C}^{4}$, the critical sets are three-spheres that are the intersection of $S^{7}$ with complex planes in $\mathbb{C}^{4}$ of the form $z^{i}=0=z^{j}$, $i\neq j$. Any two such three spheres meet in a circle $C_{i}$ given by $\lvert z^{i}\rvert=1$, $z^{k}=0$, for $k\neq i$. The $T^{3}$-quotient of the critical points in $\nu^{-1}(0)$, thus form a graph with vertices corresponding to the circles $C_{i}$, and edges corresponding to the three spheres. Thus the graph has the form claimed.

Let $q=(1/2,0,1/2,0,1/2,0,0,\epsilon/2)$, $\epsilon\in\{\pm 1\}$ be our preferred point in $\nu^{-1}(\epsilon/4)$. Then $q$ is a critical point for $\nu$, and we may compute the Hessian $\operatorname{Hess}$ of $\nu$ at $q$ as follows: for $i=0,\dots,7$, let $X_{i}=\partial_{i}-x^{i}E$, which is the projection of $\partial_{i}$ to the tangent bundle $TS^{7}$ of $S^{7}$. At a general point, the Hessian of $\nu$ is given by $\operatorname{Hess}(X_{i},X_{j})=X_{i}(X_{j}\nu)-(\nabla_{X_{i}}X_{j})\nu$, where $\nabla$ is the Levi-Civita connection on $S^{7}$. At a critical point, such as $q$, the last term is zero, since $Y_{q}\nu=0$ for all tangent vectors $Y$. Thus $\operatorname{Hess}(X_{i},X_{j})_{q}=X_{i}(X_{j}\nu)$. A direct computation gives that the matrix of $\operatorname{Hess}_{q}$ with respect to the spanning set $X_{0},X_{2},X_{4},\epsilon X_{7},X_{1},X_{3},-X_{5},\epsilon X_{6}$ is

[TABLE]

where $1_{n\times m}$ is the $(n\times m)$-matrix with all entries equal to $1$, and $\mathrm{Id}_{n}$ is the $n\times n$ identity matrix. It follows that $\operatorname{Hess}_{q}$ has rank $4$. As the dimension of the $T^{3}$-orbit through $q$ is of dimension $3$, it follows that the rank of $\operatorname{Hess}_{q}$ is maximal. Since $q$ is a maximum or minimum for $\nu$, the equivariant Morse lemma then implies that there are coordinates $(y^{1},\dots,y^{4},z^{1},z^{2},z^{3})$ on $S^{7}$ near $q$, so that $\nu(y,z)=\epsilon\sum_{j=1}^{4}(y^{i})^{2}=\epsilon\lVert y\rVert^{2}$. Thus the $T^{3}$-orbit through $q$ has a tubular neighbourhood of the form $T^{3}\times\mathbb{R}^{4}$, where $\nu|_{\mathbb{R}^{4}}=\epsilon\lVert\cdot\rVert^{2}$. Thus for some small $\delta$, each $s$ with $\lvert s\rvert\in I_{\delta}=(1/4-\delta,1/4)$ has that $\nu^{-1}(s)$ is equivariantly diffeomorphic to $T^{3}\times S^{3}\subset T^{3}\times\mathbb{R}^{4}$, and $\nu^{-1}(s)/T^{3}$ is diffeomorphic to $S^{3}$. As there are no critical values for $\nu$ in $(-1/4,0)\cup(0,1/4)$, the diffeomorphism type of $\nu^{-1}(t)$ and $\nu^{-1}(t)/T^{3}$ for $\lvert t\rvert\in(0,1/4)$ is the same as that for $\lvert s\rvert\in I_{\delta}$, and the result follows. ∎

Remark 1.5*.*

That the sets given here as the maximum and minimum of $\nu$ are the only $T^{3}$-invariant associative submanifolds of round $S^{7}$ was already noted in [25] as a direct consequence of [21]. So Proposition 1.1 already gives that there are no other non-zero critical values.

Remark 1.6*.*

In [23], Kawai studies the seven-sphere with its squashed metric and $\mathrm{G}_{2}$-structure, and in particular $T^{3}$-invariant associative submanifolds in [23, Proposition 6.1]. Using the computations of that paper, and adjusting the choice of linear coordinates, one finds that the multi-moment map is a non-zero constant multiple of the function $\nu$ given in (1.7). Thus, after scaling, the results of Proposition 1.4, also apply to the squashed metric, and the determined associative submanifolds agree.

1.2. Special classes

As described in the introduction, there are several classes of nearly parallel $\mathrm{G}_{2}$-geometries on seven-dimensional manifolds $M$, characterised by the dimension of the space of Killing spinors [2, 17]. Let us give a few concrete details, to clarify the symmetry considerations. A general geometric reference is [7].

$3$-Sasaki manifolds. The eight-dimensional metric cone $C=\mathbb{R}\times M$, $g_{C}=dr^{2}+r^{2}g$ is hyperKähler, with closed two-forms $\omega_{I}^{C},\omega_{J}^{C},\omega_{K}^{C}$, related by $\omega_{A}^{C}=g_{C}(A\cdot{},{}\cdot{})$, for $A=I,J,K$, and with $A^{2}=-\mathrm{Id}$ and $IJ=K=-JI$. Writing $\iota\colon M\to\{1\}\times M\subset C$ for the inclusion, and $E=r\partial_{r}$ for the Euler vector field on $C$, we get one-forms $\eta_{A}=\iota^{*}(E\mathbin{\lrcorner}\omega_{A}^{C})$ and two-forms $\omega_{A}=\iota^{*}(\omega_{A}^{C})$, satisfying $d\eta_{A}=2\omega_{A}$, for $A=I,J,K$. The vector fields dual to $\eta_{A}$ are the restrictions of $AE$, and generate a locally free action of $\mathrm{SU}(2)$ acting as rotation on $\mathrm{Span}\{\eta_{I},\eta_{J},\eta_{K}\}$ and $\mathrm{Span}\{\omega_{I},\omega_{J},\omega_{K}\}$.

Bielawski [6] showed that the symmetry group of the $3$-Sasaki $(M,g,\eta_{I},\eta_{J},\eta_{K})$ has rank at most $2$, when $\eta_{A}$ are all preserved individually and that all examples are obtained as $3$-Sasaki quotients of higher-dimensional spheres. Boyer & Galicki [7] gave explicit constructions showing that are infinitely many compact seven-dimensional examples, and the cohomology of these was computed by Hepworth [22]. Note that the $T^{2}$-action in these examples commutes with the above action of $\mathrm{SU}(2)$.

Fixing the choice $A=I$, there is a nearly parallel $\mathrm{G}_{2}$-structure with

[TABLE]

When the $3$-Sasaki structure has $T^{2}$-symmetry, this $\mathrm{G}_{2}$-structures gets $T^{3}=T^{2}\times S^{1}$-symmetry, where $S^{1}\subset\mathrm{SU}(2)$ is the subgroup preserving $I$. The action of $\mathrm{SU}(2)$ on $\varphi$, now gives a two-dimensional family of nearly parallel $\mathrm{G}_{2}$-structures with the same metric.

On the other hand Friedrich et al. [17] showed there is another $\mathrm{G}_{2}$-structure. Regarding $\mathcal{H}=\ker\eta_{I}\cap\ker\eta_{J}\cap\ker\eta_{K}$ as a horizontal distribution transverse to the $\mathrm{SU}(2)$-orbits, one may make a canonical variation. Explicitly, putting $\beta_{A}=\omega_{A}-\eta_{BC}$, for $(ABC)=(IJK)$, the three-form

[TABLE]

is a $\mathrm{G}_{2}$-form with metric $\tilde{g}=a^{2}(\eta_{I}^{2}+\eta_{J}^{2}+\eta_{K}^{2})+b^{2}g_{\mathcal{H}}$. One may check that this $\mathrm{G}_{2}$-structure is nearly parallel only for $a=3/5$, $b=3/\sqrt{5}$, see also [23], indeed this is the squashed metric of the previous section. This time $\tilde{\varphi}$ is preserved by the $\mathrm{SU}(2)$-action, so if the $3$-Sasaki manifold has $T^{2}$-symmetry, then the canonical variation $\tilde{\varphi}$ has $T^{2}\times\mathrm{SU}(2)$-symmetry, and is preserved by any maximal subgroup $T^{3}$-symmetry. The cone on $\tilde{g}$ has full holonomy $\mathrm{Spin}(7)$.

Sasaki-Einstein manifolds. Here the eight-dimensional metric cone is Calabi-Yau, so carries a Kähler form $\omega_{C}$, with complex structure $I$, and a complex volume form $\Omega_{C}$. Note that $S^{1}$ generated by $IE$, acts on $\Omega_{C}$ as multiplication by $e^{it}$. The Kähler induces a one-form $\eta=\iota^{*}(E\mathbin{\lrcorner}\omega_{C})$ on $M$, with $d\eta=2\omega$, for $\omega=\iota^{*}(\omega_{C})$. One may regard $\omega$ as a Hermitian form on $\mathcal{H}=\ker\eta$. Putting $\Omega=\iota^{*}(E\mathbin{\lrcorner}\Omega_{C})$ gives a $(3,0)$-form on $\mathcal{H}$ that satisfies $d\Omega=4i\eta\wedge\Omega$. The geometry on $M$ is Sasaki-Einstein, in particular the induced metric is Einstein with positive scalar curvature. A nearly parallel $\mathrm{G}_{2}$-structure compatible with this metric is given by

[TABLE]

But there are circle of these structures, obtained by replacing $\Omega$ by $e^{it}\Omega$, cf. [15].

One may construct examples, where Kähler structure of the cone has $T^{4}$-symmetry, by starting with methods of toric geometric geometry [27, 11, 18]. Such a $T^{4}$ action, necessarily acts non-trivially on $\Omega_{C}$, and the stabiliser gives a $T^{3}$-action that preserves the Calabi-Yau structure and the nearly parallel $\mathrm{G}_{2}$-structure. This construction includes the $3$-Sasaki examples as a special case.

2. Torus-reduction

Let $s$ be a non-zero regular value for $\nu$. We now wish to study the geometry of the principal bundle $\nu^{-1}(s)\to\nu^{-1}(s)/T^{3}$. We first write the nearly parallel $\mathrm{G}_{2}$-structure on $M$ in terms of the invariant forms $d\nu$, $\theta_{i}$, $\alpha_{j}$. Then we understand the geometry of the orbit space $\nu^{-1}(s)/T^{3}$ imposed by the nearly parallel condition.

By inverting the system of identities from via (1.4) and (1.6), we get the expressions of the vectors $e^{1},\dots,e^{7}$ in terms of $d\nu,\theta_{i},\alpha_{j}$, for all $i,j$. One can then write the nearly parallel $\mathrm{G}_{2}$-structure (1.2)–(1.3), the induced metric, and the volume form in terms of the invariant one-forms. The results are the following:

[TABLE]

where $h=\det H$, $\rho=h-\nu^{2}$, $\mathrm{vol}_{\theta}=\theta_{123}$, $\mathrm{vol}_{\alpha}=\alpha_{123}$, $({\star}_{\gamma}\gamma)_{i}=\gamma_{jk}$, for $(ijk)=(123)$, and

[TABLE]

so

[TABLE]

Note that $\tau$ can also be written two other forms

[TABLE]

where $\operatorname{adj}H$ is the adjugate of $H$, so the matrix satisfying $H\operatorname{adj}H=h\mathrm{Id}_{3}$. It will be convenient to write

[TABLE]

On the other hand, differentiating the expressions for $\alpha_{i}$ in (1.5), we obtain

[TABLE]

and thus

[TABLE]

We also note

[TABLE]

We now wish to compare terms in $d\varphi=4\,{\star\varphi}$ and write an equivalent formulation of this identity in terms of the invariant one-forms $d\nu$, $\theta_{i}$, $\alpha_{j}$. First, one can compute $d\varphi$ explicitly from (2.3):

[TABLE]

where $(\gamma\barwedge\delta)_{i}=\gamma_{j}\wedge\delta_{k}$, for $(ijk)=(123)$. Using (2.7), we see that the first two terms of $d\varphi$ agree with the first three terms of $4\,{\star\varphi}$, so that $d\varphi-4\,{\star\varphi}$ only includes terms of degrees [math] and $1$ in $\theta$. The terms of degree [math] give

[TABLE]

and those of degree $1$ yield

[TABLE]

as $\alpha\barwedge\tau=\tau\barwedge\alpha$, by the symmetry of $H^{-1}$. Thus

[TABLE]

Let $s$ be a regular value for $\nu$. We are now ready to study the geometry of the torus-bundle $\nu^{-1}(s)\to\nu^{-1}(s)/T^{3}$. The vertical space at each point is generated by $U_{1}$, $U_{2}$, $U_{3}$, so the triple $\theta=(\theta_{1},\theta_{2},\theta_{3})^{T}$ is a connection one-form of the bundle. One checks that $\mathcal{L}_{U_{\ell}}\alpha_{i}=0$ and $\alpha_{i}(U_{\ell})=0$ for all $\ell=1,2,3$. So the one-forms $\alpha_{i}$, $i=1,2,3$, are basic and descend to a global coframe on $\nu^{-1}(s)/T^{3}$. Write $d_{3}$ for the exterior derivative on the base. Note that $H$ is also $T^{3}$-invariant, so $H$, $\beta$, $\sigma$ and $\tau$ also descend to the base. Pulling back (2.9) to $\nu^{-1}(s)$, gives

[TABLE]

On the other hand, (2.8) gives no relation on $\nu^{-1}(s)/T^{3}$.

Restricting the identities in (2.7) to $\nu^{-1}(s)$, and writing $d_{6}$ for the exterior derivative on this submanifold, we get the curvature two-form $d_{6}\theta$ on the three-manifold $\nu^{-1}(s)/T^{3}$. For $s\neq 0$, we have, omitting pull-backs,

[TABLE]

so

[TABLE]

and the cohomology class $[d_{6}\theta]$ is that of $-\tau/s$.

We can thus formulate the following result.

Proposition 2.1**.**

Let $s\neq 0$ be a regular value for $\nu$. On the three-manifold $\nu^{-1}(s)/T^{3}$, the two-form $\tau=(\tau_{1},\tau_{2},\tau_{3})^{T}$ of (2.5) is closed with components linearly independent at each point. The cohomology class $[\tau/s]$ is integral and the two-form $\sigma=H\tau$ of (2.6) is also closed. As a principal $T^{3}$-bundle over the base, $\nu^{-1}(s)$ has curvature given by (2.10). ∎

3. Inverse construction

Let $Q$ be a three-manifold equipped with a coframe $\alpha=(\alpha_{1},\alpha_{2},\alpha_{3})^{T}$ and a symmetric positive-definite matrix $H=(H_{ij})_{i,j=1,2,3}$ of functions $H_{ij}\in C^{\infty}(Q)$. In particular, $h=\det H>0$. Assume that there is an $s\in\mathbb{R}$ with $0<s^{2}<h$; this is always the case if $Q$ is compact, and write $\rho=h-s^{2}$. We define $\tau$ by (2.5) and assume that $\tau$ and $\sigma=H\tau$ are closed, and that $[\tau/s]$ is integral in $H^{2}(Q,\mathbb{R}^{3})$. Here $\mathbb{R}^{3}$ is the Lie algebra of $T^{3}$, and integrality is that elements with values in the integral lattice in this Lie algebra, have integral periods. We put $\beta=H\alpha$.

Let $\pi\colon E\to Q$ be a principal $T^{3}$-bundle with connection-one form $\theta$ satisfying (2.10). This bundle exists, since $[d\theta]=[\tau/s]$, which was assumed to be integral. We regard the total space $E$ as the level set $\nu^{-1}(s)$ of the previous section. Consider $M$ an open neighbourhood of $\{s\}\times E\subset\mathbb{R}\times E$ that is $\mathbb{R}$-connected, meaning that $M\cap(\mathbb{R}\times\{p\})$ is an open interval for each $p\in E$. We construct a $\mathrm{G}_{2}$-structure on $M$ via the tensors (2.1)–(2.4), with $\alpha,H,\theta$ being dependent on $s$ and with $\nu$ replaced by $s$ throughout. The nearly parallel condition is now equivalent to (2.8) and (2.9) on $M$, again with $\nu$ replaced by $s$.

Remark 3.1*.*

The $\mathrm{G}_{2}$-structure is determined solely by the three-form (2.3). The set of $\mathrm{G}_{2}$ three-forms is open in $\Omega^{3}(M)$, so any small deformation of $\alpha$, $H$ and $\theta$ yields a new three-form, even if the deformation of $H$ is not symmetric.

The metric and volume form are determined from the three-form by (1.1). If $H$ is not symmetric, then $d\nu$ is no longer orthogonal to $\theta$, and thus $g$, $\mathrm{vol}$, and hence $\star\varphi$, are not given by the expressions (2.1), (2.2) and (2.4).

Thus the above statement that the nearly parallel condition is equivalent to (2.8) and (2.9) relies on $H$ being symmetric even when it varies with $s$.

The differential $d$ on $M$ splits as the sum of the differential on $E$ and the one in the remaining direction: we write $d=d_{6}+d_{s}$, where $d_{6}$ is the differential on $E$ and $d_{s}\gamma=ds\wedge\gamma^{\prime}$, with $\gamma^{\prime}$ the derivative of $\gamma$ with respect to $s$, so $\gamma^{\prime}=\partial_{s}\mathbin{\lrcorner}d\gamma$. Note that with this choice, we have $(\gamma\wedge\delta)^{\prime}=\gamma^{\prime}\wedge\delta+\gamma\wedge\delta^{\prime}$ and $d_{6}(\gamma^{\prime})=(d_{6}\gamma)^{\prime}$, for any $s$-dependent differential forms $\gamma$ and $\delta$. Also note that $\partial_{s}=(1/16\rho)d\nu^{\sharp}$ is not of constant length in $M$.

In this notation, (2.7) gives

[TABLE]

thus determining $\theta^{\prime}$ from the flow of $\alpha$ and $H$. Equation (2.8) implies

[TABLE]

and (2.9) gives

[TABLE]

Differentiating (2.7) on $M$, and substituting for $d\theta$, we have

[TABLE]

Substituting into (3.3), using $\sigma^{\prime}=(H\tau)^{\prime}=H^{\prime}\tau+H\tau^{\prime}$, gives

[TABLE]

Theorem 3.2**.**

Let $(Q,h_{Q})$ be a smooth oriented Riemannian $3$-manifold with a global coframe $\alpha_{1},\alpha_{2},\alpha_{3}$. Write $h_{Q}=\alpha^{T}H\alpha$ with $H=(H_{ij})$, where $\alpha=(\alpha_{1},\alpha_{2},\alpha_{3})^{T}$. Assume there is an $s_{0}\in\mathbb{R}$ with $0<s_{0}^{2}<h=\det H$. For each cyclic permutation $(ijk)$ of $(123)$, put

[TABLE]

where $\rho=h-s_{0}^{2}$ and $\beta_{p}=(H\alpha)_{p}=\sum_{a=1}^{3}H_{pa}\alpha_{a}$.

Suppose for all $k=1,2,3$, that $\sigma_{k}$ and $\tau_{k}$ are closed and the cohomology class $[\tau/s_{0}]\in H^{2}(Q,\mathbb{R}^{3})$, $\tau=(\tau_{1},\tau_{2},\tau_{3})^{T}$ is integral. Let $E\to Q$ be a principal $T^{3}$-bundle $E\to Q$ with connection one-form $(\theta_{1},\theta_{2},\theta_{3})$ such that $d_{6}\theta_{k}=(d\alpha_{k}-\tau_{k})/s_{0}$.

Then there exists an open $\mathbb{R}$-connected neighbourhood $M$ of $\{s_{0}\}\times E$ in $\mathbb{R}\times E$ on which there is a unique solution to the flow equations (3.1), (3.2), (3.3), (3.4), (3.5) with the given initial data at $s=s_{0}$. Via (2.1)–(2.4), with $\nu$ the $\mathbb{R}$-parameter, this solution defines the unique nearly parallel $\mathrm{G}_{2}$-structure on $M$ that is $T^{3}$-invariant and descends to the given data $(\alpha,h_{Q})$ on $Q$ at $s=s_{0}$.

Proof.

From equations (3.3) and (3.5), we can deduce a system of first order differential equations for the evolution of $\alpha$ and $H$. Solutions to this smaller system will determine solutions to the flow equations. To make this more explicit, write

[TABLE]

for some $A,B,K^{p},R\in C^{\infty}(Q,M_{3}(\mathbb{R}))$. Note that $K_{ia}^{p}=K_{ai}^{p}$, by the symmetry of $H$, and $R_{ia}=K_{ic}^{b}-K_{ib}^{c}$, for $(abc)=(123)$.

Consider the equation (3.5). We have

[TABLE]

Thus

[TABLE]

is determined algebraically by the data $(s,\alpha,H,d_{3}\alpha)$ on $Q$.

Before considering $\alpha^{\prime}$, note that

[TABLE]

so $h^{\prime}=\operatorname{Tr}(H^{\prime}\operatorname{adj}H)$, where $(\operatorname{adj}H)_{ai}=H_{jb}H_{kc}-H_{jc}H_{kb}$, when $(ijk)=(123)=(abc)$. Since $H\operatorname{adj}H=h\mathrm{Id}_{3}$, we conclude that

[TABLE]

because $\operatorname{Tr}R=0$, by $\operatorname{Tr}R=\sum_{(ijk)=(123)}K_{ik}^{j}-K_{ij}^{k}=\sum_{(ijk)=(123)}K_{ki}^{j}-K_{ij}^{k}=0$.

Turning to $\alpha^{\prime}$, first note that

[TABLE]

with $W=-A^{T}+(\operatorname{Tr}A)\mathrm{Id}_{3}$. So

[TABLE]

But ${\star}_{\alpha}\alpha=(\rho/4h)\sigma$, so equation (3.3) gives

[TABLE]

From (3.7), we have

[TABLE]

Thus $d_{3}h\wedge H\alpha=F\,{\star}_{\alpha}\alpha$ with

[TABLE]

for $(pqr)=(123)$. Note that

[TABLE]

We now have

[TABLE]

with

[TABLE]

Hence, from (3.9), we get

[TABLE]

Thus (3.6), $\alpha^{\prime}=A\alpha$, with $A$ given by (3.11), and $d_{3}$-derivatives of these equations, gives a first order system of differential equations for $H$, $\alpha$, $d_{3}\alpha$ and $d_{3}H$. The right-hand side of the system is smooth in this data away from $s=0$, $h=0$ and $\rho=0$, so is locally Lipschitz.

We need to check consistency of the system. Note that (3.4) is a consequence of (3.3) and (3.5). Also (3.2) follows directly from (3.8). Furthermore, (3.3) and (3.4) show that the conditions $d_{3}\sigma=0=d_{3}\tau$ are preserved by the flow.

It remains to see that $H^{\prime}$ is symmetric. For $X\in C^{\infty}(Q,M_{3}(\mathbb{R}))$, write $X^{\natural}\in C^{\infty}(Q,\mathbb{R}^{3})$ for the vector with

[TABLE]

where $(ijk)=(123)$.

Note that for $\gamma\in\Omega^{1}(Q,\mathbb{R}^{3})$ and $X,Y\in C^{\infty}(Q,M_{3}(\mathbb{R}))$, we have

[TABLE]

when $(ijk)=(123)$. Thus

[TABLE]

From $d_{3}\sigma=0$, we have

[TABLE]

This gives

[TABLE]

Similarly $d_{3}\tau=0$ gives

[TABLE]

and hence

[TABLE]

Now (3.6) gives

[TABLE]

But

[TABLE]

So

[TABLE]

by (3.12), and $H^{\prime}$ is symmetric.

Thus we have a consistent system, and start data $(\alpha,H)$ at $s_{0}$ determines a unique solution. ∎

4. Examples and consequences

The above results are formulated with respect to fixed generators $U_{1},U_{2},U_{3}$ for the $T^{3}$-action. In general, a change of basis is given by a constant element of $\mathrm{GL}(3,\mathbb{R})$.

Lemma 4.1**.**

If we change the generating vector fields by $\tilde{U}=PU$, for $P\in\mathrm{GL}(3,\mathbb{R})$, then the quantities expressing the nearly parallel $\mathrm{G}_{2}$-structure change by

[TABLE]

Proof.

The expression for $\tilde{\nu}$ follows directly from the definition. For $\tilde{H}$, we have $\tilde{H}_{ij}=\sum_{a,b=1}^{3}P_{ia}P_{jb}g(U_{a},U_{b})$ and the claimed formula. Now, writing $\tilde{\theta}=\tilde{P}\theta$, we have $\delta_{ij}=\tilde{\theta}_{i}(\tilde{U}_{j})=\sum_{a,b=1}^{3}\tilde{P}_{ia}P_{jb}\theta_{a}(U_{b})=\sum_{a=1}^{3}\tilde{P}_{ia}P_{ja}$, thus $\tilde{P}=(P^{T})^{-1}$ and $\tilde{\nu}\tilde{\theta}=(\operatorname{adj}P)^{T}\nu\theta$, from which the expression for $\tilde{\alpha}$ follows. Note that $\tilde{\beta}=\tilde{H}\tilde{\alpha}=(\det P)P\beta$ and the transformation rule for $\tau=(4/\rho)\,{\star}_{\beta}\beta$ follows. Finally, we use $\sigma=H\tau$, to get $\tilde{\sigma}$. ∎

As a consequence, we can always arrange that $H=\mathrm{Id}_{3}$ at a given point. If $H=\mathrm{Id}_{3}$ globally on $Q$, then the two conditions $d_{3}\sigma=0$ and $d_{3}\tau=0$ are the same.

Proposition 4.2**.**

Let $Q$ be a connected oriented three-manifold, and suppose $\eta=(\eta_{1},\eta_{2},\eta_{3})^{T}$ be an $\mathbb{R}^{3}$-valued closed two-form on $Q$ whose components are linearly independent at each point, and whose cohomology class is integral. Then there is a nearly parallel $\mathrm{G}_{2}$-manifold $M$ with $T^{3}$-symmetry, and a regular value $s_{0}\neq 0$ of the multi-moment map, such that either $\eta$ or $-\eta$ is the form $\tau/s_{0}$ on $Q$ of Proposition 2.1.

In fact, we will show that there is one such solution for each $s_{0}$ with $s_{0}^{2}\in(0,1)$.

Proof.

We will generate initial data on $Q$ as in Theorem 3.2, with $H=\mathrm{Id}_{3}$. Thus $h=1$ and $0<s_{0}^{2}<1$. Then $\sigma=\tau=\epsilon s_{0}\eta$, $\epsilon\in\{\pm 1\}$ and these are $d_{3}$-closed by assumption. Write $c_{0}^{2}=1-s_{0}^{2}=\rho$. We need to find a coframe $\alpha$, so that $\sigma=(4/c_{0}^{2})\,{\star}_{\alpha}\alpha$. This is equivalent to ${\star}_{\alpha}\alpha=\epsilon(c_{0}^{2}s_{0}/4)\eta\eqqcolon\epsilon\hat{c}\eta$.

As $\eta_{i}$, for $i=1,2,3$, is a non-zero two-form at each point, we have that each $\ker\eta_{i}$ is a one-dimensional distribution. Fixing a background metric $g_{Q}$ on $Q$, there are locally defined unit length vector fields $Y_{1},Y_{2},Y_{3}$ such that $Y_{i}\mathbin{\lrcorner}\eta_{i}=0$, $i=1,2,3$. Indeed at each point there are two choices $\pm Y_{i}$. Locally, let $\gamma=(\gamma_{1},\gamma_{2},\gamma_{3})$ be the dual coframe to $(Y_{1},Y_{2},Y_{3})$. Then we may use the orientation of $Q$ to fix the choice of sign for $Y_{i}$ by requiring $\gamma_{i}\wedge\eta_{i}>0$ for $i=1,2,3$. Hence $Y_{i}$ and $\gamma_{i}$ are globally defined.

Now $Y_{i}\mathbin{\lrcorner}\eta_{i}=0$ implies for $(ijk)=(123)$ that $\eta_{i}=f_{i}\gamma_{j}\wedge\gamma_{k}$ for some functions $f_{i}\in C^{\infty}(Q)$. Thus $\eta=f^{T}\,{\star}_{\gamma}\gamma$. We wish to take $\alpha_{i}=a_{i}\gamma_{i}$ for some functions $a_{i}$. We need to choose $a_{i}$ so that

[TABLE]

Multiplying (4.1) by $a_{i}$, we get

[TABLE]

But the left-hand side is independent of $(ijk)=(123)$, so $a_{i}=b/f_{i}$ for some function $b$ independent of $i$. Putting this into (4.1), we get $b^{2}=\epsilon\hat{c}f_{1}f_{2}f_{3}$. From linear independence of the $\eta_{i}$, we have $f_{1}f_{2}f_{3}$ is nowhere vanishing, so the product has a fixed sign, and we take $\epsilon$ to be this sign. Thus $\alpha_{i}=(\epsilon\hat{c}f_{1}f_{2}f_{3})^{1/2}\gamma_{i}/f_{i}$ satisfies ${\star}_{\alpha}\alpha=\hat{c}\epsilon\eta$, and we have the initial data needed to apply Theorem 3.2. ∎

Remark 4.3*.*

The sign change needed in the above proof can also be obtained by swapping $\eta_{2}$ and $\eta_{3}$.

Remark 4.4*.*

Since $\sigma=H\tau$, the two triples of two-forms $\sigma$ and $\tau$ determine $H$. One can then formulate a version of Proposition 4.2 in terms of $s_{0}$, $\sigma$ and $\tau$. However, the triples need to be restricted so that $H$ becomes symmetric and positive definite.

Example 4.5*.*

Consider $Q=\mathbb{R}^{3}$ as an Abelian Lie group with left-invariant data $(\alpha,H=(H_{ij}),s_{0}>0)$ for the inverse construction. We have $d_{3}\alpha_{i}=0$ and $h$ and $H$ are constant on $Q$. We may change basis for the action so that $H(s_{0})=\mathrm{Id}_{3}$, and then use $e=\alpha(s_{0})$ as a fixed basis for $T^{*}_{0}Q\cong\mathbb{R}^{3}$. Note that $0<s_{0}<1$.

Consider possible evolution of the data in the form $H(s)=r(s)\mathrm{Id}_{3}$ and $\alpha(s)=u(s)e$ for some functions $r$ and $u$ of $s$. Note that $d_{3}H=0=d_{3}\alpha$, so $B=0=K=R$. Equation (3.6) gives $r^{\prime}=-r/s$, so $r=s_{0}/s$. We have from $\alpha^{\prime}=A\alpha$ with $A=u^{\prime}\mathrm{Id}_{3}$ given by (3.11), and $F$ by (3.10) is zero, so

[TABLE]

so

[TABLE]

where $c_{0}=(1-s_{0})^{1/2}>0$. One now checks that the flow equations are satisfied, so this is the unique solution with the given initial conditions, and all left-invariant solutions on $\mathbb{R}^{3}$ are obtained in this way.

The resulting nearly parallel $\mathrm{G}_{2}$-structure from (2.1)–(2.4) has metric and three-form

[TABLE]

with

[TABLE]

Extending the algebra by adding $p^{i}$, $i=1,2,3$, with $dp^{i}=d_{6}\theta_{i}$, we have

[TABLE]

where $x^{\prime}e^{i}=\theta_{i}^{\prime}$. This structure may thus be regarded as defined on $(0,s_{0}^{3/5})\times G$, where $\mathfrak{g}$ is the nilpotent algebra $(0,0,0,23,31,12)$, and $G$ acts with cohomogeneity one.

Other first candidates for the inverse construction are homogeneous manifolds $Q$ of dimension three with left-invariant data. As they have a global left-invariant coframe, such spaces $Q$ are necessarily Lie groups. Bianchi’s classification of three-dimensional Lie algebras then yields all suitable Lie groups. Our requirement of having a triple of linearly independent closed two-forms, then implies that the Lie algebra is unimodular. The only potential cases where one could get compact examples of nearly parallel $\mathrm{G}_{2}$-structures via our inverse construction are for $\mathfrak{g}=\mathbb{R}^{3}$ and $\mathfrak{su}(2)$. We have already investigated the first case above. When the curvatures are integral we may get solutions over $Q=T^{3}$, but the metrics are incomplete. For $Q=\mathrm{SU}(2)$, one obtains nearly parallel $\mathrm{G}_{2}$-structures with a cohomogeneity-one action of $\mathrm{SU}(2)\times T^{3}$, see a special case below. This can not yield a complete metric: by Myers’ Theorem, the seven-dimensional manifold would be compact with finite fundamental group, which implies there are two special orbits, but at each special orbit at most one factor of $T^{3}$ collapses, so the fundamental group remains infinite.

Example 4.6*.*

Let us consider the concrete case of $Q=SU(2)$. Let $e^{1},e^{2},e^{3}$ be the standard left-invariant coframe on $SU(2)$ with $de^{i}=e^{jk}$. Take $\alpha=u(s)e$ and $H=r(s)\mathrm{Id}_{3}$. Then we have $d_{3}H=0$ and $d_{3}\alpha=u(s)\,{\star}_{e}e$, so $B=(1/u(s))\mathrm{Id}_{3}$, and $u(s)A=u^{\prime}(s)\mathrm{Id}_{3}$. Equations (3.6) and (3.11) give

[TABLE]

Using the initial conditions $s_{0}=1/2$, $r(s_{0})=1=u(s_{0})$, one can verify numerically that there is a smooth solution on $(0,s_{\max})$ for some $s_{\max}>s_{0}$ with $\lim_{s\uparrow s_{\max}}u(s)=0$. The resulting nearly parallel $\mathrm{G}_{2}$-structure is

[TABLE]

with

[TABLE]

Noting that the $s$-derivative of the coefficient of $e^{jk}$ in $d_{6}\theta$ is indeed the coefficient of $e_{i}$ in $\theta_{i}^{\prime}$, we have

[TABLE]

where $dp^{i}=0$. Thus $p^{1},p^{2},p^{3},e^{1},e^{2},e^{3}$ can be taken as an invariant coframe for $T^{3}\times SU(2)$, and the (incomplete) nearly parallel $\mathrm{G}_{2}$-structure has symmetry group of rank $4$.

Note that this is not the general left-invariant solution on $Q=\mathrm{SU}(2)$, unlike in the case of $Q=\mathbb{R}^{3}$; for general initial data, we can not simultaneously have $H=r\mathrm{Id}_{3}$ and $\alpha=ue$ at $s=s_{0}$.

Corollary 4.7**.**

There exist incomplete nearly parallel $\mathrm{G}_{2}$-structures that have $T^{4}$-symmetry.

Taking $Q$ compact with just a circle symmetry, and using circle invariant data for the inverse construction, may yield examples with non-trivial stabilisers at some points of $\nu^{-1}(0)$, hence potential compact solutions with $T^{4}$-symmetry.