# Associative Submanifolds of
Squashed $3$-Sasakian Manifolds

Gavin Ball and Jesse Madnick

(September 9, 2022)

###### Abstract

Every compact $3$-Sasakian $7$-manifold $M$ admits a canonical $2$-parameter
family of co-closed $\mathrm{G}_{2}$-structures $\varphi_{a,b}$ for $a,b>0$,
as well as a foliation by $\varphi_{a,b}$-associative $3$-folds whose leaf
space $X$ is a positive quaternion-Kähler $4$-orbifold. We prove that
associative $3$-folds in $(M,\varphi_{a,b})$ that are ruled by a certain type
of geodesic are in correspondence with pseudo-holomorphic curves in the
almost-complex $8$-manifold $Z\times S^{2}$, where $Z$ is the twistor space of
$X$ equipped with its strict nearly-Kähler structure.
As an application, we construct infinitely many topological types of non-
trivial, compact associative $3$-folds in the squashed $7$-spheres
$(S^{7},\varphi_{a,b})$ and squashed exceptional Aloff-Wallach spaces
$(N_{1,1},\varphi_{a,b})$. Topologically, our examples are circle bundles over
a genus $g$ surface, for any $g\geq 0$.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Squashed $3$-Sasakian $7$-Manifolds
  4. 4 Associatives in Squashed $3$-Sasakian $7$-Manifolds
  5. 5 Construction of Compact Associatives

##  1 Introduction

Let $(M,\varphi)$ be a $7$-manifold with a $\mathrm{G}_{2}$-structure
$\varphi$. An _associative $3$-fold_ is a $3$-dimensional submanifold
$N\subset M$ for which

$\varphi|_{N}=\mathrm{vol}_{N}.$

Associative submanifolds are conjectured to play an important role in defining
an enumerative invariant of torsion-free $\mathrm{G}_{2}$-structures counting
$\mathrm{G}_{2}$-instantons [12, 13, 34, 22, 19]. In this direction, it is
desirable to have a large supply of associative submanifolds, especially
examples that exist in continuous families of coclosed
$\mathrm{G}_{2}$-structures $\varphi_{t}$. In general, associative $3$-folds
are difficult to construct, making conjectures hard to test. In this work, we
construct examples of compact $3$-folds in squashed 3-Sasakian 7-manifolds $M$
that are associative with respect to a $2$-parameter family of
$\mathrm{G}_{2}$-structures.
A second motivation for this work comes from $\mathrm{Spin}(7)$ geometry.
Among the co-closed $\mathrm{G}_{2}$-structures, there is a distinguished
class of _nearly-parallel $\mathrm{G}_{2}$-structures_, defined by the
property that their metric cone is a $\mathrm{Spin}(7)$-manifold. Associative
submanifolds in nearly-parallel $7$-manifolds $M$ provide models for conically
singular Cayley manifolds in the cone
$(\mathrm{C}(M),g_{\mathrm{C}})=(\mathbb{R}^{+}\times M,dr^{2}+r^{2}g)$.
Examples of associative submanifolds in nearly-parallel $7$-manifolds have
been constructed by Fox [15], Lotay [26], Kawai [24], and the authors [4].
Gauge theory on nearly-parallel $7$-manifolds has been studied by Ball and
Oliveira [5], Waldron [33], Lotay and Oliveira [27], and Singhal [32].

A Riemannian $7$-manifold $(M,g)$ is _$3$ -Sasakian_ if its metric cone
$(\text{C}(M),g_{\mathrm{C}})$ is hyperkähler. Compact examples include the
$7$-sphere $S^{7}$, the Aloff-Wallach space
$N_{1,1}=\mathrm{SU}(3)/S^{1}_{(1,1)}$, and the $7$-manifolds
$S(p_{1},p_{2},p_{3})$ constructed by Boyer-Galicki-Mann [8]. Since
$\mathrm{C}(M)$ is hyperkähler, it admits a triple $(I_{1},I_{2},I_{3})$ of
$g_{\mathrm{C}}$-orthogonal complex structures satisfying $I_{1}I_{2}=I_{3}$
for which the associated $2$-forms $(\omega_{1},\omega_{2},\omega_{3})$ are
closed. Therefore, viewing $M$ as the link $\\{1\\}\times
M\subset\mathrm{C}(M)$, we see that $TM$ splits as

$TM=\mathsf{A}\oplus\mathsf{C},\ \
\mathsf{A}:=\text{span}(I_{1}(\partial_{r}),I_{2}(\partial_{r}),I_{3}(\partial_{r})),\
\ \mathsf{C}:=\mathsf{A}^{\perp}.$

On $M$, we define $1$-forms $\alpha_{j}:=\omega_{j}(\partial_{r},\cdot)$ and
$2$-forms $\Omega_{j}:=\omega_{j}|_{\mathsf{C}}$. These yield a canonical
$2$-parameter family of co-closed $\mathrm{G}_{2}$-structures on $M$ via

$\varphi_{a,b}:=a^{3}\,\alpha_{123}-ab^{2}\sum\alpha_{j}\wedge\Omega_{j}$

whose induced Riemannian metrics are

$g_{a,b}:=\left.a^{2}g\right|_{\mathsf{A}}+\left.b^{2}g\right|_{\mathsf{C}}.$

That $\varphi_{a,b}$ is co-closed follows from calculations in [16, $\S$5].
See also [1].
We refer to $(M,\varphi_{a,b})$ as a _squashed $3$-Sasakian $7$-manifold_. The
family of co-closed $\mathrm{G}_{2}$-structures $\varphi_{a,b}$ exhibits some
interesting features. For example, $\varphi_{a,b}$ is nearly-parallel if and
only if $b^{2}=5a^{2}$, and the metric $g_{1/\sqrt{5},1}$ is properly nearly-
parallel. Moreover, while $\varphi_{1,1}$ is not nearly-parallel, it is
nevertheless _isometric_ to a nearly-parallel $\mathrm{G}_{2}$-structure
$\widehat{\varphi}$ as its induced metric $g=g_{1,1}$ is $3$-Sasakian. See
$\S$3 for details.

In this work, we study associative $3$-folds in $(M,\varphi_{a,b})$, where $M$
is compact. The simplest examples are the _canonical leaves_ , i.e., those
associatives that are tangent to $\mathsf{A}$. Indeed, the distribution
$\mathsf{A}\subset TM$ is integrable, yielding a _canonical foliation_
$\mathcal{F}_{A}$ of $M$. The canonical leaves turn out to be totally-
geodesic, $\varphi_{a,b}$-associative $3$-folds in $M$ with constant positive
sectional curvature. It is known that the leaf space $X:=M/\mathcal{F}_{A}$ is
a positive quaternion-Kähler $4$-orbifold, and that the projection $h\colon
M\to X$ is a principal $\mathrm{SU}(2)$\- or $\mathrm{SO}(3)$-orbibundle [7,
$\S$13.3.13]. For example, when $M=S^{7}$, the leaf space $X=S^{4}$, the map
$h\colon S^{7}\to S^{4}$ is the quaternionic Hopf fibration, and the canonical
leaves are the Hopf $3$-spheres.
To obtain new examples, we consider a more general geometric ansatz, studying
those associatives in $(M,\varphi_{a,b})$ that are ruled by _Hopf circles_ ,
those closed geodesics that lie in a canonical leaf. In other words, a Hopf
circle is an orbit of the locally free $S^{1}$-action on $M$ generated by the
the Reeb field
$A_{w}:=w_{1}I_{1}(\partial_{r})+w_{2}I_{2}(\partial_{r})+w_{3}I_{3}(\partial_{r})$,
for any $w=(w_{1},w_{2},w_{3})\in S^{2}$. Each quotient space $M/\langle
A_{w}\rangle$ is diffeomorphic to the twistor space $Z$ of $X$, and the
projection $p_{w}\colon M\to Z$ is a principal $S^{1}$-orbibundle [7,
$\S$7.5.1, $\S$13.3.1]. For example, when $M=S^{7}$, we have
$Z=\mathbb{CP}^{3}$, and each $p_{w}\colon S^{7}\to\mathbb{CP}^{3}$ is a
complex Hopf fibration.
The space of Hopf circles in $M$ is diffeomorphic to $Z\times S^{2}$, so any
Hopf-ruled $3$-dimensional submanifold of $M$ (not necessarily associative)
can be viewed as a surface in $Z\times S^{2}$. Conversely, any surface
$\Sigma\subset Z\times S^{2}$ gives a corresponding Hopf-ruled $3$-fold
$\Gamma(\Sigma)\subset M$. We then ask which surfaces in $Z\times S^{2}$ yield
_associative_ Hopf-ruled $3$-folds in $M$. The Correspondence Theorem 1.1
provides the answer: it is the $J$-holomorphic curves in $Z\times S^{2}$.
Here, $J$ is the product almost-complex structure on $Z\times S^{2}$, where
$Z$ carries its standard strict nearly-Kähler structure.
Recalling that that $Z\to X$ is an $S^{2}$-bundle, we view $Z\times S^{2}\to
X$ as an $(S^{2}\times S^{2})$-bundle in the obvious way. We can now state the
result:

######  Theorem 1.1 (Correspondence Theorem).

Let $(M^{7},\varphi_{a,b})$ be a compact squashed $3$-Sasakian $7$-manifold,
where $a,b>0$ are fixed. Let $Z$ be the twistor space of
$X:=M/\mathcal{F}_{A}$.

  1. (a)

Every Hopf-ruled associative $3$-fold in $(M^{7},\varphi_{a,b})$ is locally of
the form $\Gamma(\Sigma)$ for some $J$-holomorphic curve $\Sigma\subset
Z\times S^{2}$.

  2. (b)

Let $\Sigma\subset Z\times S^{2}$ be a $J$-holomorphic curve.

    1. 1.

If $\Sigma$ does not lie in an $(S^{2}\times S^{2})$-fiber, then there exists
a discrete set $D\subset\Sigma$ such that $\Gamma(\Sigma-D)\subset M$ is a
Hopf-ruled associative $3$-fold.

    2. 2.

If $\Sigma$ lies in an $(S^{2}\times S^{2})$-fiber, then $\Gamma(\Sigma)$ is a
subset of a canonical leaf.

The proof of Theorem 1.1(b) gives more than the statement. Indeed, let
$\Sigma_{g}$ be a compact surface of genus $g\geq 0$. In $\S$4.4, we observe
that if $f\colon\Sigma_{g}\to Z$ is an immersed pseudo-holomorphic curve in
$Z$ that is horizontal for the twistor fibration $\tau\colon Z\to X$, then for
any holomorphic map $w\colon\Sigma_{g}\to S^{2}$, the surface
$(f,w)\colon\Sigma_{g}\to Z\times S^{2}$ is a $J$-holomorphic curve that does
not lie in an $(S^{2}\times S^{2})$-fiber and has $D=\O$. Consequently,
$\Gamma(f,w)\subset M$ is a compact associative $3$-fold in $M$ diffeomorphic
to an $S^{1}$-bundle over $\Sigma_{g}$.

As an application, we consider the case of $M=S^{7}$ equipped with any of its
$\mathrm{Sp}(2)\mathrm{Sp}(1)$-invariant $\mathrm{G}_{2}$-structures
$\varphi_{a,b}$. For each $g\geq 0$, Bryant [10] has constructed an embedded
horizontal pseudo-holomorphic curve $f\colon\Sigma_{g}\to\mathbb{CP}^{3}=Z$.
Taking $w\colon\Sigma_{g}\to S^{2}$ to be a constant map $w=w_{0}$ yields the
_trivial_ associative $3$-fold
$\Gamma(f,w_{0})=p_{w_{0}}^{-1}(f(\Sigma_{g}))\subset S^{7}$. On the other
hand, taking $w$ to be a non-constant holomorphic map yields the first non-
trivial compact $\varphi_{a,b}$-associative $3$-folds in $S^{7}$. That is:

######  Theorem 1.2.

Fix $a,b>0$. For every $g\geq 0$, there exists a non-trivial compact
associative $3$-fold in $(S^{7},\varphi_{a,b})$ diffeomorphic to an
$S^{1}$-bundle over a genus $g$ surface.

Note that non-trivial compact $\widehat{\varphi}$-associative $3$-folds in
$S^{7}$ have been constructed in [15] and [26], where $\widehat{\varphi}$ is
the $\mathrm{Spin}(7)$-invariant $\mathrm{G}_{2}$-structure. See $\S$5.1 for a
more detailed discussion.

Similarly, we may apply our theory to $M=N_{1,1}$, in which case
$Z=\mathrm{SU}(3)/\mathrm{T}^{2}$. In this case, horizontal pseudo-holomorphic
curves in $\mathrm{SU}(3)/\mathrm{T}^{2}$ may be constructed as lifts of
holomorphic curves in $\mathbb{CP}^{2}.$ Every compact Riemann surface
$\Sigma$ can be realized as an immersed holomorphic curve in
$\mathbb{CP}^{2}.$ Taking $w:\Sigma\to S^{2}$ to be a non-constant holomorphic
map, we obtain:

######  Theorem 1.3.

Fix $a,b>0$. For every $g\geq 0$, there exists a non-trivial compact
associative $3$-fold in $(N_{1,1},\varphi_{a,b})$ diffeomorphic to an
$S^{1}$-bundle over a genus $g$ surface.

In sum, we can obtain non-trivial compact $\varphi_{a,b}$-associative
$3$-folds in a squashed $3$-Sasakian $7$-manifold $M$ by constructing
horizontal pseudo-holomorphic curves in the corresponding nearly-Kähler
orbifold $Z$. For this reason, it would be desirable to have a large supply of
such pseudo-holomorphic curves in twistor spaces $Z$ other than
$\mathbb{CP}^{3}$ and $\mathrm{SU}(3)/\mathrm{T}^{2}$. Such twistor spaces are
orbifolds rather than manifolds, and it seems likely that techniques from
complex algebraic geometry would prove useful in attacking this problem.

Note that Theorem 1.1(b) applies to all values of the squashing parameters
$a,b>0$. That is, for any $J$-holomorphic curve $\Sigma\subset Z\times S^{2}$,
there are corresponding Hopf-ruled $\varphi_{a,b}$-associative $3$-folds
$\Gamma(\Sigma-D)\subset M$ for every $a,b>0$. This raises the following
question: For a fixed $J$-holomorphic curve $\Sigma\subset Z\times S^{2}$, to
what extent does the deformation theory of the $\varphi_{a,b}$-associative
$3$-fold $\Gamma(\Sigma-D)\subset M$ depend on the squashing parameters? This
is an interesting direction for further investigation.

Finally, we comment briefly on the organization of this work. Although $\S$2
consists of preliminary material, $\S$2.2 contains several novelties. In
particular, Proposition 2.5 does not seem to be widely known. The latter
motivates the concept of “striped” associative $3$-folds (see Definition 2.7),
which likely has applications beyond this work.
In $\S$3, we study squashed 3-Sasakian $7$-manifolds $(M,\varphi_{a,b})$ via
the method of moving frames. In $\S$3.3, we show that the space of Hopf
circles is diffeomorphic to $Z\times S^{2}$, and we equip this space with a
canonical $2$-parameter family of almost-Hermitian structures
$(h_{a,b},J,\omega_{a,b})$. Our study of associative $3$-folds in
$(M,\varphi_{a,b})$ begins in $\S$4.

Acknowledgements: We thank Benjamin Aslan and Benoit Charbonneau for
clarifying comments. The second author thanks McKenzie Wang and Chung-Jun Tsai
for conversations, and thanks McMaster University and the National Center for
Theoretical Sciences for their support.

###  1.1 Discussion: A Geometric View of the Correspondence Theorem

We wish to explain in heuristic terms the geometric idea behind the
Correspondence Theorem 1.1 in the case of the $7$-sphere equipped with one of
the co-closed $\mathrm{G}_{2}$-structures $\varphi_{a,b}$. Viewing
$M=S^{7}\subset\mathbb{H}^{2}$ and $X=S^{4}\cong\mathbb{HP}^{1}$, note that
each fiber of the quaternionic Hopf fibration $h\colon S^{7}\to S^{4}$ is a
$3$-sphere given by the intersection of $S^{7}$ with a quaternionic line $H$.
There is an $S^{2}$-family of identifications $w\colon
H\xrightarrow{\cong}\mathbb{C}^{2}$, each of which yields a complex Hopf
fibration $p_{w}\colon S^{7}\to\mathbb{CP}^{3}$. A Hopf circle in $S^{7}$ is
an $S^{1}$-fiber $p_{w}^{-1}(z)$ for some fixed $w\in S^{2}$ and
$z\in\mathbb{CP}^{3}$.
Now, for every immersed pseudo-holomorphic curve
$f\colon\Sigma\to\mathbb{CP}^{3}$ and every fixed $w\in S^{2}$, the pre-image
$p_{w}^{-1}(f(\Sigma))\subset S^{7}$ is a trivial associative $3$-fold in
$S^{7}$, where “pseudo-holomorphic” is meant with respect to the strict
nearly-Kähler structure on $\mathbb{CP}^{3}$. A natural question, then, is
whether one can generalize the class of trivial associatives by allowing the
ruling direction $w\in S^{2}$ to vary with $z\in\Sigma$ in some fashion. In
other words, which pairs of “directrix surface”
$f\colon\Sigma\to\mathbb{CP}^{3}$ and “ruling function” $w\colon\Sigma\to
S^{2}$ result in an immersed Hopf-ruled $3$-fold

$\bigcup_{z\in\Sigma}p^{-1}_{w(z)}(f(\Sigma))\subset S^{7}$ (1.1)

that is associative? Theorem 1.1 answers this by relating
$\varphi_{a,b}$-associatives in $S^{7}$ of the form (1.1) to pairs
$(f,w)\colon\Sigma\to\mathbb{CP}^{3}\times S^{2}$ consisting of a pseudo-
holomorphic map $f$ and holomorphic map $w$ for which $\text{rank}(f,w)=2$.
The technical point is that certain pairs of directrix surface $f$ and ruling
function $w$ yield Hopf-ruled $3$-folds $\Gamma(f,w)\subset S^{7}$ that are
not immersed. This sort of issue is endemic to the theory of ruled
submanifolds, occurring even in the classical case of ruled surfaces in
$\mathbb{R}^{3}$. Indeed, while one would like to say that any function
$\Gamma\colon\mathbb{R}^{2}\to\mathbb{R}^{3}$ of the form
$\Gamma(s,t):=\mathbf{f}(s)+t\mathbf{w}(s)$ with
$\mathbf{f}^{\prime}\cdot\mathbf{w}=0$ and $\|\mathbf{w}\|=1$ parametrizes a
ruled surface, certain pairs of ($\mathbf{f},\mathbf{w}$) result in
degenerations. For example, taking $\mathbf{f}(s)=\mathbf{0}$ and
$\mathbf{w}(s)=\frac{1}{\sqrt{2}}(\cos(s),\sin(s),1)$, the resulting map
$\Gamma$ parameterizes a cone in $\mathbb{R}^{3}$, but is not an immersion on
$\\{(s,t)\colon t=0\\}$. In our situation, assuming that $f$ is pseudo-
holomorphic, that $w$ is holomorphic, and that the image of $(f,w)$ doesn’t
lie in an $(S^{2}\times S^{2})$-fiber, Theorem 1.1(b) shows that one only need
remove a discrete set $D$ from $\Sigma$ to obtain an immersed Hopf-ruled
associative $\Gamma(\Sigma-D)\subset S^{7}$.

###  1.2 Terminology

(1) Given a Riemannian $n$-manifold $(M,g)$, the _(metric) cone over $(M,g)$_
is the Riemannian $(n+1)$-manifold

$(\mathrm{C}(M),g_{\mathrm{C}}):=(\mathbb{R}^{+}\times M,dr^{2}+r^{2}g).$

We always identify $M$ with the _link_ $\\{1\\}\times M\subset\text{C}(M)$.

(2) On a $7$-manifold $M$, an _$\mathrm{SO}(4)$ -structure_ is a pair
$(\varphi,\mathsf{A})$ consisting of a $\mathrm{G}_{2}$-structure
$\varphi\in\Omega^{3}(M)$ and a distribution $\mathsf{A}\subset TM$ of
associative $3$-planes. The following two related concepts are fundamental to
this work, both of which are discussed in $\S$2.2.

  * •

When $M$ carries an $\mathrm{SO}(4)$-structure $(\varphi,\mathsf{A})$, an
associative $3$-fold $N\subset M$ is called _striped_ if
$\dim(T_{x}N\cap\mathsf{A}_{x})=1$ for each $x\in N$.

  * •

An $\mathrm{SO}(4)$-structure $(\varphi,\mathsf{A})$ on $M$ is called
_quaternion-Sasakian_ if the induced $\mathrm{Sp}(2)\mathrm{Sp}(1)$-structure
on $\mathrm{C}(M)$ is quaternion-Kähler. Quaternion-Sasakian $7$-manifolds are
virtually the same as $3$-Sasakian $7$-manifolds, but entail slightly less
structure. See $\S$2.2 for details.

##  2 Preliminaries

###  2.1 $\mathrm{G}_{2}$-Structures and Associative $3$-folds

Let $M^{7}$ be a smooth $7$-manifold. A _$\mathrm{G}_{2}$ -structure_ on $M$
is a $3$-form $\varphi\in\Omega^{3}(M)$ such that at each $x\in M$, the
symmetric bilinear form
$B_{\varphi}|_{x}\colon\mathrm{Sym}^{2}(T^{*}_{x}M)\to\Lambda^{7}(T^{*}_{x}M)$
given by
$B_{\varphi}(u,v):=\textstyle\frac{1}{6}(\iota_{u}\varphi)\wedge(\iota_{v}\varphi)\wedge\varphi$
is definite. This is equivalent to the condition that at each $x\in M$, there
exists a frame $(e_{1},\ldots,e_{7})$ at $x$ for which

$\varphi|_{x}=e^{123}+e^{145}+e^{167}+e^{246}-e^{257}-e^{347}-e^{356}$

where $(e^{1},\ldots,e^{7})$ is the dual coframe and $e^{ijk}:=e^{i}\wedge
e^{j}\wedge e^{k}$. It is well-known that a $7$-manifold admits a
$\mathrm{G}_{2}$-structure if and only if it is orientable and spin. A
$\mathrm{G}_{2}$-structure $\varphi\in\Omega^{3}(M)$ naturally induces a
Riemannian metric $g_{\varphi}$ and orientation
$\mathrm{vol}_{\varphi}\in\Omega^{7}(M)$ via

$\displaystyle\textstyle g_{\varphi}(u,v)\,\mathrm{vol}_{\varphi}$
$\displaystyle=\textstyle\frac{1}{6}(\iota_{u}\varphi)\wedge(\iota_{v}\varphi)\wedge\varphi$

Two $\mathrm{G}_{2}$-structures $\varphi$, $\widehat{\varphi}$ on $M$ are
called _isometric_ if $g_{\varphi}=g_{\widehat{\varphi}}$.

A $\mathrm{G}_{2}$-structure $\varphi\in\Omega^{3}(M)$ is said to be co-closed
if $d\ast\varphi=0$. All of the $\mathrm{G}_{2}$-structures considered in this
work will be co-closed. If $\varphi$ is co-closed, then it is well-known that

$d\varphi=\tau_{0}\ast\\!\varphi+\ast\tau_{3}$

for a unique function $\tau_{0}\in\Omega^{0}(M)$ and $3$-form
$\tau_{3}\in\Omega^{3}_{27}(M)=\Gamma(\Lambda^{3}_{27}(M))$, where here

$\Lambda^{3}_{27}(M):=\left\\{\gamma\in\Lambda^{3}(T^{*}M)\colon\gamma\wedge\varphi=0\text{
and }\gamma\wedge\ast\varphi=0\right\\}\\!.$

A $\mathrm{G}_{2}$-structure $\varphi$ is called nearly-parallel if
$d\varphi=\lambda\ast\\!\varphi$ for some for some non-zero constant
$\lambda\in\mathbb{R}$. It is clear that every nearly-parallel
$\mathrm{G}_{2}$-structure is co-closed. If $\varphi$ is a nearly-parallel
$\mathrm{G}_{2}$-structure, then the induced metric $g_{\varphi}$ is Einstein
with $\text{Scal}(g_{\varphi})=\frac{21}{8}\lambda^{2}>0$.

An associative $3$-plane is an oriented $3$-dimensional subspace $E\subset
T_{x}M$ such that $\varphi|_{E}=\mathrm{vol}_{E}$. An _associative $3$-fold_
is an oriented submanifold $\Sigma^{3}\subset(M^{7},\varphi)$ for which each
tangent space $T_{x}\Sigma\subset T_{x}M$ is an associative $3$-plane — i.e.,
a submanifold satisfying $\varphi|_{\Sigma}=\pm\mathrm{vol}_{\Sigma}$.
Associative $3$-folds are fundamental sub-objects in
$\mathrm{G}_{2}$-geometry, and are formally analogous to holomorphic curves in
Calabi-Yau $3$-folds.

######  Remark 2.1.

Let $(M^{7},\varphi)$ be a $7$-manifold with a co-closed
$\mathrm{G}_{2}$-structure $\varphi$. It is natural to ask under what
conditions an associative $3$-fold $\Sigma\subset M$ is a minimal submanifold.
To answer this, consider the $\mathrm{G}_{2}$-vector bundle $TM|_{\Sigma}$.
The splitting $TM|_{\Sigma}=T\Sigma\oplus N\Sigma$ reduces the structure group
to $\mathrm{SO}(4)\leq\mathrm{G}_{2}$, and so each
$\Lambda^{k}(T^{*}M|_{\Sigma})$ is an $\mathrm{SO}(4)$-vector bundle as well.
In particular, is shown in [3] that
$\left.\Lambda^{3}_{27}(M)\right|_{\Sigma}$ decomposes into
$\mathrm{SO}(4)$-irreducible vector subbundles

$\left.\Lambda^{3}_{27}(M)\right|_{\Sigma}=(\Lambda^{3}_{27})_{0,0}\oplus(\Lambda^{3}_{27})_{0,4}\oplus(\Lambda^{3}_{27})_{1,1}\oplus(\Lambda^{3}_{27})_{1,3}\oplus(\Lambda^{3}_{27})_{2,2}$

of ranks $1,5,4,8$, and $9$, respectively. Moreover, there is a natural
$\mathrm{SO}(4)$-vector bundle isomorphism $\natural\colon
N\Sigma\to(\Lambda^{3}_{27})_{1,1}$. In this language, the mean curvature
vector field $H\in\Gamma(N\Sigma)$ of the associative $\Sigma\subset M$ is
given by

$H^{\natural}=\textstyle-\frac{\sqrt{3}}{2}(\tau_{3})_{1,1}$

Thus, the generic associative $3$-fold in $(M,\varphi)$ is not minimal.
However, if the co-closed $\mathrm{G}_{2}$-structure satisfies $\tau_{3}=0$
(e.g., if it is nearly-parallel or torsion-free), then every associative in
$M$ is minimal.

###  2.2 $\mathrm{SO}(4)$-Structures and Associative $3$-folds

The $7$-manifolds $M$ that we consider in this work will carry an additional
bit of data beyond the co-closed $\mathrm{G}_{2}$-structure. Namely:

######  Definition 2.2.

Let $M^{7}$ be a smooth $7$-manifold. An _$\mathrm{SO}(4)$ -structure_ on $M$
is a pair $(\varphi,\mathsf{A})$ consisting of a $\mathrm{G}_{2}$-structure
$\varphi\in\Omega^{3}(M)$ and an associative $3$-plane distribution
$\mathsf{A}\subset TM$. The geometry of $\mathrm{SO}(4)$-structures has been
studied in [2, $\S$7 – $\S$11].

If $M$ admits an $\mathrm{SO}(4)$-structure $(\varphi,\mathsf{A})$, then its
metric cone $(\mathrm{C}(M),g_{\mathrm{C}})$ admits a compatible
$\mathrm{Spin}^{h}(4)$-structure, where
$\mathrm{Spin}^{h}(4)=(\mathrm{SU}(2)\times\mathrm{SU}(2)\times\mathrm{SU}(2))/\mathbb{Z}_{2}$.
That is, $\mathrm{C}(M)$ admits a conical $\mathrm{Spin}(7)$-structure
$\Phi\in\Omega^{4}(\mathrm{C}(M))$ together with a preferred Cayley $4$-plane
distribution $\mathsf{K}\subset T(\mathsf{C}(M))$ via

$\displaystyle\Phi$ $\displaystyle=r^{3}dr\wedge\varphi+r^{4}\ast\\!\varphi$
$\displaystyle\mathsf{K}=\mathsf{A}\oplus\text{span}(\partial_{r}).$

More importantly for this work, since
$\mathrm{Spin}^{h}(4)\leq\mathrm{Sp}(2)\mathrm{Sp}(1)$, the $8$-manifold
$\text{C}(M)$ also admits a conical $\mathrm{Sp}(2)\mathrm{Sp}(1)$-structure.
To be explicit, note that the volume form of $M$ decomposes as
$\mathrm{vol}=\mathrm{vol}_{\mathsf{A}}\wedge\ast\mathrm{vol}_{\mathsf{A}}$,
where $\mathrm{vol}_{\mathsf{A}}\in\Omega^{3}(M)$ denotes the volume form of
$\mathsf{A}$ and all Hodge stars are taken on $M$. Then the $4$-form on
$\Theta\in\Omega^{4}(\mathsf{C}(M))$ given by

$\displaystyle\Theta$
$\displaystyle=r^{3}dr\wedge\left(4\mathrm{vol}_{\mathsf{A}}-\varphi\right)+r^{4}\\!\ast\\!(4\mathrm{vol}_{\mathsf{A}}-\varphi)$

is a conical $\mathrm{Sp}(2)\mathrm{Sp}(1)$-structure.
Interestingly, one does not need the full $\mathrm{Spin}^{h}(4)$-structure on
$\mathrm{C}(M)$ to recover the $\mathrm{SO}(4)$-structure on $M$. Indeed, if
$\mathrm{C}(M)$ carries merely a conical
$\mathrm{Sp}(2)\mathrm{Sp}(1)$-structure $\Theta\in\Omega^{4}(\text{C}(M))$,
then $M$ inherits an $\mathrm{SO}(4)$-structure $(\varphi,\mathsf{A})$ via

$\displaystyle\varphi$
$\displaystyle=4\mathrm{vol}_{\mathsf{A}}-\ast_{M}\Theta|_{M}$
$\displaystyle\mathsf{A}=\text{span}(I_{1}(\partial_{r}),I_{2}(\partial_{r}),I_{3}(\partial_{r})),$

where $(I_{1},I_{2},I_{3})$ are locally-defined almost-complex structures on
$\mathrm{C}(M)$ satisfying the quaternionic relations ($I_{1}I_{2}=I_{3}$,
etc.) that arise from the $\mathrm{Sp}(2)\mathrm{Sp}(1)$-structure. This
motivates the following:

######  Definition 2.3.

An $\mathrm{SO}(4)$-structure $(\varphi,\mathsf{A})$ on $M$ is called
_quaternion-Sasakian_ if the induced $\mathrm{Sp}(2)\mathrm{Sp}(1)$-structure
on $\mathrm{C}(M)$ is quaternion-Kähler.

######  Proposition 2.4.

If $(\varphi,\mathsf{A})$ is a quaternion-Sasakian structure on $M$, then
(a) $\varphi$ is isometric to a nearly-parallel $\mathrm{G}_{2}$-structure.
(b) $\varphi$ is co-closed.
(c) $\mathsf{A}$ is integrable.

We will prove (a) in $\S$2.4. The proofs of (b) and (c), along with a second
proof of (a), are in $\S$3.2. More generally, we will show that if
$(\varphi,\mathsf{A})$ is quaternion-Sasakian, then $\varphi$ belongs to a
canonical $2$-parameter family of co-closed $\mathrm{G}_{2}$-structures
$\varphi_{a,b}$ with $\varphi_{1,1}=\varphi$. Note that this generalization of
(b) is implicit in [16, $\S$5], and that part (c) is a standard fact from
$3$-Sasakian geometry. However, the language of quaternion-Sasakian structures
is new.

####  2.2.1 Associative $3$-folds in $\mathrm{SO}(4)$-Structures

The Grassmannian of associative $3$-planes in $\mathbb{R}^{7}$ is the
symmetric space $\text{Asoc}=\mathrm{G}_{2}/\mathrm{SO}(4)$. In order to study
associative 3-folds in 7-manifolds with $\mathrm{SO}(4)$-structure, it is
important to understand the $\mathrm{SO}(4)$-action on Assoc. This action is
the isotropy action of a rank two symmetric space, so it has cohomogeneity
two. The principal orbits are equivariantly diffeomorphic to the $6$-manifold
$\mathrm{SO}(4)/(\mathbb{Z}_{2}\times\mathbb{Z}_{2})$. The $2$-dimensional
orbit space can be viewed as a right triangle in $\mathbb{R}^{2}$, as
demonstrated in the following proposition.

######  Proposition 2.5.

Let $\mathrm{SO}(4)$ act on $\mathbb{R}^{7}$ as the stabilizer of the pair
$(\varphi,A),$ where

$\displaystyle\varphi$
$\displaystyle=e^{123}+e^{145}+e^{167}+e^{246}-e^{257}-e^{347}-e^{356}$
$\displaystyle A$ $\displaystyle=\operatorname{span}(e_{1},e_{2},e_{3}).$

Any associative 3-plane $P<\mathbb{R}^{7}$ is $\mathrm{SO}(4)$-equivalent to
the associative 3-plane

$P_{s,r}=\operatorname{span}\\!\left(\cos\\!\left(2s\right)e_{1}+\sin\\!\left(2s\right)e_{6},\
\cos\\!\left(s-r\right)e_{2}+\sin\\!\left(s-r\right)e_{5},\
\cos\\!\left(s+r\right)e_{3}+\sin\\!\left(s+r\right)e_{4}\right)\\!,$

with $s$ and $r$ unique subject to the constraints $s\geq 0,$ $3s\leq r,$
$r\leq\pi/2.$

###### Proof.

The intersection $P\cap A^{\perp}$ has dimension at least one. The
$\mathrm{SO}(4)$-action is transitive on the unit sphere in $A^{\perp},$ so by
acting by an element of $\mathrm{SO}(4)$ we may assume that $e_{7}\in P\cap
A^{\perp}$. The subgroup of $\mathrm{SO}(4)$ stabilizing $e_{7}$ is
$\mathrm{O}(3),$ which splits $\mathbb{R}^{7}$ as $A\oplus B\oplus e_{7},$
where $B=\operatorname{span}(e_{4},e_{5},e_{6}).$

We now consider $P$ as a subspace of $A\oplus B$ under the action of
$\mathrm{O}(3).$ It follows from the general theory of Jordan angles [21]
(also known as principal angles) that $P$ is $\mathrm{O}(3)$-equivalent to the
3-plane

$P_{\gamma_{1},\gamma_{2},\gamma_{3}}=\operatorname{span}\left(\cos(\gamma_{1})e_{1}+\sin(\gamma_{1})e_{6},\cos(\gamma_{2})e_{2}-\sin(\gamma_{2})e_{5},\cos(\gamma_{3})e_{3}-\sin(\gamma_{3})e_{4}\right)$

with $\gamma_{1},\gamma_{2},\gamma_{3}$ unique subject to
$0\leq\gamma_{1}\leq\gamma_{2}\leq\gamma_{3}\leq\pi/2.$ The condition that
$P_{\gamma_{1},\gamma_{2},\gamma_{3}}$ be associative is
$\gamma_{1}+\gamma_{2}+\gamma_{3}=\pi.$ We may parameterize such triples by
$\gamma_{1}=2s,$ $\gamma_{2}=s+r,$ $\gamma_{3}=\pi-s+r.$ The conditions
$0\leq\gamma_{1}\leq\gamma_{2}\leq\gamma_{3}\leq\pi/2$ become $s\geq 0,$
$3s\leq r,$ $r\leq\pi/2.$ ∎

$(0,0)$$(\frac{\pi}{2},0)$$(\frac{\pi}{2},\frac{\pi}{6})$$s=0$ Figure 1: Orbit
space of the $\mathrm{SO}(4)$-action on
$\text{Asoc}=\mathrm{G}_{2}/\mathrm{SO}(4)$ in the $rs$-plane.

If $N$ is an associative 3-fold in a 7-manifold with
$\mathrm{SO}(4)$-structure $(M,\varphi,\mathsf{A}),$ then Proposition 2.5
defines two functions $r\colon N\to\left[0,\pi/2\right]$ and $s\colon
N\to\left[0,\pi/6\right]$ that describe the position of the tangent planes
$T_{x}N$ relative to the associative distribution $\mathsf{A}_{x}$.

######  Proposition 2.6.

Let $(M,\varphi,\mathsf{A})$ be a $7$-manifold with an
$\mathrm{SO}(4)$-structure, let $N\subset M$ be an associative $3$-fold, and
let $x\in N$. The following are equivalent:
(i) $\dim(T_{x}N\cap\mathsf{A}_{x})\geq 1$.
(ii) $\dim(T_{x}N\cap\mathsf{A}_{x})=1$ or $T_{x}N=\mathsf{A}_{x}$.
(iii) $s(x)=0$.
Moreover, $\dim(T_{x}N\cap\mathsf{A}_{x})=1$ if and only if $s(x)=0$ and
$r(x)\neq 0$.

######  Definition 2.7.

Let $(M,\varphi,\mathsf{A})$ be a $7$-manifold with an
$\mathrm{SO}(4)$-structure. An associative $3$-fold $N\subset M$ is called
_striped_ if either of the following equivalent conditions holds:
(i) $\dim(T_{x}N\cap\mathsf{A}_{x})=1$ for each $x\in\Sigma$.
(ii) The function $s\colon N\to\left[0,\pi/6\right]$ vanishes identically and
$r\colon N\to\left[0,\pi/2\right]$ is nowhere vanishing.

When $M$ is a quaternion-Sasakian $7$-manifold or one of its “squashings” (as
defined in $\S$3.1.1), we will show in Proposition 4.2 that an associative
submanifold of $M$ is striped if and only if it is “ruled by Hopf circles” and
not tangent to $\mathsf{A}$.

###  2.3 $\mathrm{Sp}(1)$-Structures and Associative $3$-folds

In this work, the $\mathrm{SO}(4)$-structures that we encounter will underlie
$\mathrm{Sp}(1)$-structures. Although we are primarily interested in the
$\mathrm{SO}(4)$-structure, it will be occasionally be useful to perform
computations in terms of the richer $\mathrm{Sp}(1)$-structure.

######  Definition 2.8.

Let $M^{7}$ be a smooth $7$-manifold. An _almost contact metric structure_ (or
_$\mathrm{U}(3)$ -structure_) is a triple
$(\langle\cdot,\cdot\rangle,\alpha,\mathsf{J})$ consisting of a Riemannian
metric $\langle\cdot,\cdot\rangle$, a $1$-form $\alpha\in\Omega^{1}(M)$, and
an endomorphism $\mathsf{J}\in\Gamma(\mathrm{End}(TM))$ such that

$\displaystyle\left.\mathsf{J}^{2}\right|_{\text{Ker}(\alpha)}$
$\displaystyle=-\mathrm{Id}$ $\displaystyle\mathsf{J}(A)$ $\displaystyle=0$
$\displaystyle\langle\mathsf{J}X,\mathsf{J}Y\rangle$ $\displaystyle=\langle
X,Y\rangle-\alpha(X)\alpha(Y),$

where $A:=\alpha^{\sharp}\in\Gamma(TM)$ is the Reeb field, so that
$\alpha(A)=1$.

An _$\mathrm{Sp}(1)$ -structure_ on $M$ consists of data
$(\langle\cdot,\cdot\rangle,(\alpha_{1},\alpha_{2},\alpha_{3}),(\mathsf{J}_{1},\mathsf{J}_{2},\mathsf{J}_{3}))$
for which each triple ($\langle\cdot,\cdot\rangle,\alpha_{p},\mathsf{J}_{p})$
is an almost contact metric structure, and

$\displaystyle\mathsf{J}_{p}\circ\mathsf{J}_{q}-\alpha_{p}\otimes A_{q}$
$\displaystyle=-\epsilon_{pqr}\mathsf{J}_{r}-\delta_{pq}\mathrm{Id}$
$\displaystyle\mathsf{J}_{p}(A_{q})$ $\displaystyle=-\epsilon_{pqr}A_{r},$

where $A_{p}:=\alpha_{p}^{\sharp}\in\Gamma(TM)$ are the Reeb fields. It is
known that a compact smooth $7$-manifold admits an $\mathrm{Sp}(1)$-structure
if and only if it is orientable and spin [16, $\S$3].

Thus, given an $\mathrm{Sp}(1)$-structure on $M$, each tangent space can be
split as $T_{x}M=\mathbb{R}A_{p}\oplus\mathrm{Ker}(\alpha_{p})$, for any
$p=1,2,3$, and each $6$-plane $\mathrm{Ker}(\alpha_{p})$ carries a special
Hermitian structure
$(\langle\cdot,\cdot\rangle,\mathsf{J}_{p},\Omega_{p},\Upsilon_{p})$, where
$\Omega_{p}:=\langle\mathsf{J}_{p}\cdot,\cdot\rangle$ and $\Upsilon_{p}$ is a
complex volume form on $\mathrm{Ker}(\alpha_{p})$ that is type $(3,0)$ with
respect to $\mathsf{J}_{p}$. In fact, the tangent spaces decompose further as
$T_{x}M=\mathbb{R}A_{1}\oplus\mathbb{R}A_{2}\oplus\mathbb{R}A_{3}\oplus\mathsf{C}$,
where the $4$-plane
$\mathsf{C}:=\mathrm{Ker}(\alpha_{1},\alpha_{2},\alpha_{3})$ carries a special
Hermitian structure in the natural way.

We make four remarks on $\mathrm{Sp}(1)$-structures. First, since
$\mathrm{Sp}(1)\leq\mathrm{SO}(4)$, every $\mathrm{Sp}(1)$-structure has an
underlying $\mathrm{SO}(4)$-structure $(\varphi,\mathsf{A})$. Explicitly,

$\displaystyle\varphi$
$\displaystyle:=\alpha_{1}\wedge\alpha_{2}\wedge\alpha_{3}-\alpha_{1}\wedge\Omega_{1}-\alpha_{2}\wedge\Omega_{2}-\alpha_{3}\wedge\Omega_{3}$
$\displaystyle\mathsf{A}$ $\displaystyle:=\text{span}(A_{1},A_{2},A_{3}).$

Second, note that an $\mathrm{Sp}(1)$-structure determines an entire
$2$-sphere of almost contact metric structures. Indeed, for each $w\in
S^{2}=\\{(w_{1},w_{2},w_{3})\in\mathbb{R}^{3}\colon|w|=1\\}$, defining
$\alpha_{w}:=w_{p}\alpha_{p}$ and $\mathsf{J}_{w}:=w_{p}\mathsf{J}_{p}$ yields
an almost contact metric structure
$(\langle\cdot,\cdot\rangle,\alpha_{w},\mathsf{J}_{w})$.
Third, if a $7$-manifold $M$ carries an $\mathrm{Sp}(1)$-structure
$(g,(\alpha_{1},\alpha_{2},\alpha_{3}),(\mathsf{J}_{1},\mathsf{J}_{2},\mathsf{J}_{3}))$,
then the Riemannian $8$-manifold $(\mathrm{C}(M),g_{\mathrm{C}})$ admits a
triple $(I_{1},I_{2},I_{3})$ of $g_{\mathrm{C}}$-compatible almost-complex
structures with $I_{1}I_{2}=I_{3}$ given by

$\displaystyle I_{p}X$
$\displaystyle=\textstyle\mathsf{J}_{p}X-\alpha_{p}(X)r\partial_{r},\ \ \text{
for }X\in TM,$ $\displaystyle\textstyle I_{p}(r\partial_{r})$
$\displaystyle=A_{p}.$

In other words, $\mathrm{C}(M)$ inherits a conical $\mathrm{Sp}(2)$-structure
(or _almost hyper-Hermitian structure_). Conversely, if an $8$-dimensional
cone $\mathrm{C}(M)$ admits an $\mathrm{Sp}(2)$-structure
$(g_{\mathrm{C}},(\omega_{1},\omega_{2},\omega_{3}),(I_{1},I_{2},I_{3}))$ for
which each $I_{p}(r\partial_{r})$ is tangent to $M$ and each
$\mathscr{L}_{r\partial_{r}}I_{p}=0$, then its link $M$ inherits an
$\mathrm{Sp}(1)$-structure via

$\displaystyle\alpha_{p}$
$\displaystyle=\textstyle\partial_{r}\,\lrcorner\,\omega_{p}$
$\displaystyle\mathsf{J}_{p}$ $\displaystyle=\begin{cases}I_{p}&\mbox{ on
}\text{Ker}(\alpha_{p})\\\ 0&\mbox{ on }\mathbb{R}A_{p}.\end{cases}$

This leads to the following definition:

######  Definition 2.9.

An $\mathrm{Sp}(1)$-structure on $M$ is called _$3$ -Sasakian_ if the induced
$\mathrm{Sp}(2)$-structure on $\mathrm{C}(M)$ is hyperkähler.

######  Remark 2.10.

If an $\mathrm{Sp}(1)$-structure on $M$ is 3-Sasakian, then its underlying
$\mathrm{SO}(4)$-structure is quaternion-Sasakian. Conversely, a quaternion-
Sasakian $\mathrm{SO}(4)$-structure together with a trivialization
$\left(w_{1},w_{2},w_{3}\right)$ of the bundle $\mathsf{A}$ gives rise to a
unique 3-Sasakian $\mathrm{Sp}(1)$-structure sharing the same metric $g$ and
with $\alpha_{i}=w_{i}^{\flat}.$

######  Remark 2.11.

On an $8$-manifold $Q$, every $\mathrm{Sp}(2)$-structure
$(g,(I_{1},I_{2},I_{3}))$ induces an $\mathrm{Sp}(2)\mathrm{Sp}(1)$-structure,
essentially by forgetting the triple $(I_{1},I_{2},I_{3})$ and retaining only
the rank $3$ subbundle
$\text{span}(I_{1},I_{2},I_{3})\subset\mathrm{End}(TQ)$. In this way, for
example, every hyperkähler structure underlies a quaternion-Kähler structure.
This passage from an $\mathrm{Sp}(2)$\- to an
$\mathrm{Sp}(2)\mathrm{Sp}(1)$-structure on an $8$-manifold is analogous to
the passage from an $\mathrm{Sp}(1)$\- to an $\mathrm{SO}(4)$-structure on a
$7$-manifold.

Fourth, we note that $7$-manifolds with $\mathrm{Sp}(1)$-structures admit
several distinguished classes of submanifolds. Here are some of them:

######  Definition 2.12.

Let $N^{k}\subset M^{7}$ be a $k$-dimensional submanifold for $k=1,3,5$.
Say $N$ is _$\mathsf{J}_{1}$ -CR_ if each tangent space $T_{x}N$ contains the
Reeb vector $A_{1}$ and is $\mathsf{J}_{1}$-invariant. An equivalent condition
is that $\mathrm{C}(N)$ is an $I_{1}$-complex submanifold of $\mathrm{C}(M)$.

######  Definition 2.13.

Let $N^{3}\subset M^{7}$ be a $3$-dimensional submanifold.

  * •

Say $N$ is _$\mathsf{J}_{1}$ -Legendrian_ if $\alpha_{1}|_{N}=0$ and
$\Omega_{1}|_{N}=0$. An equivalent condition is that $\mathrm{C}(N)$ is an
$I_{1}$-Lagrangian $4$-fold in $\mathrm{C}(M)$.

  * •

Say $N$ is _$\mathsf{J}_{1}$ -special Legendrian_ if $\alpha_{1}|_{N}=0$,
$\Omega_{1}|_{N}=0$, and $\text{Im}(\Upsilon_{1})|_{N}=0$. An equivalent
condition is that $\mathrm{C}(N)$ is an $I_{1}$-special Lagrangian $4$-fold in
$\mathrm{C}(M)$.

  * •

Say $N$ is _$\mathsf{J}_{1}$ -complex Legendrian_ if it is $\mathsf{J}_{1}$-CR
and $\mathsf{J}_{2}$-Legendrian and $\mathsf{J}_{3}$-Legendrian. An equivalent
condition is that $\mathrm{C}(N)$ is an $I_{1}$-complex Lagrangian $4$-fold in
$\mathrm{C}(M)$.

The relevance of these submanifolds to our work comes from the following basic
observation:

######  Proposition 2.14.

Let $N^{3}\subset M^{7}$ be a $3$-dimensional submanifold, where $M$ carries
an $\mathrm{Sp}(1)$-structure. If $N$ is $\mathsf{J}_{w}$-CR or
$\mathsf{J}_{w}$-special Legendrian for some $w\in S^{2}$, then $N$ is an
associative $3$-fold (with respect to the underlying
$\mathrm{G}_{2}$-structure on $M$).

When $M$ carries an $\mathrm{Sp}(1)$-structure, we are interested in
constructing associative $3$-folds in $M$ that are _neither_ CR nor special
Legendrian with respect to any $\mathsf{J}_{w}$ for $w\in S^{2}$. The compact
examples we construct are obtained by “twisting” complex Legendrian $3$-folds
by a meromorphic function.

###  2.4 Nearly-Parallel $\mathrm{G}_{2}$-Structures

We now return to $\mathrm{G}_{2}$ geometry. Within the class of co-closed
$\mathrm{G}_{2}$-structures, the subclass of nearly-parallel
$\mathrm{G}_{2}$-structures is particularly important in view of its
connection to $\mathrm{Spin}(7)$ geometry. Indeed, as the following
proposition shows, nearly-parallel $\mathrm{G}_{2}$-structures arise naturally
on links of $8$-dimensional cones equipped with conical torsion-free
$\mathrm{Spin}(7)$-structures.

######  Proposition 2.15 ([6]).

Let $M^{7}$ be a smooth $7$-manifold. There is a bijection:

$\displaystyle\\{\text{nearly-parallel }\mathrm{G}_{2}\text{-str. on }M\text{
with }\lambda=4\\}$ $\displaystyle\longleftrightarrow\\{\text{conical torsion-
free }\mathrm{Spin}(7)\text{-str. on }\mathrm{C}(M)\\}$ $\displaystyle\varphi$
$\displaystyle\longmapsto r^{3}\,dr\wedge\varphi+r^{4}\ast\\!\varphi$
$\displaystyle\textstyle\left.(\partial_{r}\,\lrcorner\,\Phi)\right|_{M}$
$\displaystyle\longmapsfrom\Phi$

Now, in light of the inclusions
$\mathrm{Sp}(2)\leq\mathrm{SU}(4)\leq\mathrm{Spin}(7)$, there is a hierarchy
of nearly-parallel metrics on $M$ according to the holonomy group of the
$8$-manifold $(\text{C}(M),g_{\mathrm{C}})$. Precisely:

######  Definition 2.16.

Let $(M^{7},g)$ be a Riemannian $7$-manifold.

  * •

Say $g$ is _nearly-parallel_ if
$\mathrm{Hol}(g_{\mathrm{C}})\leq\mathrm{Spin}(7)$. Such metrics are of the
form $g=g_{\varphi}$ for some nearly-parallel $\mathrm{G}_{2}$-structure
$\varphi$ on $M$. We say $g$ is _proper nearly-parallel_ if
$\mathrm{Hol}(g_{\mathrm{C}})=\mathrm{Spin}(7)$.

  * •

Say $g$ is _Sasaki-Einstein_ if
$\text{Hol}(g_{\mathrm{C}})\leq\mathrm{SU}(4)$. Such metrics underlie a
Sasaki-Einstein $\mathrm{SU}(3)$-structure on $M$.

  * •

Say $g$ is _3-Sasakian_ if $\text{Hol}(g_{\mathrm{C}})\leq\mathrm{Sp}(2)$.
Such metrics underlie a $3$-Sasakian structure on $M$.

######  Remark 2.17.

Note that the structure groups
$\mathrm{Sp}(1)\leq\mathrm{SU}(3)\leq\mathrm{G}_{2}$ are the stabilizers of a
unit vector in $\mathbb{R}^{8}$ under the standard actions of
$\mathrm{Sp}(2)\leq\mathrm{SU}(4)\leq\mathrm{Spin}(7)$, respectively.

With that above terminology in place, we now prove:

###### Proof of Proposition 2.4(a).

Let $(\varphi,\mathsf{A})$ be a quaternion-Sasakian structure on $M$. By
definition, the $8$-manifold $\mathsf{C}(M)$ admits a conical quaternion-
Kähler structure, so the cone metric $g_{\mathrm{C}}$ is quaternion-Kähler.
Since any conical Einstein metric is Ricci-flat, it follows that
$g_{\mathrm{C}}$ is hyperkähler, and hence $g$ is $3$-Sasakian, whence $g$ is
nearly-parallel. We conclude there exists a nearly-parallel
$\mathrm{G}_{2}$-structure $\widehat{\varphi}$ whose induced metric is $g$.
That is, $\varphi$ is isometric to the nearly-parallel $\widehat{\varphi}$. ∎

In the remainder of this section, we briefly discuss the two extreme ends of
the hierarchy, namely $3$-Sasakian $7$-manifolds and proper nearly-parallel
$7$-manifolds. Our discussion of $3$-Sasakian $7$-manifolds follows [7,
$\S$13.2 – $\S$13.5].

####  2.4.1 $3$-Sasakian $7$-Manifolds

Let $M^{7}$ be a compact $7$-manifold equipped with a $3$-Sasakian structure
$(\langle\cdot,\cdot\rangle,(\alpha_{1},\alpha_{2},\alpha_{3})$,
$(\mathsf{J}_{1},\mathsf{J}_{2},\mathsf{J}_{3}))$. As we now explain, the
geometry of $M$ can be understood in terms of a $3$-dimensional foliation
$\mathcal{F}_{A}\subset M$ and an $S^{2}$-family of $1$-dimensional foliations
$\mathcal{F}_{w}\subset\mathcal{F}_{A}\subset M$.
Since $M$ is compact, the Reeb fields $A_{1},A_{2},A_{3}$ are complete.
Moreover, they are an orthonormal set of Killing vector fields, and define a
locally free $\mathrm{Sp}(1)$-action on $M$. In turn, this yields a quasi-
regular $3$-dimensional _canonical foliation_ $\mathcal{F}_{A}$ on $M$ whose
leaf space $X:=M/\mathcal{F}_{A}$ is a compact $4$-orbifold. We refer to the
leaves of $\mathcal{F}_{A}$ as _canonical leaves_.

######  Theorem 2.18 ([7, $\S$13.3.11, 13.3.13]).

Let $M$ be a $3$-Sasakian $7$-manifold such that the Reeb fields
$A_{1},A_{2},A_{3}$ are complete.
(a) The canonical leaves are totally-geodesic, spherical space forms
$S^{3}/\Gamma$ of constant curvature, where $\Gamma\leq\mathrm{Sp}(1)$ is a
finite subgroup. Moreover, the generic leaves are either $S^{3}$ or
$\mathrm{SO}(3)$.
(b) The leaf space $X=M/\mathcal{F}_{A}$ admits the structure of a positive
quaternion-Kähler $4$-orbifold such that the natural projection $h\colon M\to
X$ is a principal $G$-orbibundle for $G=\mathrm{SU}(2)$ or $\mathrm{SO}(3)$,
and a Riemannian orbifold submersion. Further,
$\alpha=(\alpha_{1},\alpha_{2},\alpha_{3})$ is a connection form for $h$.

Interestingly, this process can be inverted:

######  Theorem 2.19 ([7, $\S$13.3.14, 13.3.16] Konishi Inversion).

Let $\hat{X}$ be a positive quaternion-Kähler $4$-orbifold, and let
$\epsilon\in H^{2}_{\mathrm{orb}}(\hat{X},\mathbb{Z}_{2})$ be its Marchiafava-
Romani class. Then:
(a) There is a principal $\mathrm{SO}(3)$-orbibundle $\hat{M}\to\hat{X}$ for
which $\hat{M}$ admits a $3$-Sasakian structure.
(b) The $\mathrm{SO}(3)$-orbibundle $\hat{M}\to\hat{X}$ lifts to a principal
$S^{3}$-orbibundle if and only if $\epsilon=0$. In this case, the $3$-Sasakian
structure on $\hat{M}$ lifts to the total space of the $S^{3}$-orbibundle.

As above, let $M$ be a compact $7$-manifold with a $3$-Sasakian structure. For
each $w\in S^{2}$, the Reeb field $A_{w}$ is a complete, unit-length, Killing
vector field, and defines a locally free $S^{1}$-action on $M$. Thus, $A_{w}$
yields a quasi-regular $1$-dimensional foliation
$\mathcal{F}_{w}\subset\mathcal{F}_{A}\subset M$ whose leaves are closed
geodesics that we call _Hopf circles_ , and whose leaf space
$Z_{w}:=M/\mathcal{F}_{w}$ is a compact $6$-orbifold. In fact:

######  Theorem 2.20 ([7, $\S$7.5.1, 13.3.1]).

Let $M$ be a compact $3$-Sasakian $7$-manifold.
(a) The projection $p_{w}\colon M\to Z_{w}$ is a principal $S^{1}$-orbibundle
with connection $1$-form $\alpha_{w}$.
(b) For $w,w^{\prime}\in S^{2}$, there is a diffeomorphism $Z_{w}\approx
Z_{w^{\prime}}$. In fact, each $Z_{w}$ may be identified with the (orbifold)
twistor space $Z$ of the quaternion-Kähler 4-orbifold $X=M/\mathcal{F}_{A}$.

######  Theorem 2.21 ([7, $\S$13.2.5, 13.5.10]).

Let $M$ be a compact $3$-Sasakian $7$-manifold. The foliation
$\mathcal{F}_{w_{0}}$ is regular for some $w_{0}\in S^{2}$ if and only if the
foliations $\mathcal{F}_{w}$ are regular for all $w\in S^{2}$. In this case,
$M$ is homogeneous.

######  Remark 2.22.

It is known that each $Z_{w}$ admits a Kähler-Einstein metric $g_{\text{KE}}$
of positive scalar curvature such that the projection
$p_{w}\colon(M,g)\to(Z_{w},g_{\text{KE}})$ is an orbifold Riemannian
submersion. However, we will instead equip $Z_{w}\approx Z$ with its standard
_strict nearly-Kähler structure_. We will be more explicit in $\S$3.3.3.

Altogether, we have a diagram:

${M}$${Z}$${X}$$\scriptstyle{h}$$\scriptstyle{p_{w}}$$\scriptstyle{\tau}$

Letting $\tau\colon Z\to X$ denote the twistor $S^{2}$-bundle over $X$, we let
$\mathsf{V}\subset TZ$ denote the vertical subbundle, and let
$\mathsf{H}\subset TZ$ denote the horizontal subbundle with respect to the
nearly-Kähler metric on $Z$. Thus, we have an orthogonal splitting
$TZ=\mathsf{V}\oplus\mathsf{H}$. A submanifold of $Z$ that is tangent to
$\mathsf{H}$ is said to be _horizontal_. In terms of the decomposition
$T_{x}M=\mathbb{R}A_{1}\oplus\mathbb{R}A_{2}\oplus\mathbb{R}A_{3}\oplus\mathsf{C}$
discussed in $\S$2.3, we have $p_{w}(\mathsf{C})=\mathsf{H}$ for any $w\in
S^{2}$.

We now turn to examples. The homogeneous $3$-Sasakian $7$-manifolds are
classified [7, $\S$13.4.6], and consist of

$\displaystyle S^{7}$ $\displaystyle=\frac{\mathrm{Sp}(2)}{\mathrm{Sp}(1)}$
$\displaystyle\mathbb{RP}^{7}$
$\displaystyle=\frac{\mathrm{Sp}(2)}{\mathrm{Sp}(1)\times\mathbb{Z}_{2}}$
$\displaystyle N_{1,1}$
$\displaystyle=\frac{\mathrm{SU}(3)}{\text{S}(\mathrm{U}(1)\times\mathrm{U}(1))}.$

A wide variety of compact _inhomogeneous_ $3$-Sasakian $7$-manifolds have been
constructed by Boyer-Galicki-Mann:

######  Theorem 2.23 ([8]).

For each $(p_{1},p_{2},p_{3})\in\mathbb{Z}^{3}_{+}$ with $1\leq p_{1}\leq
p_{2}\leq p_{3}$, there exists a compact, simply-connected $3$-Sasakian
$7$-manifold called $S(p_{1},p_{2},p_{3})$. Among them are infinitely many
inhomogeneous $7$-manifolds that are not homotopy equivalent.

Finally, we return to submanifolds.

######  Proposition 2.24.

Let $M^{7}$ be a $3$-Sasakian $7$-manifold, and let $N^{3}\subset M^{7}$ be a
$3$-dimensional submanifold. Fix $w\in S^{2}$ and consider the
$S^{1}$-orbibundle $p_{w}\colon M\to Z$, where $Z$ carries its standard strict
nearly-Kähler $\mathrm{SU}(3)$-structure. Then:
(a) $N$ is $\mathsf{J}_{w}$-CR if and only if $N=p_{w}^{-1}(S)$ for some
pseudo-holomorphic curve $S\subset Z$.
(b) $N$ is $\mathsf{J}_{w}$-complex Legendrian if and only if
$N=p_{w}^{-1}(S)$ for some horizontal pseudo-holomorphic curve $S\subset Z$.

####  2.4.2 Proper Nearly-Parallel $7$-Manifolds

At the other end of the hierarchy are the proper nearly-parallel
$7$-manifolds, about which far less is known. The compact, simply-connected,
homogeneous ones have been classified by Friedrich, Kath, Moroianu, and
Semmelmann [16], and consist of

$\displaystyle S^{7}$
$\displaystyle=\frac{\mathrm{Sp}(2)\mathrm{Sp}(1)}{\mathrm{SO}(4)}$
$\displaystyle B$ $\displaystyle=\frac{\mathrm{SO}(5)}{\mathrm{SO}(3)}$
$\displaystyle N_{k,\ell}$
$\displaystyle=\frac{\mathrm{SU}(3)}{S^{1}_{(k,\ell)}}$

In the same paper, the authors constructed _inhomogeneous_ proper nearly-
parallel $7$-manifolds by “squashing” the metric of a $3$-Sasakian
$7$-manifold. Precisely:

######  Theorem 2.25 ([16], [17]).

Let $M$ be a $7$-manifold equipped with a $3$-Sasakian structure
$(g,(\alpha_{1},\alpha_{2},\alpha_{3})$,
$(\mathsf{J}_{1},\mathsf{J}_{2},\mathsf{J}_{3}))$. Orthogonally split
$TM=\mathsf{A}\oplus\mathsf{C}$, where
$\mathsf{A}=\mathrm{span}(A_{1},A_{2},A_{3})$ and $A_{1},A_{2},A_{3}$ are the
Reeb fields. For $t>0$, define the metrics

$g_{t}(X,Y)=\begin{cases}t^{2}g(X,Y)&\mbox{if }X,Y\in\mathsf{A}\\\
g(X,Y)&\mbox{if }X\in\mathsf{C}\text{ or }Y\in\mathsf{C}.\end{cases}$

Then:
(a) The metric $g_{t}$ is Einstein if and only if $t=1$ or
$t=\frac{1}{\sqrt{5}}$.
(b) The metric $g_{{1}/{\sqrt{5}}}$ is proper nearly-parallel.

######  Remark 2.26.

The squashed metrics $g_{t}$ do not require the full $3$-Sasakian structure
for their construction, but simply the underlying quaternion-Sasakian
structure.

######  Example 2.27.

  1. (i)

Take $M=S^{7}$ with its standard $\mathrm{Sp}(2)$-invariant 3-Sasakian
structure. The underlying metric $g=g_{1}$ is the ($\mathrm{SO}(8)$-invariant)
round metric. The proper nearly-parallel metric $g_{{1}/{\sqrt{5}}}$ is the
$\mathrm{Sp}(2)\mathrm{Sp}(1)$-invariant Jensen metric [20] and is induced by
an $\mathrm{Sp}(2)\mathrm{Sp}(1)$-invariant nearly-parallel
$\mathrm{G}_{2}$-structure $\varphi$. The $7$-manifold
$(S^{7},\varphi,g_{{1}/{\sqrt{5}}})$ is often called the _squashed 7-sphere_.

  2. (ii)

Take $M=N_{1,1}$ with its standard $\mathrm{SU}(3)$-invariant 3-Sasakian
structure. The proper nearly-parallel metric $g_{{1}/{\sqrt{5}}}$ is an
$(\mathrm{SU}(3)\times\mathrm{SO}(3))$-invariant Einstein metric induced by an
$(\mathrm{SU}(3)\times\mathrm{SO}(3))$-invariant nearly-parallel
$\mathrm{G}_{2}$-structure $\varphi$. We refer to $7$-manifold
$(N_{1,1},\varphi,g_{{1}/{\sqrt{5}}})$ as a _squashed exceptional Aloff-
Wallach space_.

  3. (iii)

Taking $M$ to be an inhomogeneous Boyer-Galicki-Mann $3$-Sasakian $7$-manifold
$S(p_{1},p_{2},$ $p_{3})$, Theorem 2.25(b) yields compact, inhomogeneous,
proper nearly-parallel $7$-manifolds.

Above, we described squashing as a deformation of the _Riemannian structure_.
However, in $\S$3.2.2, we will view squashing as a deformation of the
underlying _$\mathrm{G}_{2}$ -structure_. A bit of care will be required in
that regard, as distinct $\mathrm{G}_{2}$-structures can induce the same
metric: see Example 3.6 and the discussion preceding it.

######  Remark 2.28.

We have discussed various $G$-structures on $7$-manifolds and their
$8$-dimensional cones. As a guide to these, we summarize some of the Lie
groups $G$ under consideration:

${\text{G}_{2}}$${\text{U}(3)}$${\text{Spin}(7)}$${\text{U}(4)}$${\text{SO}(4)}$${\text{SU}(3)}$${\text{Sp}(2)\text{Sp}(1)}$${\text{SU}(4)}$${\text{U}(2)}$${\text{Sp}(2)\text{U}(1)}$${\text{Sp}(1)}$${\text{Sp}(2)}$

Notice that although $\mathrm{Sp}(2)\mathrm{Sp}(1)$ is not a subgroup of
$\mathrm{Spin}(7)$, it is nevertheless the case that the
$\mathrm{Sp}(2)\mathrm{Sp}(1)$-stabilizer of a unit vector $\mathbf{v}\in
S^{7}\subset\mathbb{H}^{2}$ is contained in the $\mathrm{Spin}(7)$-stabilizer
of $\mathbf{v}$, i.e.:

$\mathrm{SO}(4)=\text{Stab}_{\mathrm{Sp}(2)\mathrm{Sp}(1)}(\mathbf{v})\leq\text{Stab}_{\mathrm{Spin}(7)}(\mathbf{v})=\mathrm{G}_{2}.$

This “accidental inclusion” relating $\mathrm{Sp}(2)\mathrm{Sp}(1)$ and
$\mathrm{Spin}(7)$ plays a key role in this work: see, e.g., $\S$2.2 and the
discussion preceding Example 3.6.

###  2.5 Hyperkähler and Quaternion-Kähler Manifolds

In this section, we quickly review the definitions of hyperkähler structures,
hyperkähler metrics, quaternion-Kähler structures, and quaternion-Kähler
metrics.

We begin by recalling the Lie groups $\mathrm{Sp}(n)$ and
$\mathrm{Sp}(n)\mathrm{Sp}(1)$. Identifying $\mathbb{R}^{4n}$ with
$\mathbb{H}^{n},$ multiplication by quaternions $i,$ $j,$ and $k$ gives rise
to three complex structures $I_{0},$ $J_{0}$ and $K_{0}$ on $\mathbb{R}^{4n}$
satisfying the quaternion relation $I_{0}J_{0}=K_{0}.$ Lowering an index using
the Euclidean metric $g$ gives three symplectic forms $\omega_{I_{0}},$
$\omega_{J_{0}},$ and $\omega_{K_{0}}$ on $\mathbb{R}^{4n}$. The Lie group
$\mathrm{Sp}(n)$ may be defined as the simultaneous
$\mathrm{GL}_{4n}(\mathbb{R})$-stabilizer of these three 2-forms and the group
$\mathrm{Sp}(n)\mathrm{Sp}(1)$ may be defined as the
$\mathrm{GL}_{4n}(\mathbb{R})$-stabilizer of the 4-form

$\Theta_{0}=\omega_{I_{0}}^{2}+\omega_{J_{0}}^{2}+\omega_{K_{0}}^{2}.$

One can check that both $\mathrm{Sp}(n)$ and $\mathrm{Sp}(n)\mathrm{Sp}(1)$
fix a positive definite inner product on $\mathbb{R}^{4n}$, so we have
$\mathrm{Sp}(n)<\mathrm{Sp}(n)\mathrm{Sp}(1)<\mathrm{SO}(4n)$. Explicitly, for
$n=2$ in an appropriate basis $(e^{0},\ldots,e^{7})$ of $\mathbb{R}^{8*}$ we
have

$\displaystyle\omega_{I_{0}}$ $\displaystyle=e^{01}+e^{23}-e^{45}-e^{67},$
$\displaystyle\omega_{J_{0}}$ $\displaystyle=e^{02}-e^{13}-e^{46}+e^{57},$
$\displaystyle\omega_{K_{0}}$ $\displaystyle=e^{03}+e^{12}+e^{47}+e^{56}.$

While this basis of $\mathbb{R}^{8\ast}$ is non-standard, it has the advantage
of aligning with common conventions in $\mathrm{G}_{2}$ geometry.

We now specialize to real dimension $8$. Let $Q$ be an $8$-manifold. An
_$\mathrm{Sp}(2)$ -structure_ on $Q$ consists of data
$(g,(I_{1},I_{2},I_{3}),(\omega_{1},\omega_{2},\omega_{3}))$, where $g$ is a
Riemannian metric, $(I_{1},I_{2},I_{3})$ are $g$-orthogonal almost-complex
structures satisfying the quaternionic relations $I_{1}I_{2}=I_{3}$, etc., and
where $\omega_{p}:=g(I_{p}\cdot,\cdot)$ are the corresponding Kähler forms. A
_hyperkähler structure_ is an $\mathrm{Sp}(2)$-structure for which each
$I_{p}$ is integrable and each $\omega_{p}$ is closed. Equivalently, it is a
torsion-free $\mathrm{Sp}(2)$-structure. A _hyperkähler metric_ on $Q$ is a
Riemanninan metric $g$ for which $\mathrm{Hol}(g)\leq\mathrm{Sp}(2)$. Such
metrics are Ricci-flat. If $(g,(I_{p}),(\omega_{p}))$ is a hyperkähler
structure, then $g$ is a hyperkähler metric. Conversely, if $g$ is a
hyperkähler metric, then $g$ arises from a hyperkähler structure.

An _$\mathrm{Sp}(2)\mathrm{Sp}(1)$ -structure_ on $Q$ is a pair
$(g,\mathsf{W})$ consisting of a Riemannian metric $g$ and a rank 3 subbundle
$\mathsf{W}\subset\mathrm{End}(TQ)$ such that locally $\mathsf{W}$ admits a
basis $\\{I,J,K\\}$ consisting of $g$-orthogonal almost-complex structures
with $IJ=K$. Equivalently, it is a $4$-form $\Theta\in\Omega^{4}(Q)$ such that
at each $x\in Q$, there is a coframe $L\colon T_{x}Q\to\mathbb{R}^{8}$ for
which $\Theta|_{x}=L^{*}\Theta_{0}$. A _quaternion-Kähler structure_ is an
$\mathrm{Sp}(2)\mathrm{Sp}(1)$-structure for which the $4$-form
$\Theta\in\Omega^{4}(Q)$ is $g$-parallel. A _quaternion-Kähler metric_ on $Q$
is a Riemanninan metric $g$ for which
$\mathrm{Hol}(g)\leq\mathrm{Sp}(2)\mathrm{Sp}(1)$. Such metrics are Einstein.
If $(g,\mathsf{W})$ is a QK structure, then $g$ is a QK metric. Conversely, if
$g$ is a QK metric, then $g$ arises from a QK structure.

Finally, we explain the relationship between the above objects. First, note
that every $\mathrm{Sp}(2)$-structure $(g,(I_{p}),(\omega_{p}))$ induces an
$\mathrm{Sp}(2)\mathrm{Sp}(1)$-structure $(g,\mathsf{W})$ by setting
$\mathsf{W}:=\text{span}(I_{1},I_{2},I_{3})$, or equivalently, setting
$\Theta:=\omega_{1}^{2}+\omega_{2}^{2}+\omega_{3}^{2}$. Moreover, if the
$\mathrm{Sp}(2)$-structure is hyperkähler, then the induced
$\mathrm{Sp}(2)\mathrm{Sp}(1)$-structure is quaternion-Kähler and $g$ is
Ricci-flat. Conversely, if a quaternion-Kähler structure $(g,\mathsf{W})$ is
such that $g$ is Ricci-flat, then $\mathrm{Hol}^{0}(g)\leq\mathrm{Sp}(2)$, so
$(g,\mathsf{W})$ is locally induced from a hyperkähler structure.
In particular, suppose that $Q=(\text{C}(M),g_{\mathrm{C}})$ is an
$8$-dimensional cone for which $g_{\mathrm{C}}$ is a QK metric. Since an
Einstein cone metric is Ricci-flat, it follows that $g_{\mathrm{C}}$ is Ricci-
flat, and hence $g_{\mathrm{C}}$ is a hyperkähler metric. We will exploit this
observation in $\S$3.2.

##  3 Squashed $3$-Sasakian $7$-Manifolds

Let $M$ be a $7$-manifold with an $\mathrm{SO}(4)$-structure
$(\varphi,\mathsf{A})$. In $\S$3.1, we explain how $\varphi$ belongs to a
canonical $2$-parameter family of “squashed” $\mathrm{G}_{2}$-structures
$\varphi_{a,b}$ with $\varphi_{1,1}=\varphi$. Then, starting in $\S$3.2, we
restrict to the situation where $(\varphi,\mathsf{A})$ is quaternion-Sasakian
and study the squashings $\varphi_{a,b}$. In that case, we show that:

  1. (i)

$\mathsf{A}$ integrates to a foliation $\mathcal{F}_{A}$ whose leaves are
associatives with constant positive curvature;

  2. (ii)

$\varphi_{a,b}$ is co-closed;

  3. (iii)

$\varphi_{a,b}$ is nearly-parallel if and only if $b^{2}=5a^{2}$; and

  4. (iv)

$\varphi_{1,1}$ is isometric to a nearly-parallel $\mathrm{G}_{2}$-structure.

Note that both (i) and (iv) follow from the discussion in $\S$2.4, but we will
give a unified exposition of (i)–(iv) in this section. Moreover, we will see
that $\varphi_{t,1}$ induces the squashed metrics $g_{t}$ of Theorem 2.25, in
which $g_{1}$ is $3$-Sasakian and $g_{1/\sqrt{5}}$ is proper nearly-parallel.
For this reason, we refer to each $(M,\varphi_{a,b},\mathsf{A})$ as a
“squashed $3$-Sasakian $7$-manifold” or “squashed quaternion-Sasakian
$7$-manifold.”

To establish the above claims, in $\S$3.2 we derive the _structure equations_
of a squashed $3$-Sasakian $7$-manifold. These equations encode not only the
geometry of $M$, but also that of auxiliary spaces associated to $M$. For this
work, the most important such auxiliary space is the $8$-orbifold $Z\times
S^{2}$, where $Z$ is the twistor space of the quaternion-Kähler $4$-orbifold
$X:=M/\mathcal{F}_{A}$. In $\S$3.3, we show that $Z\times S^{2}$ parametrizes
the Hopf circles in $M$, a class of closed geodesics that we will use to
construct associative $3$-folds in $M$. We also show that $Z\times S^{2}$ has
a natural $2$-parameter family of almost-Hermitian structures
$(h_{a,b},J,\omega_{a,b})$; these will play a role in Theorem 1.1.

###  3.1 $\mathrm{SO}(4)$-Structures and their Squashings

Let $M$ be a $7$-manifold with an $\mathrm{SO}(4)$-structure
$(\varphi,\mathsf{A}$). As we now explain, $(\varphi,\mathsf{A})$ determines
an $\mathrm{SO}(4)$-subbundle of the coframe bundle of $M$. Indeed, by the
discussion in $\S$2.2, the cone $Q:=\text{C}(M)=\mathbb{R}^{+}\times M$ admits
a conical $\mathrm{Sp}(2)\mathrm{Sp}(1)$-structure $\Theta\in\Omega^{4}(Q)$
via

$\Theta=r^{3}dr\wedge\left(4\,\mathrm{vol}_{\mathsf{A}}-\varphi\right)+r^{4}\\!\ast\\!(4\,\mathrm{vol}_{\mathsf{A}}-\varphi).$
(3.1)

Since the Lie group $\mathrm{Sp}(2)\mathrm{Sp}(1)$ acts transitively on the
unit sphere $S^{7}\subset\mathbb{H}^{2}$, in an open neighborhood of any point
$x\in M$, there exists a coframe $(e^{1},\ldots,e^{7})$ of $M$ for which

$\displaystyle\mathsf{A}$ $\displaystyle=\text{span}(e_{1},e_{2},e_{3})$
$\displaystyle\mathsf{C}$ $\displaystyle=\text{span}(e_{4},e_{5},e_{6},e_{7})$
(3.2) $\displaystyle\left.\Theta\right|_{M}$
$\displaystyle=3e^{4567}-e^{12}\wedge(e^{45}+e^{67})-e^{23}\wedge(e^{46}-e^{57})-e^{31}\wedge(-e^{47}-e^{56}).$

Let $\mathcal{B}\subset FM$ denote the bundle of coframes for which (3.1)
holds. Since the $\mathrm{Sp}(2)\mathrm{Sp}(1)$-stabilizer of a point in
$S^{7}$ is $\mathrm{SO}(4)$, it follows that $\pi\colon\mathcal{B}\to M$ is an
$\mathrm{SO}(4)$-bundle. In terms of a local $\mathrm{SO}(4)$-coframe
$(e^{1},\ldots,e^{7})$, the $\mathrm{G}_{2}$-structure
$\varphi\in\Omega^{3}(M)$ is given by

$\displaystyle\varphi$
$\displaystyle=e^{123}+e^{1}\wedge(e^{45}+e^{67})+e^{2}\wedge(e^{46}-e^{57})+e^{3}\wedge(-e^{47}-e^{56}).$

We may also reconstruct the $4$-form on $Q=\mathbb{R}^{+}\times M$ in terms of
a local $\mathrm{SO}(4)$-coframe $(e^{1},\ldots,e^{7})$ via

$\Theta=r^{3}dr\wedge\left[3\,e^{123}-e^{1}\wedge(e^{45}+e^{67})-e^{2}\wedge(e^{46}-e^{57})-e^{3}\wedge(-e^{47}-e^{56})\right]+r^{4}\Theta|_{M}$

In computations, we will frequently work on the total space $\mathcal{B}$ or
$\mathcal{B}\times\mathbb{R}$ and omit pullbacks from the notation. For
example, let
$(\alpha,\beta)=(\alpha_{1},\alpha_{2},\alpha_{3},\beta_{1},\beta_{2},\beta_{3},\beta_{4})\in\Omega^{1}(\mathcal{B};\mathbb{R}^{7})$
denote the tautological $1$-form. Pulled back to
$\mathcal{B}\times\mathbb{R}$, the Riemannian metrics on $M$ and $Q$ are,
respectively,

$\displaystyle g$
$\displaystyle=\alpha_{1}^{2}+\alpha_{2}^{2}+\alpha_{3}^{2}+\beta_{1}^{2}+\beta_{2}^{2}+\beta_{3}^{2}+\beta_{4}^{2}$
$\displaystyle g_{Q}$
$\displaystyle=dr^{2}+r^{2}(\alpha_{1}^{2}+\alpha_{2}^{2}+\alpha_{3}^{2}+\beta_{1}^{2}+\beta_{2}^{2}+\beta_{3}^{2}+\beta_{4}^{2})$

Similarly, continuing to omit pullbacks from the notation, the conical
$\mathrm{Sp}(2)\mathrm{Sp}(1)$-structure $\Theta$ is

$\Theta=r^{3}\,dr\wedge\chi_{3}+r^{4}\chi_{4}$

where

$\displaystyle\chi_{3}$
$\displaystyle:=3\,\alpha_{123}-\alpha_{1}\wedge(\beta_{12}+\beta_{34})-\alpha_{2}\wedge(\beta_{13}-\beta_{24})-\alpha_{3}\wedge(-\beta_{14}-\beta_{23})$
$\displaystyle\chi_{4}$
$\displaystyle:=3\,\beta_{1234}-\alpha_{23}\wedge(\beta_{12}+\beta_{34})-\alpha_{31}\wedge(\beta_{13}-\beta_{14})-\alpha_{12}\wedge(-\beta_{14}-\beta_{23}).$

####  3.1.1 Squashed $\mathrm{G}_{2}$-Structures

On the total space $\mathcal{B}$, the set of $\mathrm{SO}(4)$-invariant
semibasic 3-forms is spanned by

$\displaystyle\Psi_{1}$ $\displaystyle=\alpha_{123},$ $\displaystyle\Psi_{2}$
$\displaystyle=\alpha_{1}\wedge\left(\beta_{12}+\beta_{34}\right)+\alpha_{2}\wedge\left(\beta_{13}-\beta_{24}\right)+\alpha_{3}\wedge\left(-\beta_{14}-\beta_{23}\right)\\!,$

while the set of $\mathrm{SO}(4)$-invariant semibasic 4-forms on $\mathcal{B}$
is spanned by

$\displaystyle\Gamma_{1}$ $\displaystyle=\beta_{1234},$
$\displaystyle\Gamma_{2}$
$\displaystyle=\alpha_{23}\wedge\left(\beta_{12}+\beta_{34}\right)+\alpha_{31}\wedge\left(\beta_{13}-\beta_{24}\right)+\alpha_{12}\wedge\left(-\beta_{14}-\beta_{23}\right)\\!.$

Let $a,b\in\mathbb{R}$ be non-zero constants and define a 3-form
$\varphi_{a,b}$ on $\mathcal{B}$ by

$\varphi_{a,b}=a^{3}\Psi_{1}+ab^{2}\Psi_{2}.$ (3.3)

The $3$-form $\varphi_{a,b}$ is $\mathrm{SO}(4)$-invariant and descends to $M$
to define a $\mathrm{G}_{2}$-structure (which will also be denoted by
$\varphi_{a,b}$). The resulting metric and 4-form will be denoted by $g_{a,b}$
and $\psi_{a,b}$ respectively. On $\mathcal{B}$,

$\displaystyle g_{a,b}$
$\displaystyle=a^{2}\left(\alpha_{1}^{2}+\alpha_{2}^{2}+\alpha_{3}^{3}\right)+b^{2}\left(\beta_{1}^{2}+\beta_{2}^{2}+\beta_{3}^{2}+\beta_{4}^{2}\right)\\!,$
$\displaystyle\psi_{a,b}$
$\displaystyle=b^{4}\Gamma_{1}+a^{2}b^{2}\Gamma_{2}.$

The $3$-forms $\varphi_{a,b}$ are the _squashed $\mathrm{G}_{2}$-structures_
associated to the $\mathrm{SO}(4)$-structure $(\varphi,\mathsf{A})$.

###  3.2 Quaternion-Sasakian $7$-Manifolds and their Squashings

Let $M$ be a $7$-manifold equipped with a quaternion-Sasakian structure
$(\varphi,\mathsf{A})$, and let $Q:=\text{C}(M)=\mathbb{R}^{+}\times M$ be the
metric cone of $M$. By definition, the conical
$\mathrm{Sp}(2)\mathrm{Sp}(1)$-structure $\Theta\in\Omega^{4}(Q)$ defined in
(3.1) is quaternion-Kähler. In this section, we equip the total space
$\mathcal{B}$ of the $\mathrm{SO}(4)$-coframe bundle of $M$ with a global
coframe and derive the corresponding structure equations — i.e., we will
express the exterior derivatives of the coframe in terms of the coframe
itself.

Let
$(\alpha,\beta)=(\alpha_{1},\alpha_{2},\alpha_{3},\beta_{1},\beta_{2},\beta_{3},\beta_{4})\in\Omega^{1}(\mathcal{B};\mathbb{R}^{7})$
denote the tautological $1$-form. This provides a basis for the semibasic (or
“horizontal”) $1$-forms on $\mathcal{B}$, and we aim to complete
$(\alpha,\beta)$ to a coframe of $\mathcal{B}$. For this, let
$\mathcal{F}_{M}$ be the $\mathrm{SO}(7)$-bundle of oriented $g$-orthonormal
coframes, and note that $\mathcal{B}\subset\mathcal{F}_{M}$. The Levi-Civita
form of $g$, denoted
$\psi_{M}\in\Omega^{1}(\mathcal{F}_{M};\mathfrak{so}(7))$, is a connection
form for $\mathcal{F}_{M}\to M$. Its restriction
$\left.\psi_{M}\right|_{\mathcal{B}}\in\Omega^{1}(\mathcal{B};\mathfrak{so}(7))$
is not a connection for $\mathcal{B}\to M$, but splits into connection terms
and semibasic terms. Indeed, noting the $\mathrm{SO}(4)$-irreducible
decomposition

$\mathfrak{so}(7)=\mathsf{V}_{0,2}\oplus\mathsf{V}_{1,1}\oplus\mathsf{V}_{1,3}\oplus\mathfrak{sp}(1)\oplus\mathfrak{sp}(1),$

where $\mathsf{V}_{0,2}$, $\mathsf{V}_{1,1}$, and $\mathsf{V}_{1,3}$ are
irreducible $\mathrm{SO}(4)$-modules of dimensions $3$, $4$, $8$,
respectively, the restriction
$\psi_{M}|_{\mathcal{B}}\in\Omega^{1}(\mathcal{B};\mathfrak{so}(7))$ splits
into blocks as

$\left.\psi_{M}\right|_{\mathcal{B}}=\left[\begin{array}[]{c|c}\left[\gamma+2\zeta\right]_{3}&\sigma+\rho\\\
\hline\cr-\sigma^{t}-\rho^{t}&\left[\zeta\right]^{+}+\left[\nu\right]^{-}\end{array}\right],$

where

$\displaystyle\left[\gamma+2\zeta\right]_{3}$
$\displaystyle=\begin{bmatrix}0&\gamma_{3}+2\zeta_{3}&-\gamma_{2}-2\zeta_{2}\\\
-\gamma_{3}-2\zeta_{3}&0&\gamma_{1}+2\zeta_{1}\\\
\gamma_{2}+2\zeta_{2}&-\gamma_{1}-2\zeta_{1}&0\end{bmatrix},$
$\displaystyle\sigma+\rho$
$\displaystyle=\left[\begin{array}[]{cccc}2\,\sigma_{{6}}-\rho_{{2}}&-2\,\sigma_{{5}}+\rho_{{1}}&2\,\sigma_{{8}}-\rho_{{4}}&-2\,\sigma_{{7}}+\rho_{{3}}\\\
-\sigma_{{1}}-\sigma_{{7}}-\rho_{{3}}&-\sigma_{{2}}+\sigma_{{8}}+\rho_{{4}}&-\sigma_{{3}}+\sigma_{{5}}+\rho_{{1}}&-\sigma_{{4}}-\sigma_{{6}}-\rho_{{2}}\\\
\sigma_{{2}}+\sigma_{{8}}+\rho_{{4}}&-\sigma_{{1}}+\sigma_{{7}}+\rho_{{3}}&\sigma_{{4}}-\sigma_{{6}}-\rho_{{2}}&-\sigma_{{3}}-\sigma_{{5}}-\rho_{{1}}\end{array}\right],$
$\displaystyle\left[\zeta\right]^{+}+\left[\nu\right]^{-}$
$\displaystyle=\begin{bmatrix}0&-\zeta_{1}-\nu_{1}&-\zeta_{2}+\nu_{2}&\zeta_{3}+\nu_{3}\\\
\zeta_{1}+\nu_{1}&0&\zeta_{3}-\nu_{3}&\zeta_{2}+\nu_{2}\\\
\zeta_{2}-\nu_{2}&-\zeta_{3}+\nu_{3}&0&\zeta_{1}+\nu_{1}\\\
-\zeta_{3}-\nu_{3}&\zeta_{2}-\nu_{2}&\zeta_{1}-\nu_{1}&0\end{bmatrix}.$

Here, $\zeta$ and $\nu$ are the components of the
$\mathfrak{so}(4)=\mathfrak{sp}(1)\oplus\mathfrak{sp}(1)$-valued _natural
connection_ on $\mathcal{B}$, whereas
$\gamma\in\Omega^{1}(\mathcal{B};\mathsf{V}_{0,2})$,
$\rho\in\Omega^{1}(\mathcal{B};\mathsf{V}_{1,1})$, and
$\sigma\in\Omega^{1}(\mathcal{B};\mathsf{V}_{1,3})$ are semi-basic $1$-forms.
The components of
$(\alpha,\beta,\zeta,\nu)\in\Omega^{1}(\mathcal{B};\mathbb{R}^{7}\oplus\mathfrak{so}(4))$
provide a coframe for $\mathcal{B}$.

Before computing the exterior derivatives $d\alpha$, $d\beta$, $d\zeta$,
$d\nu$, we consider the Levi-Civita connection
$\psi_{Q}\in\Omega^{1}(\mathcal{F}_{Q};\mathfrak{so}(8))$ of $g_{Q}$, where
$\mathcal{F}_{Q}\to Q$ is the $\mathrm{SO}(8)$-bundle of oriented
$g_{Q}$-orthonormal coframes. Restricting to the link $M\simeq\\{1\\}\times
M\subset Q$, the $1$-form $\psi_{Q}$ is given by

$\psi_{Q}=\left[\begin{array}[]{c|c|c}0&-\alpha&-\beta\\\
\hline\cr\alpha&\left[\gamma+2\zeta\right]_{3}&\sigma+\rho\\\
\hline\cr\beta&-\sigma^{t}-\rho^{t}&\left[\zeta\right]^{+}+\left[\nu\right]^{-}\end{array}\right]\\!.$

Since the $4$-form $\Theta\in\Omega^{4}(Q)$ is parallel with respect to the
Levi-Civita connection on $Q$, it follows that $\psi_{Q}$ must take values in
the Lie algebra $\mathfrak{sp}(2)\oplus\mathfrak{sp}(1)$. Therefore,
$\gamma=\alpha,$ $\rho=\beta$ and $\sigma=0$. We may now compute:

######  Proposition 3.1.

The _first structure equations_ are given by:

$\displaystyle d\begin{bmatrix}\alpha_{1}\\\ \alpha_{2}\\\
\alpha_{3}\end{bmatrix}$
$\displaystyle=-\begin{bmatrix}0&2\zeta_{3}&-2\zeta_{2}\\\
-2\zeta_{3}&0&2\zeta_{1}\\\
2\zeta_{2}&-2\zeta_{1}&0\end{bmatrix}\wedge\begin{bmatrix}\alpha_{1}\\\
\alpha_{2}\\\
\alpha_{3}\end{bmatrix}+2\begin{bmatrix}\alpha_{2}\wedge\alpha_{3}-\beta_{1}\wedge\beta_{2}-\beta_{3}\wedge\beta_{4}\\\
\alpha_{3}\wedge\alpha_{1}-\beta_{1}\wedge\beta_{3}+\beta_{2}\wedge\beta_{4}\\\
\alpha_{1}\wedge\alpha_{2}+\beta_{1}\wedge\beta_{4}+\beta_{2}\wedge\beta_{3}\end{bmatrix},$
(3.4a) $\displaystyle d\begin{bmatrix}\beta_{1}\\\ \beta_{2}\\\ \beta_{3}\\\
\beta_{4}\end{bmatrix}$
$\displaystyle=-\begin{bmatrix}0&-\zeta_{1}-\nu_{1}&-\zeta_{2}+\nu_{2}&\zeta_{3}+\nu_{3}\\\
\zeta_{1}+\nu_{1}&0&\zeta_{3}-\nu_{3}&\zeta_{2}+\nu_{2}\\\
\zeta_{2}-\nu_{2}&-\zeta_{3}+\nu_{3}&0&\zeta_{1}+\nu_{1}\\\
-\zeta_{3}-\nu_{3}&\zeta_{2}-\nu_{2}&\zeta_{1}-\nu_{1}&0\end{bmatrix}\wedge\begin{bmatrix}\beta_{1}\\\
\beta_{2}\\\ \beta_{3}\\\
\beta_{4}\end{bmatrix}+\begin{bmatrix}\alpha_{1}\wedge\beta_{2}+\alpha_{2}\wedge\beta_{3}-\alpha_{3}\wedge\beta_{4}\\\
-\alpha_{1}\wedge\beta_{2}-\alpha_{3}\wedge\beta_{3}-\alpha_{2}\wedge\beta_{4}\\\
\alpha_{2}\wedge\beta_{2}+\alpha_{3}\wedge\beta_{3}+\alpha_{1}\wedge\beta_{4}\\\
\alpha_{3}\wedge\beta_{2}+\alpha_{2}\wedge\beta_{3}-\alpha_{1}\wedge\beta_{4}\end{bmatrix}.$
(3.4b)

Further, there exist functions $W_{ij}\colon\mathcal{B}\to\mathbb{R}$ with
$W_{11}+W_{22}+W_{33}=0$ such that the _second structure equations_ hold:

$\displaystyle d\begin{bmatrix}\zeta_{1}\\\ \zeta_{2}\\\
\zeta_{3}\end{bmatrix}$
$\displaystyle=2\begin{bmatrix}\zeta_{2}\wedge\zeta_{3}\\\
\zeta_{3}\wedge\zeta_{1}\\\ \zeta_{1}\wedge\zeta_{2}\end{bmatrix},$ (3.4c)
$\displaystyle d\begin{bmatrix}\nu_{1}\\\ \nu_{2}\\\ \nu_{3}\end{bmatrix}$
$\displaystyle=2\begin{bmatrix}\nu_{2}\wedge\nu_{3}\\\ \nu_{3}\wedge\nu_{1}\\\
\nu_{1}\wedge\nu_{2}\end{bmatrix}+\begin{bmatrix}2+W_{11}&W_{12}&W_{13}\\\
W_{12}&2+W_{22}&W_{23}\\\
W_{13}&W_{23}&2+W_{33}\end{bmatrix}\begin{bmatrix}-\beta_{1}\wedge\beta_{2}+\beta_{3}\wedge\beta_{4}\\\
\beta_{1}\wedge\beta_{3}+\beta_{2}\wedge\beta_{4}\\\
\beta_{1}\wedge\beta_{4}-\beta_{2}\wedge\beta_{4}\end{bmatrix}.$ (3.4d)

###### Proof.

By the Fundamental Lemma of Riemannian Geometry, we have

$d\begin{bmatrix}\alpha\\\
\beta\end{bmatrix}=-\psi_{M}\wedge\begin{bmatrix}\alpha\\\
\beta\end{bmatrix}\\!.$

This implies the first structure equations. Taking exterior derivatives of the
first structure equations, noting that $d(d\alpha)=0$ and $d(d\beta)=0$ and
applying Cartan’s Lemma, yields the second structure equations. ∎

######  Remark 3.2.

The structure equations (3.4) characterize quaternion-Sasakian structures in
the following sense. If $P$ is a 13-manifold endowed with a coframe
$\left(\alpha,\beta,\zeta,\nu\right)$ and a function $W\colon
P\to\mathrm{Sym}^{2}_{0}\left(\mathbb{R}^{3}\right)$ satisfying (3.4), then
the equations $\alpha=\beta=0$ define an integrable plane field of codimension
7 on $P.$ Let $N$ be the leaf space of the resulting foliation. It is
straightforward to check that the pair $\left(\varphi,\mathsf{A}\right)$ given
by

$\displaystyle\varphi$
$\displaystyle=\alpha_{123}+\alpha_{1}\wedge(\beta_{12}+\beta_{34})+\alpha_{2}\wedge(\beta_{13}-\beta_{24})+\alpha_{3}\wedge(-\beta_{14}-\beta_{23})$
$\displaystyle\mathsf{A}$ $\displaystyle=\left\\{\beta=0\right\\}$

descend to $N$ to define a quaternion-Sasakian structure. The space $P$ may be
identified with an open subset of the $\mathrm{SO}(4)$-structure $\mathcal{B}$
associated to this quaternion-Sasakian structure.

The 1-form $\zeta\in\Omega^{1}(\mathcal{B};\mathfrak{sp}(1))$ defines a flat
connection on the subbundle $\mathsf{A}\subset TM$. The vector bundle
$\mathsf{A}$ is trivial: it is spanned by the three linearly independent non-
vanishing vector fields $I_{1}(\partial_{r})$, $I_{2}(\partial_{r})$,
$I_{3}(\partial_{r})$.

######  Remark 3.3.

Choosing a fixed $\zeta$-flat trivialization of $\mathsf{A}$ defines an
$\mathrm{Sp}(1)$-subbundle $\mathcal{B}^{\prime}\subset\mathcal{B}$ on which
$\zeta=0$. It is possible to define and work with 3-Sasakian structures using
the subbundle $\mathcal{B}^{\prime}$ alone. However, for our purposes it is
preferable to work directly with the $\mathrm{SO}(4)$-structure $\mathcal{B}$
because the $\mathrm{G}_{2}$-structures we define on $M$ will naturally be
invariant under diffeomorphisms of $M$ preserving $\mathcal{B}.$ The group of
diffeomorphisms of $M$ preserving $\mathcal{B}$ is strictly larger than the
group preserving $\mathcal{B}^{\prime}$ and our techniques will allow us to
determine when two associative submanifolds are equivalent up to the action of
the larger group, something that would not be possible if we worked only with
a fixed $\zeta$-flat trivialization of $\mathsf{A}$ and the resulting
$\mathrm{Sp}(1)$-structure $\mathcal{B}^{\prime}$.

####  3.2.1 The $4$-Dimensional Quotient

We now observe that the associative distribution $\mathsf{A}\subset TM$ is
integrable. Indeed, consider the distribution $\beta=0$ on the $13$-manifold
$\mathcal{B}$. The structure equations (3.4) show both that $\beta=0$ descends
to $M$, where it coincides with $\mathsf{A}$, and furthermore that
$\mathsf{A}$ is integrable. The resulting associative foliation of $M$ is the
canonical foliation $\mathcal{F}_{A}\subset M$ discussed in $\S$2.4.1.
Moreover, it follows from the $d\alpha$ and $d\zeta$ equations in (3.4) that
the induced metric on any canonical leaf $F\subset M$ has constant positive
curvature, which also accords with the remarks in $\S$2.4.1.

Let $X=M/\mathcal{F}_{A}$ denote the 4-dimensional leaf space of the
distribution $\mathsf{A}$ on $M$, and let $h\colon M\to X$ denote the quotient
map. With respect to the projection $h\circ\pi\colon\mathcal{B}\to X$, the
$1$-forms $\beta_{1},\beta_{2},\beta_{3},\beta_{4}$ provide a basis for the
semi-basic forms. It follows from the structure equations (3.4) that the
symmetric 2-tensor

$\beta_{1}^{2}+\beta_{2}^{2}+\beta_{3}^{2}+\beta_{4}^{2}$

on $\mathcal{B}$ descends to $X$ to define a Riemannian metric $g_{X}$ on $X$.
The equations for $d\nu,d\zeta$ and $d\alpha$ in (3.4) imply that $g_{X}$ is
self-dual Einstein with positive scalar curvature. The quaternion-Sasakian
$7$-manifold $M$ may be locally identified with the $\mathrm{SO}(3)$-frame
bundle of the rank 3 Riemannian vector bundle $\Lambda^{2}_{+}(T^{*}X)\to X$
of self-dual $2$-forms on $X$.

The construction of the preceding paragraph can be reversed. Let $(X,g_{X})$
be a Riemannian 4-orbifold, and set
$\mathcal{B}:=\mathcal{F}_{X}\times\mathrm{SO}(3)$, where $\mathcal{F}_{X}$ is
the orthonormal coframe bundle of $X$. There is an $\mathrm{SO}(4)$-action on
$\mathcal{B}$ defined as follows. Letting
$\rho\colon\mathrm{SO}(4)\to\mathrm{SO}(3)$ denote the
$\mathrm{SO}(4)$-representation on
$\Lambda^{2}_{+}\mathbb{R}^{4}\simeq\mathbb{R}^{3}$, an element
$k\in\mathrm{SO}(4)$ acts on $\mathcal{F}_{X}\times\mathrm{SO}(3)$ by the
standard action on the first factor and via left-multiplication by $\rho(k)$
on the second factor. The quotient $M:=\mathcal{B}/\mathrm{SO}(4)$ is exactly
the $\mathrm{SO}(3)$-frame bundle of the rank 3 bundle
$\Lambda^{2}_{+}(T^{*}X)\to X$.
If $g_{X}$ is a self-dual positive Einstein metric, then we can endow $M$ with
a quaternion-Sasakian $\mathrm{SO}(4)$-structure in the following way. The
space $\mathcal{B}$ carries the following vector-valued 1-forms
$\alpha,\beta,\zeta$, and $\nu$:

  * •

$\beta$ is the pullback of tautological 1-form on $\mathcal{F}_{X}$

  * •

$\zeta$ is the pullback of the left-invariant Maurer-Cartan form on
$\mathrm{SO}(3)$, and

  * •

$\alpha=\mu_{2}-\zeta$, $\nu=\mu_{1}$, where $\mu_{1},\mu_{2}$ denote the two
components of the $(\mathfrak{sp}(1)\oplus\mathfrak{sp}(1))$-valued Levi-
Civita form of $g_{X}$.

Note that $\alpha$ and $\beta$ are semibasic for the projection
$\mathcal{B}\to M$. Moreover, these forms satisfy the hypotheses and
conclusion of Proposition 3.1, and therefore by the discussion in Remark 3.2
endow $M$ with a quaternion-Sasakian $\mathrm{SO}(4)$-structure.

####  3.2.2 Squashed $3$-Sasakian $7$-Manifolds

We continue to let $M$ be a $7$-manifold with a quaternion-Sasakian structure
$(\varphi,\mathsf{A})$. Let $\varphi_{a,b}$ denote the associated squashed
$\mathrm{G}_{2}$-structures (3.3) with corresponding $4$-forms $\psi_{a,b}$
and induced metrics $g_{a,b}$. Using the structure equations (3.4), we may
compute

$\displaystyle d\,\varphi_{a,b}$
$\displaystyle=-2\frac{a^{2}+b^{2}}{ab^{2}}\psi_{a,b}-2\frac{b^{2}\left(5a^{2}-b^{2}\right)}{a}\Gamma_{1},$
$\displaystyle d\,\psi_{a,b}$ $\displaystyle=0.$

Therefore, each $\mathrm{G}_{2}$-structure $\varphi_{a,b}$ is co-closed.
Moreover, $\varphi_{a,b}$ is nearly-parallel if and only if $b^{2}=5a^{2}$.

######  Definition 3.4.

Let $M$ be a $7$-manifold with a quaternion-Sasakian
$\mathrm{SO}(4)$-structure $(\varphi,\mathsf{A})$. We refer to any member of
the $2$-parameter family $(M,\varphi_{a,b},\mathsf{A})$ as a _squashed
quaternion-Sasakian $7$-manifold_, or by abuse of language, as a _squashed
$3$-Sasakian $7$-manifold_.

######  Remark 3.5.

Renaming $a=t$ and setting $b=1$, we obtain a $1$-parameter family of
$\mathrm{G}_{2}$-structures $\varphi_{t}:=\varphi_{t,1}$ that induce the
squashed metrics $g_{t}=g_{t,1}$ of Theorem 2.25. Note that $\varphi_{t}$ is
nearly-parallel if and only if $t=\frac{1}{\sqrt{5}}$.

$\ b=a$$b=\sqrt{5}a$$\varphi_{1,1}$ Figure 2: The family of co-closed
$\mathrm{G}_{2}$-structures $\varphi_{a,b}$ for $a,b>0$.

Now, since the $\mathrm{SO}(4)$-structure $(\varphi,\mathsf{A})$ is
quaternion-Sasakian, Proposition 2.4(a) implies that the co-closed
$\mathrm{G}_{2}$-structure $\varphi=\varphi_{1,1}$ is _isometric_ to some
nearly-parallel $\mathrm{G}_{2}$-structure $\widehat{\varphi}$. We can now
explicitly exhibit such a $\mathrm{G}_{2}$-structure. Indeed, by choosing a
trivialization of $\mathsf{A}$ as in Remark 3.3, the $3$-form
$\widehat{\varphi}\in\Omega^{3}(\mathcal{B}^{\prime})$ given by

$\widehat{\varphi}:=\alpha_{123}-\alpha_{1}\wedge\left(\beta_{12}+\beta_{34}\right)+\alpha_{2}\wedge\left(\beta_{13}-\beta_{24}\right)+\alpha_{3}\wedge\left(-\beta_{14}-\beta_{23}\right),$

which differs from $\varphi_{1,1}$ by a sign, descends to a nearly-parallel
$\mathrm{G}_{2}$-structure on $M$ that induces $g=g_{1,1}$. This subtlety is
related to the fact that $\mathrm{Sp}(2)\mathrm{Sp}(1)$ is not a subgroup of
$\mathrm{Spin}(7)$ (whereas $\mathrm{Sp}(2)$ is a subgroup of both). The
following example may help clarify the situation.

######  Example 3.6.

Let $M=S^{7}$ carry its standard $\mathrm{Sp}(2)$-invariant $3$-Sasakian
structure. There is a unique $\mathrm{Spin}(7)$-invariant
$\mathrm{G}_{2}$-structure $\widehat{\varphi}\in\Omega^{3}(S^{7})$ that
induces the round metric of constant curvature $1$. This
$\mathrm{G}_{2}$-structure is nearly-parallel and is called the _standard
$\mathrm{G}_{2}$-structure_ on $S^{7}$. Note that $\widehat{\varphi}$ does
_not_ belong to the family of co-closed,
$\mathrm{Sp}(2)\mathrm{Sp}(1)$-invariant $\mathrm{G}_{2}$-structures
$\varphi_{a,b}$, but _is_ isometric to $\varphi_{1,1}$.

Let $\mu,\theta$ denote rescaled forms on $\mathcal{B}$ adapted to the
$\mathrm{G}_{2}$-structure $\varphi_{a,b}$:

$\mu=a\alpha,\>\>\>\>\theta=b\beta.$

In terms of the rescaled forms, we have

$\displaystyle\varphi_{a,b}$
$\displaystyle=\mu_{123}+\mu_{1}\wedge\left(\theta_{12}+\theta_{34}\right)+\mu_{2}\wedge\left(\theta_{13}-\theta_{24}\right)+\mu_{3}\wedge\left(-\theta_{14}-\theta_{23}\right),$
$\displaystyle\psi_{a,b}$
$\displaystyle=\theta_{1234}+\mu_{23}\wedge\left(\theta_{12}+\theta_{34}\right)+\mu_{31}\wedge\left(\theta_{13}-\theta_{24}\right)+\mu_{12}\wedge\left(-\theta_{14}-\theta_{23}\right),$
$\displaystyle g_{a,b}$
$\displaystyle=\mu_{1}^{2}+\mu_{2}^{2}+\mu_{3}^{3}+\theta_{1}^{2}+\theta_{2}^{2}+\theta_{3}^{2}+\theta_{4}^{2}.$

The structure equations (3.4) become

$\displaystyle d\begin{bmatrix}\mu_{1}\\\ \mu_{2}\\\ \mu_{3}\end{bmatrix}$
$\displaystyle=-\begin{bmatrix}0&2\zeta_{3}&-2\zeta_{2}\\\
-2\zeta_{3}&0&2\zeta_{1}\\\
2\zeta_{2}&-2\zeta_{1}&0\end{bmatrix}\wedge\begin{bmatrix}\mu_{1}\\\
\mu_{2}\\\
\mu_{3}\end{bmatrix}+\frac{2}{a}\begin{bmatrix}\mu_{2}\wedge\mu_{3}\\\
\mu_{3}\wedge\mu_{1}\\\
\mu_{1}\wedge\mu_{2}\end{bmatrix}+\frac{2}{b^{2}}\begin{bmatrix}-\theta_{1}\wedge\theta_{2}-\theta_{3}\wedge\theta_{4}\\\
-\theta_{1}\wedge\theta_{3}+\theta_{2}\wedge\theta_{4}\\\
\theta_{1}\wedge\theta_{4}+\theta_{2}\wedge\theta_{3}\end{bmatrix},$ (3.5a)
$\displaystyle d\begin{bmatrix}\theta_{1}\\\ \theta_{2}\\\ \theta_{3}\\\
\theta_{4}\end{bmatrix}$
$\displaystyle=-\begin{bmatrix}0&-\zeta_{1}-\nu_{1}&-\zeta_{2}+\nu_{2}&\zeta_{3}+\nu_{3}\\\
\zeta_{1}+\nu_{1}&0&\zeta_{3}-\nu_{3}&\zeta_{2}+\nu_{2}\\\
\zeta_{2}-\nu_{2}&-\zeta_{3}+\nu_{3}&0&\zeta_{1}+\nu_{1}\\\
-\zeta_{3}-\nu_{3}&\zeta_{2}-\nu_{2}&\zeta_{1}-\nu_{1}&0\end{bmatrix}\wedge\begin{bmatrix}\theta_{1}\\\
\theta_{2}\\\ \theta_{3}\\\
\theta_{4}\end{bmatrix}+\frac{1}{a}\begin{bmatrix}\mu_{1}\wedge\theta_{2}+\mu_{2}\wedge\theta_{3}-\mu_{3}\wedge\theta_{4}\\\
-\mu_{1}\wedge\theta_{2}-\mu_{3}\wedge\theta_{3}-\mu_{2}\wedge\theta_{4}\\\
\mu_{2}\wedge\theta_{2}+\mu_{3}\wedge\theta_{3}+\mu_{1}\wedge\theta_{4}\\\
\mu_{3}\wedge\theta_{2}+\mu_{2}\wedge\theta_{3}-\mu_{1}\wedge\theta_{4}\end{bmatrix},$
(3.5b) and  $\displaystyle d\begin{bmatrix}\zeta_{1}\\\ \zeta_{2}\\\
\zeta_{3}\end{bmatrix}$
$\displaystyle=2\begin{bmatrix}\zeta_{2}\wedge\zeta_{3}\\\
\zeta_{3}\wedge\zeta_{1}\\\ \zeta_{1}\wedge\zeta_{2}\end{bmatrix},$ (3.5c)
$\displaystyle d\begin{bmatrix}\nu_{1}\\\ \nu_{2}\\\ \nu_{3}\end{bmatrix}$
$\displaystyle=2\begin{bmatrix}\nu_{2}\wedge\nu_{3}\\\ \nu_{3}\wedge\nu_{1}\\\
\nu_{1}\wedge\nu_{2}\end{bmatrix}+\frac{1}{b^{2}}\begin{bmatrix}2+W_{11}&W_{12}&W_{13}\\\
W_{12}&2+W_{22}&W_{23}\\\
W_{13}&W_{23}&2+W_{33}\end{bmatrix}\begin{bmatrix}-\theta_{1}\wedge\theta_{2}+\theta_{3}\wedge\theta_{4}\\\
\theta_{1}\wedge\theta_{3}+\theta_{2}\wedge\theta_{4}\\\
\theta_{1}\wedge\theta_{4}-\theta_{2}\wedge\theta_{4}\end{bmatrix}.$ (3.5d)

###  3.3 The Space of Hopf Circles

Let $M$ be a compact $7$-manifold. In $\S$2.4.1, we equipped $M$ with a
$3$-Sasakian structure $(\langle\cdot,\cdot\rangle$,
$(\alpha_{1},\alpha_{2},\alpha_{3})$,
$(\mathsf{J}_{1},\mathsf{J}_{2},\mathsf{J}_{3}))$ and defined a “Hopf circle”
to be a leaf of the quasi-regular $1$-dimensional foliation
$\mathcal{F}_{w}\subset\mathcal{F}_{A}\subset M$, for any $w\in S^{2}$. We
remarked that each projection $p_{w}\colon M\to M/\mathcal{F}_{w}$ is a
principal $S^{1}$-orbibundle, and each leaf space $M/\mathcal{F}_{w}$ may be
identified with the (orbifold) twistor space $Z$ of $X$.
In this section, we equip $M$ with less data, namely that of a squashed
quaternion-Sasakian structure $(\varphi_{a,b},\mathsf{A})$. In $\S$3.3.1, we
give two alternate characterizations of Hopf circles that are better suited to
$\mathrm{SO}(4)$-geometry. We also show that the space of Hopf circles may be
identified with the $8$-orbifold $Z\times S^{2}$. Then, in $\S$3.3.2, we endow
$Z\times S^{2}$ with a natural almost-Hermitian structure
$(h_{a,b},J,\omega_{a,b})$, and in $\S$3.3.3 we relate
$(h_{1,1},J,\omega_{1,1})$ to the strict nearly-Kähler structure on $Z$.

####  3.3.1 Hopf Circles

Let $M$ be a compact $7$-manifold equipped with a squashed quaternion-Sasakian
structure $(\varphi_{a,b},\mathsf{A})$. Recall that the distribution
$\mathsf{A}$ is integrable, yielding a canonical foliation $\mathcal{F}_{A}$
whose leaves, called _canonical leaves_ , are totally-geodesic associative
$3$-folds $F\subset M$ of constant positive sectional curvature.

######  Definition 3.7.

A _Hopf circle_ in $(M,\varphi_{a,b},\mathsf{A})$ is a closed geodesic that
lies in a canonical leaf.

######  Proposition 3.8.

Let $M$ be a compact squashed quaternion-Sasakian 7-manifold. The space of
Hopf circles in $M$ is diffeomorphic to $Z\times S^{2}$, where $Z$ is the
twistor space of the 4-orbifold $X=M/\mathcal{F}_{A}.$

###### Proof.

A Hopf circle is a closed geodesic in $M$ lying entirely in a fixed canonical
leaf. Given a point $m\in M$ and a unit vector $w\in\mathsf{A}_{m}$, there is
a unique Hopf circle passing through $m$ with tangent vector $w$. Working in a
fixed $\zeta$-flat trivialization of $\mathsf{A}$, this geodesic is a fiber of
the projection $p_{w}\colon M\to Z_{w}$ of Theorem 2.20. Now, define an
equivalence relation $\sim$ on $M\times S^{2}$ by declaring that
$(m_{1},w_{1})\sim(m_{2},w_{2})$ if and only if $w_{1}=w_{2}$ in this
trivialization and $m_{1}$ and $m_{2}$ lie in the same fiber of
$p_{w_{1}}\colon M\to Z_{w_{1}}$. Then $(m_{1},w_{1})$ and $(m_{2},w_{2})$
define the same Hopf circle if and only if $(m_{1},w_{1})\sim(m_{2},w_{2})$.
Consequently, the space of Hopf circles is diffeomorphic to $(M\times
S^{2})/\\!\sim\ =Z\times S^{2}$. ∎

We now study Hopf circles in $M$ in terms of $\mathrm{SO}(4)$-frames. For
this, we continue with the setup of the previous section. That is, we let
$\pi\colon\mathcal{B}\to M$ be the $\mathrm{SO}(4)$-coframe bundle, let
$(\mu,\theta)\in\Omega^{1}(\mathcal{B};\mathbb{R}^{3}\oplus\mathbb{R}^{4})$
denote the squashed tautological $1$-form, and let
$(\zeta,\nu)\in\Omega^{1}(\mathcal{B};\mathfrak{sp}(1)\oplus\mathfrak{sp}(1))$
denote the natural connection. Thus, $(\mu,\theta,\zeta,\nu)$ is a global
coframe of the $13$-manifold $\mathcal{B}$. We now consider the following
differential ideal $\mathcal{I}$ on $\mathcal{B}$ consisting of $8$ linearly
independent $1$-forms

$\mathcal{I}:=\langle\theta_{1},\theta_{2},\theta_{3},\theta_{4},\mu_{2},\mu_{3},\zeta_{2},\zeta_{3}\rangle.$

By the structure equations (3.4), the ideal $\mathcal{I}$ is Frobenius.
Consequently, $\mathcal{B}$ is foliated by the $5$-dimensional integral
submanifolds of $\mathcal{I}$. Moreover, any such maximal integral $5$-fold in
$\mathcal{B}$ projects via $\pi\colon\mathcal{B}\to M$ to a closed geodesic in
$M$ that lies in a canonical leaf $F\subset M$ of the associative distribution
$\mathsf{A}$.
Conversely, if $\gamma\colon\left[a,b\right]\to M$ is a closed geodesic in $M$
lying entirely in a single canonical leaf $F,$ then define the subset
$\mathcal{Q}(\gamma)\subset\gamma^{*}\mathcal{B}$ consisting of
$\mathrm{SO}(4)$-coframes $T_{m}M\to\mathbb{R}^{7}$ sending the unit tangent
vector of $\gamma$ to $e_{1}$, a $\mathrm{U}(2)$-subbundle of
$\gamma^{*}\mathcal{B}$. We have $\theta_{1}=\ldots=\theta_{4}=0$ and
$\mu_{2}=\mu_{3}=0$ on this subset by virtue of the frame adaptation, and the
geodesic condition implies $\zeta_{2}=\zeta_{3}=0$ on $\mathcal{Q}(\gamma)$.
Therefore, $\mathcal{Q}(\gamma)$ is a maximal 5-dimensional integral
submanifold of $\mathcal{I}$. In summary, we have proved:

######  Proposition 3.9.

Let $M$ be a compact squashed quaternion-Sasakian 7-manifold. A closed curve
in $M$ is a Hopf circle if and only if it is the image under
$\pi\colon\mathcal{B}\to M$ of a maximal integral $5$-fold of $\mathcal{I}$.

On the $13$-manifold $\mathcal{B}$, the vector fields dual to
$\mu_{1},\zeta_{1},\nu_{1},\nu_{2},\nu_{3}$ generate a locally free
${\mathrm{U}(1)\times\mathrm{U}(2)}$-action, thereby yielding a principal
$(\mathrm{U}(1)\times\mathrm{U}(2))$-orbibundle
$\lambda\colon\mathcal{B}\to\mathcal{B}/(\mathrm{U}(1)\times\mathrm{U}(2))$.
The maximal 5-dimensional integral submanifolds of the ideal $\mathcal{I}$ are
easily seen to be the orbits of ${\mathrm{U}(1)\times\mathrm{U}(2)}$ on
$\mathcal{B}$. Consequently, Propositions 3.8 and 3.9 imply that
$\mathcal{B}/(\mathrm{U}(1)\times\mathrm{U}(2))$ can be identified with the
space of Hopf circles, $Z\times S^{2}$.

In summary, $\mathcal{B}$ admits the structure of a double fibration via the
principal $\mathrm{SO}(4)$-bundle $\pi\colon\mathcal{B}\to M$ and the
principal $(\mathrm{U}(1)\times\mathrm{U}(2))$-orbibundle
$\lambda\colon\mathcal{B}\to Z\times S^{2}$. Moreover, the eight $1$-forms
$\beta_{1},\beta_{2},\beta_{3},\beta_{4}$ and
$\alpha_{2},\alpha_{3},\zeta_{2},\zeta_{3}$ are $\lambda$-semibasic, whereas
$\alpha_{1},\zeta_{1},\nu_{1},\nu_{2},\nu_{3}$ are connection $1$-forms for
$\lambda$. Altogether, we have a diagram

${\mathcal{B}}$${M}$${Z\times
S^{2}}$${X}$$\scriptstyle{\pi}$$\scriptstyle{\lambda}$$\scriptstyle{h}$$\scriptstyle{q}$

where $h\colon M\to X$ is the canonical fibration, and $q\colon Z\times
S^{2}\to X$ is the obvious $(S^{2}\times S^{2})$-bundle.

####  3.3.2 Almost-Complex Structure on $Z\times S^{2}$

We now describe how the space $Z\times
S^{2}=\mathcal{B}/(\mathrm{U}(1)\times\mathrm{U}(2))$ may be equipped with a
naturally defined 2-parameter family of almost-Hermitian structures
$(h_{a,b},J,\omega_{a,b})$. Define $\mathbb{C}$-valued 1-forms $\pi_{i}$ on
$\mathcal{B}$ by

$\displaystyle\pi_{1}$ $\displaystyle=\mu_{2}+i\mu_{3},$
$\displaystyle\pi_{2}$
$\displaystyle=\sqrt{2}\left(\theta_{1}+i\theta_{2}\right),$
$\displaystyle\pi_{3}$
$\displaystyle=\sqrt{2}\left(\theta_{3}+i\theta_{4}\right),$
$\displaystyle\pi_{4}$ $\displaystyle=\zeta_{2}+i\zeta_{3}.$

The space of $(1,0)$-forms for the almost-complex structure $J$ consists of
precisely the $\mathbb{C}$-valued 1-forms on $Z\times S^{2}$ whose pullbacks
to $\mathcal{B}$ lie in
$\mathrm{span}_{\mathbb{C}}(\pi_{1},\pi_{2},\pi_{3},\pi_{4}).$ This property
characterizes $J.$ Note that $J$ does not depend on $a$ or $b.$

The Riemannian metric $h_{a,b}$ on $Z\times S^{2}$ is defined by declaring
$(\pi_{1},\pi_{2},\pi_{3},\pi_{4})$ to be a $\mathrm{U}(4)$-frame, i.e.:

$\displaystyle h_{a,b}$
$\displaystyle=\pi_{1}\circ\overline{\pi}_{1}+\pi_{2}\circ\overline{\pi}_{2}+\pi_{3}\circ\overline{\pi}_{3}+\pi_{4}\circ\overline{\pi}_{4}$
$\displaystyle=\mu_{2}^{2}+\mu_{3}^{2}+\zeta_{2}^{2}+\zeta_{3}^{2}+2(\theta_{1}^{2}+\theta_{2}^{2}+\theta_{3}^{2}+\theta_{4}^{2}).$

The non-degenerate $2$-form $\omega_{a,b}$ making $(h_{a,b},J,\omega_{a,b})$
an almost-Hermitian triple is then given by

$\displaystyle\omega$
$\displaystyle=\textstyle\frac{i}{2}\left(\pi_{1}\wedge\overline{\pi}_{1}+\pi_{2}\wedge\overline{\pi}_{2}+\pi_{3}\wedge\overline{\pi}_{3}+\pi_{4}\wedge\overline{\pi}_{4}\right)$
$\displaystyle=\mu_{2}\wedge\mu_{3}+\zeta_{2}\wedge\zeta_{3}+2(\theta_{1}\wedge\theta_{2}+\theta_{3}\wedge\theta_{4}).$

The structure equations (3.4) imply

$\displaystyle d\pi_{1}$
$\displaystyle\equiv-\tfrac{a}{b^{2}}\,\overline{\pi_{2}\wedge\pi_{3}}$
$\displaystyle d\pi_{2}$
$\displaystyle\equiv-\tfrac{1}{a}\,\overline{\pi_{3}\wedge\pi_{1}}-\overline{\pi_{3}\wedge\pi_{4}},$
$\displaystyle d\pi_{4}$ $\displaystyle\equiv 0$ $\displaystyle d\pi_{3}$
$\displaystyle\equiv-\tfrac{1}{a}\,\overline{\pi_{1}\wedge\pi_{2}}+\overline{\pi_{2}\wedge\pi_{4}},$

modulo $(\pi_{1},\pi_{2},\pi_{3},\pi_{4}).$ Therefore, the almost-complex
structure $J$ is not integrable.

####  3.3.3 Strict Nearly-Kähler Structure on $Z\times\\{w\\}$

For every $w\in S^{2},$ the 6-fold $Z_{w}=Z\times\\{w\\}$ is an almost-complex
sub-orbifold of $Z\times S^{2}.$ The almost-Hermitian structure
$(h_{1,1},J,\omega_{1,1})$ on $Z\times S^{2}$ restricts to $Z_{w}$ to define a
strict nearly-Kähler structure: setting $a=b=1$ and restricting to $Z_{w},$ we
have $\pi_{4}=\overline{\pi_{4}}=0$ and

$\displaystyle d\omega$ $\displaystyle=-3\,\mathrm{Im}\Upsilon,$
$\displaystyle d\,\mathrm{Re}\Upsilon$ $\displaystyle=-2\omega^{2},$

where $\Upsilon=\pi_{1}\wedge\pi_{2}\wedge\pi_{3}$. The nearly Kähler
structure on $Z_{w}$ is independent of $w\in S^{2}$ and is equivalent to the
standard nearly Kähler structure on $Z$ viewed as the twistor space of the
positive self-dual Einstein 4-orbifold $X$.

##  4 Associatives in Squashed $3$-Sasakian $7$-Manifolds

We now begin our study of associative $3$-folds in squashed $3$-Sasakian
$7$-manifolds $(M,\varphi_{a,b},\mathsf{A})$. In Proposition 4.2, we
characterize the striped associatives in $M$ — recall from Definition 2.7 that
“striped” is a condition on the Gauss map — as those that are locally “Hopf-
ruled.” Precisely:

######  Definition 4.1.

A $3$-dimensional submanifold $N\subset M$, not necessarily associative, is
_Hopf-ruled_ if there exists a smooth fibration $N\to S$ to a surface $S$
whose fibers are Hopf circles in $M$.

In $\S$4.3, we set up a formalism for Hopf ruled 3-folds (not necessarily
associative) in $M$ that essentially identifies them with surfaces in $Z\times
S^{2}$, the space of Hopf circles. We then ask which surfaces in $Z\times
S^{2}$ correspond to the _associative_ Hopf-ruled $3$-folds in $M$. In
$\S$4.4, we answer this question by establishing Theorem 1.1: the
$J$-holomorphic curves in $(Z\times S^{2},h_{a,b},J,\omega_{a,b})$ correspond
to Hopf-ruled associative $3$-folds in $(M,\varphi_{a,b})$.

###  4.1 Adapted Frames for Associatives

Let $(M,\varphi_{a,b},\mathsf{A})$ be a squashed $3$-Sasakian $7$-manifold. In
this section, we study associative $3$-folds in $M$ with respect to the
$\mathrm{SO}(4)$-structure $(\varphi_{a,b},\mathsf{A})$ by the method of
moving frames. We begin by adapting $\mathrm{SO}(4)$-frames to the
associative.
Let $\mathcal{B}\to M$ be the bundle of $\mathrm{SO}(4)$-coframes as in
$\S$3.1, and let $u\colon N\to M$ be an associative $3$-fold. At each point
$p\in N$, the tangent space $T_{p}N$ is an element of the associative
Grassmannian
$\mathrm{Gr}_{\text{ass}}(T_{p}M,\varphi_{a,b})\cong\mathrm{G}_{2}/\mathrm{SO}(4)$.
By Proposition 2.5, there exists a coframe
$\left(\mu_{1},\mu_{2},\mu_{3},\theta_{1},\theta_{2},\theta_{3},\theta_{4}\right)\in\mathcal{B}_{p}$
and numbers $r\in[0,\pi/2]$ and $s\in[0,\pi/6]$ such that $T_{p}N$ is defined
by the equations

$\displaystyle-\sin(2s)\mu_{1}$ $\displaystyle+\cos(2s)\theta_{3}=0,$ (4.1)
$\displaystyle-\sin(s+r)\mu_{2}$ $\displaystyle+\cos(s+r)\theta_{2}=0,$
$\displaystyle-\sin(s-r)\mu_{3}$ $\displaystyle+\cos(s-r)\theta_{1}=0,$
$\displaystyle\theta_{4}$ $\displaystyle=0.$

As $p\in N$ varies, we obtain a pair of functions $r\colon N\to[0,\pi/2]$ and
$s\colon N\to[0,\pi/6]$ together with a subset $\mathcal{B}_{1}$ of the pulled
back bundle $u^{*}(\mathcal{B})$ consisting of all coframes satisfying (4.1).

If we assume that the tangent spaces of the associative have uniform
$\mathrm{SO}(4)$-stabilizer $\mathrm{G}$, then $\mathcal{B}_{1}$ defines a
principal $\mathrm{G}$-bundle over $N$ and we have a diagram

${{\mathrm{G}}}$${{\mathrm{SO}(4)}}$${{\mathrm{SO}(4)}}$${{\mathcal{B}_{1}}}$${{u^{*}(\mathcal{B})}}$${{\mathcal{B}}}$${N}$${N}$${M.}$

For example, if $u\colon N\to M$ is striped and $r\in(0,\pi/2)$, then the
tangent spaces all have stabilizer
$\mathrm{G}\cong\text{O}(2)\leq\mathrm{SO}(4)$ for a suitable embedding of
$\text{O}(2)\hookrightarrow\mathrm{SO}(4)$, and $\mathcal{B}_{1}\to N$ is a
principal $\text{O}(2)$-bundle.

###  4.2 Striped Associative $3$-folds

Let $u\colon N\to M$ be a _striped_ associative submanifold with adapted
coframe bundle $\mathcal{B}_{1}\to N$. By Definition 2.7, we have $s=0$
identically on $\mathcal{B}_{1}$, so that

$\displaystyle-\sin(r)\mu_{2}+\cos(r)\theta_{2}$ $\displaystyle=0,$
$\displaystyle\theta_{3}$ $\displaystyle=0,$
$\displaystyle\sin(r)\mu_{3}+\cos(r)\theta_{1}$ $\displaystyle=0,$
$\displaystyle\theta_{4}$ $\displaystyle=0.$

Let us write $\cos(r)=x_{1}$ and $\sin(r)=x_{2},$ so that
$x_{1}^{2}+x_{2}^{2}=1,$ and define 1-forms
$\epsilon_{1},\epsilon_{2},\epsilon_{3}$ on $\mathcal{B}_{1}$ by

$\displaystyle\epsilon_{1}$ $\displaystyle=\mu_{1}$
$\displaystyle\epsilon_{2}$ $\displaystyle=x_{1}\mu_{2}-x_{2}\theta_{2},$
$\displaystyle\epsilon_{3}$ $\displaystyle=x_{1}\mu_{3}+x_{2}\theta_{1}.$

The $1$-forms $\epsilon_{1},\epsilon_{2},\epsilon_{3}$ are semi-basic for the
projection $\mathcal{B}_{1}\to N$. The metric and volume form on $N$ induced
from $M$ pull back to $\mathcal{B}_{1}$ to be

$g_{N}=\epsilon_{1}^{2}+\epsilon_{2}^{2}+\epsilon_{3}^{2},\>\>\>\>\mathrm{vol}_{N}=\epsilon_{123}.$

The equations (4.1) together with the structure equations (3.5) imply

$\displaystyle d\epsilon_{1}$
$\displaystyle=\left(-4f_{1}x_{1}^{2}+2\frac{a}{b^{2}}x_{1}^{2}-4\frac{1}{a}x_{1}^{2}-2\frac{a}{b^{2}}\right)\epsilon_{2}\wedge\epsilon_{3},$
(4.2a) $\displaystyle d\epsilon_{2}$
$\displaystyle=-\left(\psi-\frac{1}{a}\epsilon_{1}\right)\wedge\epsilon_{3}-2f_{2}x_{1}^{2}\epsilon_{2}\wedge\epsilon_{3}+\frac{x_{1}}{x_{2}}y\,\epsilon_{1}\wedge\epsilon_{2},$
$\displaystyle d\epsilon_{3}$
$\displaystyle=\left(\psi-\frac{1}{a}\epsilon_{1}\right)\wedge\epsilon_{2}+2f_{3}x_{1}^{2}\epsilon_{3}\wedge\epsilon_{2}+\frac{x_{1}}{x_{2}}y\,\epsilon_{1}\wedge\epsilon_{3},$
$\displaystyle dx_{1}$
$\displaystyle=x_{2}y\,\epsilon_{1}-2x_{1}x_{2}^{2}f_{3}\,\epsilon_{2}+2x_{1}x_{2}^{2}f_{2}\,\epsilon_{3},$
$\displaystyle dx_{2}$
$\displaystyle=-x_{1}y\,\epsilon_{1}+2x_{1}^{2}x_{2}f_{3}\,\epsilon_{2}-2x_{1}^{2}x_{2}f_{2}\,\epsilon_{3},$
and $\displaystyle\zeta_{1}$
$\displaystyle=\tfrac{1}{2}\psi+f_{1}\epsilon_{1}+f_{2}\epsilon_{2}+f_{3}\epsilon_{3},$
$\displaystyle\nu_{1}$ $\displaystyle=-\tfrac{3}{2}\psi-
f_{1}\epsilon_{1}-f_{2}\epsilon_{2}-f_{3}\epsilon_{3}.$ (4.2b)
$\displaystyle\zeta_{2}$
$\displaystyle=-\frac{x_{1}}{2a}\left(2af_{1}+3\right)\epsilon_{2}+\frac{y}{2x_{2}}\epsilon_{3},$
$\displaystyle\nu_{2}$ $\displaystyle=p_{2}\epsilon_{2}+p_{3}\epsilon_{3},$
$\displaystyle\zeta_{3}$
$\displaystyle=-\frac{y}{2x_{2}}\epsilon_{2}-\frac{x_{1}}{2a}\left(2af_{1}+3\right)\epsilon_{3},$
$\displaystyle\nu_{2}$ $\displaystyle=-p_{3}\epsilon_{2}+p_{2}\epsilon_{3},$

where $y,$ $f_{1},f_{2},f_{3},$ $p_{2},p_{3}$ are functions defined on
$\mathcal{B}_{1}$ and $\psi$ is a connection 1-form for the
$\mathrm{S}^{1}$-structure $\mathcal{B}_{1}$ on $N.$

######  Proposition 4.2.

Let $u\colon N\to(M,\varphi_{a,b})$ be an associative $3$-fold. Then:

$N\text{ is striped or locally a subset of a canonical leaf }\iff N\text{ is
locally ruled by Hopf circles.}$

###### Proof.

$(\Longleftarrow)$ Suppose $N$ is locally ruled by Hopf circles. Then for each
$p\in N$, we have $\dim(T_{p}N\cap\mathsf{A}_{p})\geq 1$. The result now
follows from Proposition 2.6 and Definition 2.7.
$(\Longrightarrow)$ If $N$ is a subset of a canonical leaf, then the result is
clear. So, suppose that $u\colon N\to M$ is striped. Equations (4.2) show that
the ideal $\langle\epsilon_{2},\epsilon_{3}\rangle$ defined on the
$4$-manifold $\mathcal{B}_{1}$ is Frobenius. They also imply that the integral
surfaces of $\langle\epsilon_{2},\epsilon_{3}\rangle$ on $\mathcal{B}_{1}$ are
integral surfaces of the ideal
$\left\langle\mu_{2},\mu_{3},\theta_{1},\theta_{2},\theta_{3},\theta_{4},\zeta_{2},\zeta_{3}\right\rangle$
defined on $u^{*}(\mathcal{B})$. Therefore, these integral surfaces project to
Hopf circles that lie in $N$. ∎

######  Remark 4.3.

We emphasize that the definition of “striped associative” makes sense in any
$7$-manifold with an $\mathrm{SO}(4)$-structure $(\varphi,\mathsf{A})$. On the
other hand, the concept of “Hopf-ruled $3$-fold” is specific to _quaternion-
Sasakian_ $\mathrm{SO}(4)$-structures and their squashings. It would be
interesting to know if striped associatives in $7$-manifolds with other types
of $\mathrm{SO}(4)$-structures admit similar geometric characterizations.

###  4.3 Hopf-Ruled $3$-folds

Let $(M,\varphi_{a,b},\mathsf{A})$ be a compact squashed $3$-Sasakian
$7$-manifold. Recall that Proposition 3.8 gives a diffeomorphism

$\displaystyle\Gamma\colon Z\times S^{2}$
$\displaystyle\to\frac{\mathcal{B}}{\mathrm{U}(1)\times\mathrm{U}(2)}\cong\\{\text{Hopf
circles in }M\\}$ $\displaystyle\Gamma(z,w)$
$\displaystyle=\pi(\lambda^{-1}(z,w)),$

where we are viewing $\mathcal{B}$ as a double fibration

${\mathcal{B}}$${M}$${Z\times
S^{2}}$$\scriptstyle{\pi}$$\scriptstyle{\lambda}$

Heuristically, this suggests correspondences between the following classes of
objects:

  1. 1.

A Hopf-ruled $3$-fold in $M$ — i.e., an immersed $3$-fold $u\colon N\to M$
admitting a fibration to a surface whose fibers are Hopf circles.

  2. 2.

An immersed $3$-fold in $M$ of the form
$\Gamma(\Sigma):=\pi(\lambda^{-1}(\Sigma))\subset M$ for some surface
$\Sigma\subset Z\times S^{2}$.

  3. 3.

A $2$-parameter family of Hopf circles — i.e., an immersed surface
$v\colon\Sigma\to Z\times S^{2}$.

  4. 4.

A “directrix” surface $f\colon\Sigma\to Z$ together with a “ruling” map
$w\colon\Sigma\to S^{2}$ for which $\text{rank}(f,w)=2$.

Guided by this intuitive picture, in this section we explain how to recast the
geometry of a Hopf-ruled $3$-fold $u\colon N\to M$ in terms of an associated
surface $v\colon\Sigma\to Z\times S^{2}$, and vice versa. The formalism that
we establish here will be used to prove the Correspondence Theorem 1.1.

####  4.3.1 From Surfaces in $Z\times S^{2}$ to Hopf-Ruled $3$-folds in $M$

Let $v\colon\Sigma\to Z\times S^{2}$ be an immersed surface. Recall that
$Z\times S^{2}$ carries the Riemannian metric $h_{a,b}$ defined in $\S$3.3.2.
Equip $\Sigma$ with the metric $h_{\Sigma}$ induced from
$v\colon\Sigma\to(Z\times S^{2},h_{a,b})$, let $\mathcal{F}_{\Sigma}\to\Sigma$
be the corresponding oriented orthonormal frame bundle, and let
$\eta=\eta_{1}+i\eta_{2}\in\Omega^{1}(\mathcal{F}_{\Sigma};\mathbb{C})$ be the
complex tautological form. Thus, the induced metric and volume form on
$\Sigma$ are

$\displaystyle h_{\Sigma}$
$\displaystyle=\eta\circ\overline{\eta}=\eta_{1}^{2}+\eta_{2}^{2}$
$\displaystyle\text{vol}_{\Sigma}$
$\displaystyle=\textstyle\frac{i}{2}\eta\wedge\overline{\eta}=\eta_{1}\wedge\eta_{2}.$

Next, noting that $v^{*}(\mathcal{B})\to\Sigma$ is a
$(\mathrm{U}(1)\times\mathrm{U}(2))$-bundle, we may define the
$(T^{2}\times\mathrm{U}(2))$-bundle

$\mathcal{G}_{\Sigma}:=\mathcal{F}_{\Sigma}\times_{\Sigma}v^{*}(\mathcal{B}).$

We have a diagram

${\mathcal{G}_{\Sigma}}$${v^{*}(\mathcal{B})}$${\mathcal{B}}$${\Sigma}$${\Sigma}$${Z\times
S^{2}}$${M}$$\scriptstyle{\lambda}$$\scriptstyle{\pi}$$\scriptstyle{v}$

Now, note that the image of $v^{*}(\mathcal{B})\hookrightarrow\mathcal{B}$ is
simply $\lambda^{-1}(\Sigma)$. Therefore, the image of the natural map
$\mathcal{G}_{\Sigma}\to M$ is equal to $(\pi\circ\lambda^{-1})(\Sigma)$.
Moreover, the image of $\mathcal{G}_{\Sigma}\to M$ is a Hopf-ruled
$3$-dimensional immersed submanifold on the set

$\mathcal{G}_{\Sigma}^{\circ}:=\\{p\in\mathcal{G}_{\Sigma}\colon\text{rank}(\mu_{1},\mu_{2},\mu_{3},\theta_{1},\theta_{2},\theta_{3},\theta_{4})|_{p}=3\\},$

i.e., the set of points on which the linear map
$(\mu_{1},\mu_{2},\mu_{3},\theta_{1},\theta_{2},\theta_{3},\theta_{4})\colon
T_{p}\mathcal{G}_{\Sigma}\to\mathbb{R}^{7}$ has rank $3$.

####  4.3.2 From Hopf-Ruled $3$-folds in $M$ to Surfaces in $Z\times S^{2}$

Let $u\colon N\to M$ be a Hopf-ruled immersed $3$-fold in $M$, and let $V$ be
the unit vector field on $N$ tangent to the ruling. Consider
$u^{*}(\mathcal{B})\to N$, which is a $\mathrm{SO}(4)$-bundle. Adapt frames to
the ruling by considering

$\mathcal{B}^{1}_{N}|_{p}:=\\{(e_{1},\ldots,e_{7})\in
u^{*}(\mathcal{B})|_{p}\colon e_{1}=V_{p}\\}.$

Note that $\mathcal{B}^{1}_{N}\to N$ is a $\mathrm{U}(2)$-bundle.
Next, equip $N$ with the Riemannian metric $g_{N}$ induced from $u\colon N\to
M$, let $\mathcal{F}_{N}\to N$ be the $\mathrm{SO}(3)$-coframe bundle, and let
$\rho=(\rho_{1},\rho_{2},\rho_{3})\in\Omega^{1}(\mathcal{F}_{N};\mathbb{R}^{3})$
be the tautological $1$-form. Again, we can adapt frames to the ruling by
considering the $\mathrm{SO}(2)$-bundle

$\mathcal{F}^{1}_{N}:=\\{f\in\mathcal{F}_{N}\colon f_{*}(V)=(1,0,0)\\}.$

Finally, define the $(\mathrm{SO}(2)\times\mathrm{U}(2))$-bundle

$\mathcal{Q}_{N}:=\mathcal{F}^{1}_{N}\times_{N}\mathcal{B}^{1}_{N}.$

Altogether, we have a diagram

${\mathcal{Q}_{N}}$${\mathcal{B}^{1}_{N}}$${u^{*}(\mathcal{B})}$${\mathcal{B}}$${N}$${N}$${N}$${M}$${Z\times
S^{2}}$$\scriptstyle{\pi}$$\scriptstyle{\lambda}$$\scriptstyle{u}$

One can check that the image of the natural map
$\mathcal{Q}_{N}\to\mathcal{B}$ is equal to $\lambda^{-1}(\Sigma)$ for some
immersed surface $\Sigma\to Z\times S^{2}$, and hence
$N=(\pi\circ\lambda^{-1})(\Sigma)$.

The above formalism established, we now explore the geometry of a Hopf-ruled
$3$-fold $u\colon N\to M$. For this, recall the $\mathbb{C}$-valued $1$-forms
$\pi_{1},\pi_{2},\pi_{3},\pi_{4}\in\Omega^{1}(\mathcal{B};\mathbb{C})$
discussed in $\S$3.3.2. In terms of these, the $\mathrm{G}_{2}$-structure and
metric on $M^{7}$ are:

$\displaystyle\varphi_{a,b}$
$\displaystyle=\textstyle\mu_{1}\wedge\frac{i}{2}(\pi_{1}\wedge\overline{\pi}_{1}+\frac{1}{2}\pi_{2}\wedge\overline{\pi}_{2}+\frac{1}{2}\pi_{3}\wedge\overline{\pi}_{3})+\frac{1}{2}\,\text{Re}(\pi_{1}\wedge\pi_{2}\wedge\pi_{3})$
(4.3) $\displaystyle g_{a,b}$
$\displaystyle\textstyle=\mu_{1}^{2}+\pi_{1}\circ\overline{\pi}_{1}+\frac{1}{2}\pi_{2}\circ\overline{\pi}_{2}+\frac{1}{2}\pi_{3}\circ\overline{\pi}_{3}.$

Pulling all quantities back to $\mathcal{Q}_{N}$, we have:

######  Lemma 4.4.

Let $\sigma=\rho_{2}+i\rho_{3}$. There exist $\mathbb{C}$-valued functions
$A_{1},\ldots,A_{4},B_{1},\ldots,B_{4}\colon\mathcal{Q}_{N}\to\mathbb{C}$ for
which

$\displaystyle\pi_{1}$ $\displaystyle=A_{1}\sigma+B_{1}\overline{\sigma}$
$\displaystyle\pi_{2}$ $\displaystyle=A_{2}\sigma+B_{2}\overline{\sigma}$
$\displaystyle\pi_{4}$ $\displaystyle=A_{4}\sigma+B_{4}\overline{\sigma}$
$\displaystyle\pi_{3}$ $\displaystyle=A_{3}\sigma+B_{3}\overline{\sigma}.$

Write $A=(A_{1},\frac{1}{\sqrt{2}}A_{2},\frac{1}{\sqrt{2}}A_{3})$ and
$B=(B_{1},\frac{1}{\sqrt{2}}B_{2},\frac{1}{\sqrt{2}}B_{3})$, viewed as vectors
in $\mathbb{C}^{3}$, and let $\langle
A,B\rangle:=A_{1}\overline{B}_{1}+\frac{1}{2}A_{2}\overline{B}_{2}+\frac{1}{2}A_{3}\overline{B}_{3}$
denote their inner product. Then on $\mathcal{Q}_{N}$,

$\displaystyle\left.\varphi_{a,b}\right|_{N}$
$\displaystyle=\frac{i}{2}\left(\|A\|^{2}-\|B\|^{2}\right)\mu_{1}\wedge\sigma\wedge\overline{\sigma}$
$\displaystyle g_{a,b\,N}$
$\displaystyle=\mu_{1}^{2}+\left(\|A\|^{2}+\|B\|^{2}\right)\sigma\circ\overline{\sigma}+2\,\mathrm{Re}\\!\left[\langle
A,B\rangle\,\sigma^{2}\right]$ $\displaystyle\mathrm{vol}_{a,b\,N}$
$\displaystyle=\frac{i}{2}\left[\left(\|A\|^{2}+\|B\|^{2}\right)^{2}-4\left|\langle
A,B\rangle\right|^{2}\right]^{1/2}\mu_{1}\wedge\sigma\wedge\overline{\sigma}.$

###### Proof.

By the definitions of $\mathcal{B}^{1}_{N}$ and $\mathcal{F}^{1}_{N}$ we have
$\mu_{1}=\rho_{1}$ on $\mathcal{Q}_{N}.$ On the one hand, the Hopf circles in
$N$ are the projections to $N$ of the integral submanifolds of the ideal
$\langle\rho_{2},\rho_{3}\rangle=\langle\sigma,\overline{\sigma}\rangle.$ On
the other hand, Hopf circles are the projections to $N$ of the integral
submanifolds of the ideal
$\left\langle\mu_{2},\mu_{3},\theta_{1},\theta_{2},\theta_{3},\theta_{4},\zeta_{2},\zeta_{3}\right\rangle.$
Therefore, on $\mathcal{Q}_{N},$ we must have

$\displaystyle\mu_{2},\mu_{3},\zeta_{2},\zeta_{3},\theta_{1},\theta_{2},\theta_{3},\theta_{4}$
$\displaystyle\equiv 0\,\text{ mod }(\sigma,\overline{\sigma}).$

This yields the existence of
$A_{1},A_{2},A_{3},A_{4},B_{1},B_{2},B_{3},B_{4}$. The remaining formulas
follow by using (4.3) to express $\varphi_{a,b}|_{N}$ and $g_{a,b\,N}$ in
terms of these functions. ∎

###  4.4 Hopf-Ruled Associatives: Correspondence Theorem

We are now in a position to prove the Correspondence Theorem 1.1. Recall that
$Z\times S^{2}$ carries the almost-Hermitian structure
$(h_{a,b},J,\omega_{a,b})$ defined in $\S$3.3.2. An immersed surface
$v\colon\Sigma\to Z\times S^{2}$ is called _$J$ -holomorphic_ if each tangent
plane $v_{*}(T_{p}\Sigma)\subset T_{v(p)}(Z\times S^{2})$ is $J$-invariant. In
practice, we will use the following alternate characterizations of
$J$-invariant $2$-planes.

######  Lemma 4.5 (Characterizations of complex lines).

Let $(V,(h,J,\omega))$ be a Hermitian vector space, and let $E\subset V$ be an
oriented real $2$-dimensional subspace. The induced metric and orientation on
$E$ endow it with a complex structure (that is not necessarily the restriction
of $J$).
(a) $E\subset V$ is $J$-invariant $\iff$ Every $\beta\in\Lambda^{2,0}(V^{*})$
satisfies $\beta|_{E}=0$.
(b) $E\subset V$ is $J$-invariant and positively oriented $\iff$ Every
$\alpha\in\Lambda^{1,0}(V^{*})$ satisfies
$\alpha|_{E}\in\Lambda^{1,0}(E^{*})$.

Finally, recall that $Z\times S^{2}$ has the structure of an $(S^{2}\times
S^{2})$-bundle $q\colon Z\times S^{2}\to X$. We can now prove:

######  Theorem 4.6 (Correspondence Theorem).

Let $(M^{7},\varphi_{a,b})$ be a compact squashed $3$-Sasakian $7$-manifold,
where $a,b>0$ are fixed.

  1. (a)

Every Hopf-ruled associative $3$-fold in $(M^{7},\varphi_{a,b})$ is locally of
the form $\Gamma(\Sigma)$ for some $J$-holomorphic curve $\Sigma\subset
Z\times S^{2}$.

  2. (b)

Let $\Sigma\subset Z\times S^{2}$ be a $J$-holomorphic curve.

    1. 1.

If $\Sigma$ does not lie in an $(S^{2}\times S^{2})$-fiber, then there exists
a discrete set $D\subset\Sigma$ such that $\Gamma(\Sigma-D)\subset M$ is a
Hopf-ruled associative $3$-fold.

    2. 2.

If $\Sigma$ lies in an $(S^{2}\times S^{2})$-fiber, then $\Gamma(\Sigma)$ is a
subset of a canonical leaf.

###### Proof.

(a) Suppose that $u\colon N^{3}\to M^{7}$ is an immersed Hopf-ruled
associative $3$-fold. Since $u$ is a Hopf-ruled $3$-fold, we may apply the
formalism of $\S$4.3.2 and perform calculations on the space
$\mathcal{Q}_{N}$. Recall also that there exists a immersed surface
$v\colon\Sigma\to Z\times S^{2}$ for which
$\lambda^{-1}(\Sigma)\subset\mathcal{B}$ is the image of the natural map
$\mathcal{Q}_{N}\to\mathcal{B}$.
Now, since $u$ is associative, we have
$\left.\varphi_{a,b}\right|_{N}=\mathrm{vol}_{a,b\,N}$. Therefore, Lemma 4.4
yields

$\left(\|A\|^{2}-\|B\|^{2}\right)^{2}=\left(\|A\|^{2}+\|B\|^{2}\right)^{2}-4\left|\langle
A,B\rangle\right|^{2}.$

Expanding and simplifying yields $\|A\|\|B\|=\left|\langle A,B\rangle\right|$,
and so $\\{A,B\\}$ is a $\mathbb{C}$-linearly dependent set in
$\mathbb{C}^{3}$. It follows that

$\displaystyle\pi_{1}\wedge\pi_{2}$ $\displaystyle=0$
$\displaystyle\pi_{1}\wedge\pi_{3}$ $\displaystyle=0$
$\displaystyle\pi_{2}\wedge\pi_{3}$ $\displaystyle=0.$

In particular, $d(\pi_{1}\wedge\pi_{2})=0$ and $d(\pi_{1}\wedge\pi_{3})=0$.
Expanding these via the structure equations (3.4), we obtain
$A_{2}B_{4}-A_{4}B_{2}=0$ and $A_{3}B_{4}-A_{4}B_{3}=0$, whence
$\pi_{2}\wedge\pi_{4}=0$ and $\pi_{3}\wedge\pi_{4}=0$. Considering the vectors
$(A_{1},A_{2},A_{3},A_{4})$ and $(B_{1},B_{2},B_{3},B_{4})$ in
$\mathbb{C}^{4}$, a linear algebra argument shows that
$A_{1}B_{4}-A_{4}B_{1}=0$, so that $\pi_{1}\wedge\pi_{4}=0$.
Altogether, we have shown that $\pi_{i}\wedge\pi_{j}=0$ for all
$i,j=1,\ldots,4$. By Lemma 4.5(a), we conclude that $v\colon\Sigma\to Z\times
S^{2}$ is a $J$-holomorphic curve.

(b) Let $v\colon\Sigma\to Z\times S^{2}$ be an immersed $J$-holomorphic curve.
Since $v\colon\Sigma\to Z\times S^{2}$ is an immersed surface, we may apply
the formalism of $\S$4.3.1 and perform calculations on the space
$\mathcal{G}_{\Sigma}$. Since $v$ is $J$-holomorphic, Lemma 4.5(b) implies
that there exist functions $W_{1}$, $W_{2}$, $W_{3}$,
$W_{4}\colon\mathcal{G}_{\Sigma}\to\mathbb{C}$ for which

$\displaystyle\pi_{1}$ $\displaystyle=W_{1}\eta$ $\displaystyle\pi_{2}$
$\displaystyle=W_{2}\eta$ $\displaystyle\pi_{3}$ $\displaystyle=W_{3}\eta$
$\displaystyle\pi_{4}$ $\displaystyle=W_{4}\eta.$

and since $\omega_{a,b}|_{\Sigma}=\mathrm{vol}_{\Sigma}$,

$|W_{1}|^{2}+|W_{2}|^{2}+|W_{3}|^{2}+|W_{4}|^{2}=1$ (4.4)

The proof will consist of five steps: (i) Show that the image of
$\mathcal{G}_{\Sigma}^{\circ}\to M^{7}$ is associative; (ii) Show that
$\mathcal{G}_{\Sigma}-\mathcal{G}_{\Sigma}^{\circ}=\\{W_{1}=W_{2}=W_{3}=0\\}$;
(iii) Compute the derivatives of $W_{1},\ldots,W_{4}$; (iv) Exhibit
$\mathcal{Y}:=\\{W_{2}=W_{3}=0\\}$ as the zero set of a holomorphic section
over $\Sigma$; (v) Complete the proof.

(i) Let $N\subset M$ be the image of $\mathcal{G}_{\Sigma}^{\circ}\to M$.
Recall that the $\mathrm{G}_{2}$-structure and metric on $M^{7}$ are

$\displaystyle\varphi_{a,b}$
$\displaystyle=\textstyle\mu_{1}\wedge\frac{i}{2}(\pi_{1}\wedge\overline{\pi}_{1}+\frac{1}{2}\pi_{2}\wedge\overline{\pi}_{2}+\frac{1}{2}\pi_{3}\wedge\overline{\pi}_{3})+\frac{1}{2}\text{Re}(\pi_{1}\wedge\pi_{2}\wedge\pi_{3})$
$\displaystyle g_{a,b}$
$\displaystyle=\textstyle\mu_{1}^{2}+\pi_{1}\circ\overline{\pi}_{1}+\frac{1}{2}\pi_{2}\circ\overline{\pi}_{2}+\frac{1}{2}\pi_{3}\circ\overline{\pi}_{3}$

Therefore, on $\mathcal{G}_{\Sigma}$:

$\displaystyle\varphi_{a,b}$
$\displaystyle=\textstyle\left(|W_{1}|^{2}+\frac{1}{2}|W_{2}|^{2}+\frac{1}{2}|W_{3}|^{2}\right)\frac{i}{2}\mu_{1}\wedge\eta\wedge\overline{\eta}$
$\displaystyle g_{a,b}|_{\mathcal{G}_{\Sigma}}$
$\displaystyle=\textstyle\mu_{1}^{2}+\left(|W_{1}|^{2}+\frac{1}{2}|W_{2}|^{2}+\frac{1}{2}|W_{3}|^{2}\right)\eta\circ\overline{\eta}$

The volume form on $N$ induced from $g_{a,b}|_{\mathcal{G}_{\Sigma}}$ is then

$\textstyle\text{vol}_{N}=\left(|W_{1}|^{2}+\frac{1}{2}|W_{2}|^{2}+\frac{1}{2}|W_{3}|^{2}\right)\frac{i}{2}\alpha_{1}\wedge\eta\wedge\overline{\eta}.$

This shows that the image of $\mathcal{G}_{\Sigma}^{\circ}\to M$ is
associative. To complete the proof, we need to construct
$\Sigma^{\circ}:=\Sigma-D$. We will do this by identifying an open dense set
$\Sigma^{\circ}\subset\Sigma$ with the property that
$\mathcal{G}_{\Sigma^{\circ}}\subset\mathcal{G}_{\Sigma}^{\circ}$.

(ii) On $\mathcal{G}_{\Sigma}$, we have

$\displaystyle\mu_{2}\wedge\mu_{3}$
$\displaystyle=\frac{i}{2}|W_{1}|^{2}\,\eta\wedge\overline{\eta}$
$\displaystyle\theta_{1}\wedge\theta_{2}$
$\displaystyle=\frac{i}{4}|W_{2}|^{2}\,\eta\wedge\overline{\eta}$
$\displaystyle\theta_{3}\wedge\theta_{4}$
$\displaystyle=\frac{i}{4}|W_{3}|^{2}\,\eta\wedge\overline{\eta}$

and

$\displaystyle\theta_{2}\wedge\theta_{4}$
$\displaystyle=\theta_{1}\wedge\theta_{3}=\textstyle\frac{1}{8}\left(W_{2}\overline{W}_{3}-\overline{W}_{2}W_{3}\right)\eta\wedge\overline{\eta}$
$\displaystyle\theta_{1}\wedge\theta_{4}$
$\displaystyle=-\theta_{2}\wedge\theta_{3}=\textstyle\frac{i}{8}\left(W_{2}\overline{W}_{3}+\overline{W}_{2}W_{3}\right)\eta\wedge\overline{\eta}$
$\displaystyle\mu_{2}\wedge\theta_{1}$
$\displaystyle=\mu_{3}\wedge\theta_{2}=\textstyle\frac{1}{4\sqrt{2}}\left(W_{1}\overline{W}_{2}-\overline{W}_{1}W_{2}\right)\eta\wedge\overline{\eta}$
$\displaystyle\mu_{2}\wedge\theta_{2}$
$\displaystyle=-\mu_{3}\wedge\theta_{1}=\textstyle\frac{i}{4\sqrt{2}}\left(W_{1}\overline{W}_{2}+\overline{W}_{1}W_{2}\right)\eta\wedge\overline{\eta}$
$\displaystyle\mu_{2}\wedge\theta_{3}$
$\displaystyle=\mu_{3}\wedge\theta_{4}=\textstyle\frac{1}{4\sqrt{2}}\left(W_{1}\overline{W}_{3}-\overline{W}_{1}W_{3}\right)\eta\wedge\overline{\eta}$
$\displaystyle\mu_{2}\wedge\theta_{4}$
$\displaystyle=-\mu_{3}\wedge\theta_{3}=\textstyle\frac{i}{4\sqrt{2}}\left(W_{1}\overline{W}_{3}+\overline{W}_{1}W_{3}\right)\eta\wedge\overline{\eta}$

It follows that
$\mathcal{G}_{\Sigma}-\mathcal{G}_{\Sigma}^{\circ}=\\{W_{1}=W_{2}=W_{3}=0\\}$.

(iii) Let $\psi$ denote the connection 1-form for the Levi-Civita connection
on $\Sigma$ associated to the metric $h_{\Sigma},$ so that $\eta$ satisfies
the the structure equation

$d\eta=i\,\psi\wedge\eta.$

Now, for each $j=1,2,3,4$, we have

$dW_{j}\wedge\eta=d(W_{j}\eta)-W_{j}\,d\eta=d\pi_{j}-iW_{j}\psi\wedge\eta.$

Calculating $d\pi_{j}$ via the structure equations (3.4), we find

$\displaystyle d(W_{1}-iaW_{4})\wedge\eta$
$\displaystyle=2i(W_{1}-iaW_{4})\alpha_{1}\wedge\eta+2i\left(W_{1}-iaW_{4}\right)\zeta_{1}\wedge\eta-i(W_{1}-iaW_{4})\psi\wedge\eta$
$\displaystyle dW_{2}\wedge\eta$ $\displaystyle=-iW_{2}\alpha_{1}\wedge\eta-
iW_{2}\zeta_{1}\wedge\eta-iW_{2}\nu_{1}\wedge\eta-
W_{3}(\nu_{2}-i\nu_{3})\wedge\eta-iW_{2}\psi\wedge\eta$ $\displaystyle
dW_{3}\wedge\eta$ $\displaystyle=-iW_{3}\alpha_{1}\wedge\eta-
iW_{3}\zeta_{1}\wedge\eta+iW_{3}\nu_{1}\wedge\eta-
W_{2}(-\nu_{2}-i\nu_{3})\wedge\eta-iW_{3}\psi\wedge\eta$ $\displaystyle
dW_{4}\wedge\eta$ $\displaystyle=2iW_{4}\zeta_{1}\wedge\eta-
iW_{4}\psi\wedge\eta$

Set $X:=W_{1}-iaW_{4}$. By Cartan’s Lemma, there exist functions
$F_{1},F_{2},F_{3},F_{4}\colon\mathcal{G}_{\Sigma}\to\mathbb{C}$ for which

$\displaystyle dX$ $\displaystyle=2iX(\alpha_{1}+\zeta_{1})-iX\psi+F_{1}\eta,$
(4.5) $\displaystyle dW_{4}$
$\displaystyle=2iW_{4}\zeta_{1}-iW_{4}\psi+F_{4}\eta,$
$\displaystyle\begin{bmatrix}dW_{2}\\\ dW_{3}\end{bmatrix}$
$\displaystyle=-\begin{pmatrix}i\nu_{1}+i\zeta_{1}&-\nu_{2}-i\nu_{3}\\\
\nu_{2}-i\nu_{3}&-i\nu_{1}+i\zeta_{1}\end{pmatrix}\begin{bmatrix}W_{2}\\\
W_{3}\end{bmatrix}+\begin{pmatrix}-i\alpha_{1}-i\psi&0\\\
0&-i\alpha_{1}-i\psi\end{pmatrix}\begin{bmatrix}W_{2}\\\
W_{3}\end{bmatrix}+\begin{bmatrix}F_{2}\\\ F_{3}\end{bmatrix}\eta.$

(iv) The first two equations of (4.5) imply that $|X|^{2}$ and $|W_{4}|^{2}$
descend to well-defined functions on $\Sigma$. Similarly, the last equation of
(4.5) implies that $|W_{2}|^{2}+|W_{3}|^{2}$ descends to a well-defined
function on $\Sigma$. We may thus consider the following set:

$\displaystyle\mathcal{Y}$ $\displaystyle:=\\{W_{2}=W_{3}=0\\}\subset\Sigma$

Let $G=\mathrm{T}^{2}\times\mathrm{U}(2),$ the structure group of the bundle
$\mathcal{G}_{\Sigma}\to\Sigma$. The vector fields dual to $\alpha_{1}$ and
$\psi$ generate the $\mathrm{T}^{2}$-action on $\mathcal{G}_{\Sigma}$ and the
vector fields dual to $\zeta_{1},\nu_{1},\nu_{2},\nu_{3}$ generate the
$\mathrm{U}(2)$-action on $\mathcal{G}_{\Sigma}$. The function
$X\colon\mathcal{G}_{\Sigma}\to\mathbb{C}$ is $G$-equivariant with respect to
the $G$-representation on $\mathbb{C}$ for which $\mathrm{U}(2)$ acts via the
determinant representation and $\mathrm{T}^{2}$ acts with weights $(2,-1)$ and
the function $W_{4}\colon\mathcal{G}_{\Sigma}\to\mathbb{C}$ is $G$-equivariant
with respect to the $G$-representation on $\mathbb{C}$ for which
$\mathrm{U}(2)$ acts via the determinant representation and $\mathrm{T}^{2}$
acts with weights $(0,-1)$. Therefore, $X$ and $W_{4}$ correspond to sections
$\Phi_{\mathcal{X}},\Phi_{\mathcal{W}_{4}}$ of associated complex line bundles
$\mathscr{L}_{\mathcal{X}},\mathscr{L}_{\mathcal{W}_{4}}$ over $\Sigma$.
Similarly, $(W_{2},W_{3})\colon\mathcal{G}_{\Sigma}\to\mathbb{C}$ is
$G$-equivariant with respect to the $G$-representation on $\mathbb{C}^{2}$
given by the tensor product of the standard $\mathrm{U}(2)$-representation
with the $\mathrm{T}^{2}$-representation having weights $(-1,-1)$ and so
corresponds to a section $\Phi_{\mathcal{H}}$ of an associated rank-$2$
complex vector bundle $\mathscr{L}_{\mathcal{H}}\to\Sigma$.
Let us equip all of these vector bundles with their Koszul-Malgrange
holomorphic structure with respect to the connection on the principal
$G$-bundle $\mathcal{G}_{\Sigma}\to\Sigma$ whose connection forms are
$\left(\alpha_{1},\psi,\zeta_{1},\nu_{1},\nu_{2},\nu_{3}\right)$. Equations
(4.5) show that the sections
$\Phi_{\mathcal{X}},\Phi_{\mathcal{Y}},\Phi_{\mathcal{H}}$ are holomorphic.
Moreover, since

$\displaystyle\mathcal{Y}$ $\displaystyle=\\{\Phi_{\mathcal{H}}=0\\}$

we see that $\mathcal{Y}$ is the vanishing locus of a holomorphic section. In
particular, $\mathcal{Y}$ is discrete or $\mathcal{Y}=\Sigma$.

(v) If $\mathcal{Y}$ is discrete, then we are done: the set $D$ is simply
$\mathcal{Y}$. We therefore assume for the rest of the proof that
$\Sigma=\mathcal{Y}.$

The differential ideal
$\langle\pi_{2},\pi_{3}\rangle=\langle\theta_{1},\theta_{2},\theta_{3},\theta_{4}\rangle$
descends to $Z\times S^{2}$ to define a Frobenius distribution. The
4-dimensional integral submanifolds of this distribution are precisely the
fibres of the projection $q\colon Z\times S^{2}\to X.$ Therefore, the
condition $\Sigma=\mathcal{Y}$ implies $\Sigma$ lies entirely in a fiber of
$q$. On the other hand, the ideal
$\langle\theta_{1},\theta_{2},\theta_{3},\theta_{4}\rangle$ descends to $M$ to
define a Frobenius distribution whose 3-dimensional integral submanifolds are
precisely the canonical leaves, i.e., the fibers of $h\colon M\to X.$ It
follows that if $\Sigma$ lies entirely in an $(S^{2}\times S^{2})$-fiber, then
$\Gamma(\Sigma)$ is a subset of a canonical leaf. ∎

The proof of Theorem 4.6(b) yields more information than is stated in the
theorem. In particular, we have:

######  Corollary 4.7.

The set $D$ consists of the points of $\Sigma$ tangent to the fibers of the
projection $q\colon Z\times S^{2}\to X.$ Therefore, if $\Sigma\subset Z\times
S^{2}$ is a compact $J$-holomorphic curve nowhere tangent to the fibers of
$Z\times S^{2}\to X,$ then $\Gamma(\Sigma)$ is a smooth compact Hopf-ruled
associative 3-fold in $M.$

Now, recall from $\S$3.3.3 that we have equipped the twistor space $Z$ with
its standard strict nearly-Kähler structure. Recall also from $\S$2.4.1 that a
submanifold of $Z$ is _horizontal_ if it is tangent to $\mathsf{H}\subset TZ$,
the distribution orthogonal to the fibers of the twistor projection
$\tau\colon Z\to X$. We observe:

######  Proposition 4.8.

Let $f\colon\Sigma\to Z$ be a horizontal immersed pseudo-holomorphic curve,
equip $\Sigma$ with the induced complex structure, and let $w\colon\Sigma\to
S^{2}$ be a holomorphic map. Then $(f,w)\colon\Sigma\to Z\times S^{2}$ is a
$J$-holomorphic curve that is nowhere tangent to the fibers of $q\colon
Z\times S^{2}\to X$.

Proposition 4.8 together with Corollary 4.7 will be the key tool in our
construction of compact associative submanifolds in the squashed 7-sphere
$S^{7}$ and squashed Aloff-Wallach space $N_{1,1}$ in §5.

######  Remark 4.9.

Let $(M,\varphi_{a,b},\mathsf{A})$ be a squashed $3$-Sasakian $7$-manifold.
What follows is a heuristic description of the Hopf-ruled associative
$3$-folds $\Gamma(\Sigma)\subset M$. Choose a trivialization of
$\mathsf{A}\subset TM$, so that the projections $p_{w}\colon M\to Z$ (for
$w\in S^{2}$) from $\S$2.4.1 are well-defined. If $\Sigma\subset Z$ is a
pseudo-holomorphic curve with respect to the strict nearly-Kähler structure on
$Z$, then for a fixed $w\in S^{2}$, the pre-image

$p_{w}^{-1}(\Sigma)=\bigcup_{z\in\Sigma}p_{w}^{-1}(z)\subset M$

is an associative $3$-fold (by the squashed analogue of Proposition 2.24(a)
followed by Proposition 2.14) that is clearly Hopf-ruled. Now, by varying the
ruling direction $w$ in a suitable fashion, we can obtain a larger class of
examples. That is, heuristically speaking, the general Hopf-ruled associative
in $M$ is a submanifold of the form

$\bigcup_{z\in\Sigma}p_{w(z)}^{-1}(\Sigma)\subset M,$

where the “directrix surface” $f\colon\Sigma\to Z$ is a pseudo-holomorphic
map, and the “ruling function” $w\colon\Sigma\to S^{2}$ is a holomorphic map.
As the Correspondence Theorem indicates, this heuristic is valid after
removing a discrete set $D$ from $\Sigma$ and avoiding any pair
$(f,w)\colon\Sigma\to Z\times S^{2}$ that lies in an $(S^{2}\times
S^{2})$-fiber of $q\colon Z\times S^{2}\to X$.

######  Remark 4.10.

Every $3$-Sasakian $(4n+3)$-manifold $M^{4n+3}$ carries a generalized
associative calibration $\phi_{K}\in\Omega^{3}(M)$, as defined by Bryant and
Harvey [9]. We expect that our construction generalizes straightforwardly to
that setting.

##  5 Construction of Compact Associatives

In this section, we prove Theorems 1.2 and 1.3. That is, we will apply
Proposition 4.8 and Corollary 4.7 to construct infinitely many topological
types of non-trivial, compact associative $3$-folds in the squashed
$7$-spheres $(S^{7},\varphi_{a,b})$ and squashed Aloff-Wallach spaces
$(N_{1,1},\varphi_{a,b})$.

###  5.1 The $7$-Sphere

We begin by considering the $7$-sphere. There are three natural families of
$\mathrm{G}_{2}$-structures on $S^{7}$, depending on whether one views the
$7$-sphere as

$\frac{\mathrm{Spin}(7)}{\mathrm{G}_{2}}\ \ \text{ or }\ \
\frac{\mathrm{Sp}(2)\mathrm{Sp}(1)}{\mathrm{SO}(4)}\ \ \text{ or }\ \
\frac{\mathrm{Sp}(2)}{\mathrm{Sp}(1)}.$

First, regarding $S^{7}=\mathrm{Spin}(7)/\mathrm{G}_{2}$, it is well-known
that $S^{7}$ admits a unique $\mathrm{Spin}(7)$-invariant
$\mathrm{G}_{2}$-structure $\widehat{\varphi}$ up to scale. In fact,
$\widehat{\varphi}$ is nearly-parallel and induces the round metric on
$S^{7}$. For the sake of context, in $\S$5.1.1 we briefly review the
literature on associative $3$-folds in $(S^{7},\widehat{\varphi})$.
Second, regarding $S^{7}=\mathrm{Sp}(2)\mathrm{Sp}(1)/\mathrm{SO}(4)$, one can
show that $S^{7}$ admits a $2$-parameter family $\varphi_{a,b}$ of
$\mathrm{Sp}(2)\mathrm{Sp}(1)$-invariant $\mathrm{G}_{2}$-structures, where
the parameters are chosen so that $\varphi_{1,1}$ induces the round metric
$g_{1,1}$ of constant curvature $1$. In fact, letting $\mathsf{A}\subset
TS^{7}$ denote the vertical bundle for the quaternionic Hopf fibration
$h\colon S^{7}\to S^{4}$, the $\mathrm{SO}(4)$-structure
$(\varphi_{1,1},\mathsf{A})$ is quaternion-Sasakian, and the $2$-parameter
family $\varphi_{a,b}$ consists of its squashings. Consequently, by the
discussion in $\S$3, each $\varphi_{a,b}$ is co-closed, and $\varphi_{a,b}$ is
nearly-parallel if and only if $b^{2}=5a^{2}$. Moreover, $\varphi_{1,1}$ is
isometric to the nearly-parallel $\widehat{\varphi}$.
Third, regarding $S^{7}=\mathrm{Sp}(2)/\mathrm{Sp}(1)$, it was recently shown
by Loubeau, Moreno, Sà Earp, and Saavedra [28] that the space of
$\mathrm{Sp}(2)$-invariant $\mathrm{G}_{2}$-structures on $S^{7}$ is
parametrized by the $10$-manifold
$\mathbb{R}^{+}\times\mathrm{GL}_{3}^{+}(\mathbb{R})$. Note that
$\widehat{\varphi}$ and $\varphi_{a,b}$ all belong to this $10$-dimensional
family. We will not consider these $\mathrm{G}_{2}$-structures here, but refer
the reader to [28].

####  5.1.1 Associative Submanifolds in the Round $7$-Sphere
$(S^{7},\widehat{\varphi})$

Associative $3$-folds in $(S^{7},\widehat{\varphi})$ have been studied in
several works:

  * •

Fox [15] proved a correspondence theorem relating _geodesically-ruled_
associative $3$-folds in $(S^{7},\widehat{\varphi})$ to $J$-holomorphic curves
in the $12$-manifold
$\mathrm{Gr}_{2}^{+}(\mathbb{R}^{8})\cong\mathrm{Spin}(7)/\mathrm{U}(3)$,
where $J$ is a particular $\mathrm{Spin}(7)$-invariant non-integrable almost-
complex structure. Using this, he constructed constructed a wealth of
geodesically-ruled associative $3$-folds in $(S^{7},\widehat{\varphi})$ from
minimal surfaces in $S^{6}=\mathrm{Spin}(7)/\mathrm{SU}(4)$ and meromorphic
functions. See also [26, $\S$6].

  * •

Lotay [26] classified the homogeneous associatives in
$(S^{7},\widehat{\varphi})$ up to $\mathrm{Spin}(7)$-congruence, building upon
work of Mashimo [29]. Moreover, for each non-constant-curvature minimal
$2$-sphere in $S^{6}$, Lotay constructed a geodesically-ruled associative in
$(S^{7},\widehat{\varphi})$ that admits an $S^{1}$-family of non-congruent,
global isometric deformations.

  * •

Kawai [23], [25] studied the deformation theory of the $8$ homogeneous
associatives in $(S^{7},\widehat{\varphi})$. This was revisited by Moreno and
Sà Earp [31] and Moore [30].

Up to $\mathrm{Spin}(7)$-congruence, the $8$ homogeneous associative $3$-folds
in $(S^{7},\widehat{\varphi})$ are:

  1. 1.

$A_{1}\approx T^{3}$ is a $T^{3}$-orbit in $S^{7}\subset\mathbb{C}^{4}$, where
$T^{3}\leq\mathrm{SU}(4)$ is the maximal torus. In fact, $A_{1}$ is special
Legendrian: it is the link of the $T^{3}$-invariant special Lagrangian cone in
$\mathbb{C}^{4}$ discovered by Harvey and Lawson [18, $\S$III.3].

  2. 2.

$A_{2}\approx S^{3}/\mathbb{Z}_{3}$ is an $\mathrm{SU}(2)$-orbit in
$S^{7}\subset\mathbb{C}^{4}$, where the $\mathrm{SU}(2)$-action on
$\mathrm{Sym}^{3}(\mathbb{C}^{2})=\mathbb{C}^{4}$ is irreducible. In fact,
$A_{2}$ is complex Legendrian: it is of the form
$A_{2}=p_{w}^{-1}(\Sigma_{0})$ for a suitable $w\in S^{2}$, where
$\Sigma_{0}\subset\mathbb{CP}^{3}$ is the Veronese $\mathbb{CP}^{1}$, which is
horizontal for the twistor fibration $\tau\colon\mathbb{CP}^{3}\to S^{4}$.

  3. 3.

$A_{3}\approx S^{3}$ is an $\mathrm{SU}(2)$-orbit in
$S^{7}\subset\mathbb{C}^{4}$, where $\mathrm{SU}(2)$ acts on
$\mathrm{Sym}^{3}(\mathbb{C}^{2})=\mathbb{C}^{4}$ via its irreducible
representation.

  4. 4.

$L_{1}\approx S^{3}$, $L_{2}\approx\mathbb{RP}^{3}$, $L_{3}\approx
S^{3}/A_{4}^{*}$, and $L_{4}\approx S^{3}/D_{3}^{*}$ are Lagrangian
submanifolds of a totally geodesic $S^{6}\subset S^{7}$. In particular,
$L_{1}$ is the link of the Lawson-Osserman coassociatve cone, and is
$\mathrm{U}(2)$-invariant. Also, $L_{2}\subset S^{5}$ is a CR $3$-fold in a
totally-geodesic $S^{5}\subset S^{6}$ that is $\mathrm{SU}(2)$-invariant.
Finally, $L_{3}$ and $L_{4}$ are orbits of irreducible the
$\mathrm{SO}(3)$-action on $\mathbb{R}^{7}$.

  5. 5.

$P_{1}\approx S^{3}$ is the totally-geodesic Hopf $3$-sphere.

####  5.1.2 Associative Submanifolds in Squashed $7$-Spheres
$(S^{7},\varphi_{a,b})$

To our knowledge, the only previous work on associative $3$-folds in the
squashed $7$-spheres $(S^{7},\varphi_{a,b})$ is that of Kawai [24], who
classifies the homogeneous associatives in the nearly-parallel
$(S^{7},\varphi_{a,a\sqrt{5}})$ for $a=3/5$ and studies their deformation
theory. Up to $\mathrm{Sp}(2)\mathrm{Sp}(1)$-congruence, he shows that the $5$
homogeneous associative $3$-folds in $(S^{7},\varphi_{a,a\sqrt{5}})$ are:

  1. 1.

$A_{1}\approx T^{3}$ is is a $T^{3}$-orbit in $S^{7}\subset\mathbb{C}^{4}$,
where $T^{3}\leq\mathrm{SU}(4)$ is the maximal torus. In fact,
$A_{1}=p_{w}^{-1}(\Sigma)$ for some $w\in S^{2}$ and some pseudo-holomorphic
curve $\Sigma\subset\mathbb{CP}^{3}$.

  2. 2.