# Homogeneous non-degenerate $3$-$(\alpha,\delta)$-Sasaki manifolds and
submersions over quaternionic Kähler spaces

Ilka Agricola, Giulia Dileo, and Leander Stecker

###### Abstract

We show that every $3$-$(\alpha,\delta)$-Sasaki manifold of dimension $4n+3$
admits a locally defined Riemannian submersion over a quaternionic Kähler
manifold of scalar curvature $16n(n+2)\alpha\delta$. In the non-degenerate
case we describe all homogeneous $3$-$(\alpha,\delta)$-Sasaki manifolds
fibering over symmetric Wolf spaces and over their non-compact dual symmetric
spaces. If $\alpha\delta>0$, this yields a complete classification of
homogeneous $3$-$(\alpha,\delta)$-Sasaki manifolds. For $\alpha\delta<0$, we
provide a general construction of homogeneous $3$-$(\alpha,\delta)$-Sasaki
manifolds fibering over non-symmetric Alekseevsky spaces, the lowest possible
dimension of such a manifold being $19$.

###### Contents

  1. 1 Introduction and basic notions
    1. 1.1 Introduction
    2. 1.2 Review of $3$-$(\alpha,\delta)$-Sasaki manifolds and their basic properties
  2. 2 The Riemannian submersion over a quaternionic Kähler base
    1. 2.1 The canonical submersion
    2. 2.2 The quaternionic Kähler structure on the base
  3. 3 Construction of homogeneous $3$-$(\alpha,\delta)$-Sasaki manifolds
    1. 3.1 Homogeneous $3$-$(\alpha,\delta)$-Sasaki manifolds over symmetric quaternionic Kähler spaces
    2. 3.2 Negative homogeneous $3$-$(\alpha,\delta)$-Sasaki manifolds over Alekseevsky spaces
    3. 3.3 Examples
  4. 4 Nomizu maps of homogeneous $3$-$(\alpha,\delta)$-Sasaki manifolds
    1. 4.1 Nomizu map of the canonical connection
    2. 4.2 Nomizu maps in the symmetric base case

x

MSC (2010): primary 53B05, 53C15, 53C25, 53D10; secondary 53C27, 32V05, 22E25.

Keywords and phrases: Almost $3$-contact metric manifold; $3$-Sasaki manifold;
$3$-$(\alpha,\delta)$-Sasaki manifold; Riemannian homogeneous space; canonical
connection; Riemannian submersion; quaternionic Kähler manifold; Wolf space;
Alekseevsky space; Nomizu map.

##  1 Introduction and basic notions

###  1.1 Introduction

Sasaki manifolds have been studied since the 1970s as an odd dimensional
counterpart to Kähler geometry. Similarly, $3$-Sasaki manifolds are considered
the $(4n+3)$-dimensional analogue to hyper-Kähler (hK) geometry. However,
while these geometries are linked via the hK cone of a $3$-Sasaki manifold,
$3$-Sasaki geometry also connects to another $4n$-dimensional geometry, namely
quaternionic Kähler (qK) manifolds. Initially shown in the regular case by
Ishihara and in full generality by C. Boyer, K. Galicki and B. Mann in ’94,
every $3$-Sasaki manifold locally admits a fibration over a qK orbifold
[BGM94]. This led to the classification of all homogeneous $3$-Sasaki
manifolds. The reverse construction is given by taking the Konishi bundle of a
positive scalar curvature qK space, i.e. the orthonormal frame bundle of the
quaternionic structure [Ko75]. For qK manifolds with negative scalar curvature
one does not obtain a $3$-Sasaki manifold but a so-called pseudo $3$-Sasaki
structure [Ta96]. This notion, however, did not gather as much traction since
it comes with a metric of semi-Riemannian signature $(4n,3)$.

More recently the first two authors investigated Riemannian almost $3$-contact
metric manifolds by means of connections with torsion [AD20]. They found
necessary and sufficient conditions for the existence of compatible
connections. Along their investigations, they discovered the more specific
class of $3$-$(\alpha,\delta)$-Sasaki manifolds connecting many examples on
which partial results were known previously. In particular, they showed that
pseudo $3$-Sasaki structures can be turned into _negative_
$3$-$(\alpha,\delta)$-Sasaki manifolds (i. e. with $\alpha\delta<0$).

This paper aims to connect both worlds and presents
$3$-$(\alpha,\delta)$-Sasaki geometry as the go-to structure above any qK
space. We quickly review all necessary notions involving
$3$-$(\alpha,\delta)$-Sasaki structures in Section 1. Using results by R.
Cleyton, A. Moroianu and U. Semmelmann [CMS18] we obtain a locally defined
Riemannian submersion over a qK space establishing the canonical connection as
the link between both geometries. This is done in Section 2. In the $3$-Sasaki
case we recover the result of Boyer, Galicki, Mann. We further show that the
scalar curvature on the base is a positive multiple of $\alpha\delta$. Thus,
for negative and degenerate $3$-$(\alpha,\delta)$-Sasaki manifolds we obtain
submersions onto qK spaces of negative scalar curvature, respectively hK
spaces. This suggests to investigate non-degenerate homogeneous
$3$-$(\alpha,\delta)$-Sasaki manifolds by looking at homogeneous qK manifolds
of non-vanishing scalar curvature. Section 3 is therefore devoted to a hands
on construction of homogeneous $3$-$(\alpha,\delta)$-Sasaki spaces over all
known homogeneous qK manifolds. This yields a construction over symmetric Wolf
spaces, deforming the description given in [DOP18] (see also [Bi96, Theorem
4]), and by similar means their non-compact duals. Additionally, homogeneous
$3$-$(\alpha,\delta)$-Sasaki manifolds over Alekseevsky spaces are constructed
using a description of the latter given by V. Cortés in [Co00]. We provide
detailed descriptions of the $7$-dimensional Aloff-Wallach space, its negative
counterpart fibering over the $4$-dimensional Wolf space
$\mathrm{SU}(3)/S(\mathrm{U}(2)\times\mathrm{U}(1))$, respectively its non-
compact dual, as well as of the homogeneous $3$-$(\alpha,\delta)$-Sasaki space
$\hat{\mathcal{T}}(1)$ in dimension $19$ sitting above the non-symmetric
Alekseevsky space $\mathcal{T}(1)$. In Section 4 we compute the Nomizu map
associated to the canonical connection, the necessary tool for any further
investigation of these spaces. In the symmetric base case we find the Nomizu
map of the Levi-Civita connection as well.

###  1.2 Review of $3$-$(\alpha,\delta)$-Sasaki manifolds and their basic
properties

We review some basic definitions and properties on almost contact metric
manifolds. This serves mainly as a reference.

An _almost contact metric structure_ on a $(2n+1)$-dimensional differentiable
manifold $M$ is a quadruple $(\varphi,\xi,\eta,g)$, where $\varphi$ is a
$(1,1)$-tensor field, $\xi$ a vector field, $\eta$ a $1$-form, $g$ a
Riemannian metric, such that

$\displaystyle\varphi^{2}=-I+\eta\otimes\xi,\quad\eta(\xi)=1,\quad\varphi(\xi)=0,\quad\eta\circ\varphi=0,$
$\displaystyle g(\varphi X,\varphi Y)=g(X,Y)-\eta(X)\eta(Y)\quad\forall
X,Y\in{\mathfrak{X}}(M).$

It follows that $\varphi$ has rank $2n$ and the tangent bundle of $M$ splits
as $TM=\mathcal{H}\oplus\langle\xi\rangle$, where $\mathcal{H}$ is the
$2n$-dimensional distribution defined by
$\mathcal{H}=\mathrm{Im}(\varphi)=\ker\eta=\langle\xi\rangle^{\perp}$. In
particular, $\eta=g(\cdot,\xi)$. The vector field $\xi$ is called the
_characteristic_ or _Reeb vector field_. The almost contact metric structure
is said to be _normal_ if
$N_{\varphi}\coloneqq[\varphi,\varphi]+d\eta\otimes\xi$ vanishes, where
$[\varphi,\varphi]$ is the Nijenhuis torsion of $\varphi$ [Bl10].

An _$\alpha$ -Sasaki manifold_ is defined as a normal almost contact metric
manifold such that $d\eta\,=\,2\alpha\Phi$, $\alpha\in\mathbb{R}^{*},$ where
$\Phi$ is the fundamental $2$-form defined by $\Phi(X,Y)=g(X,\varphi Y)$. For
$\alpha=1$, this is a _Sasaki manifold_. The $1$-form $\eta$ of an
$\alpha$-Sasaki structure is a _contact form_ , in the sense that
$\eta\wedge(d\eta)^{n}\neq 0$ everywhere on $M$. The Reeb vector field is
always Killing.

An _almost $3$-contact metric manifold_ is a differentiable manifold $M$ of
dimension $4n+3$ endowed with three almost contact metric structures
$(\varphi_{i},\xi_{i},\eta_{i},g)$, $i=1,2,3$, sharing the same Riemannian
metric $g$, and satisfying the following compatibility relations

$\varphi_{k}=\varphi_{i}\varphi_{j}-\eta_{j}\otimes\xi_{i}=-\varphi_{j}\varphi_{i}+\eta_{i}\otimes\xi_{j},\quad\xi_{k}=\varphi_{i}\xi_{j}=-\varphi_{j}\xi_{i},\quad\eta_{k}=\eta_{i}\circ\varphi_{j}=-\eta_{j}\circ\varphi_{i}$

for any even permutation $(ijk)$ of $(123)$ [Bl10]. The tangent bundle of $M$
splits into the orthogonal sum $TM=\mathcal{H}\oplus\mathcal{V}$, where
$\mathcal{H}$ and $\mathcal{V}$ are respectively the _horizontal_ and the
_vertical_ distribution, defined by

$\mathcal{H}\,\coloneqq\,\bigcap_{i=1}^{3}\ker\eta_{i},\qquad\mathcal{V}\,\coloneqq\,\langle\xi_{1},\xi_{2},\xi_{3}\rangle.$

In particular $\mathcal{H}$ has rank $4n$ and the three Reeb vector fields
$\xi_{1},\xi_{2},\xi_{3}$ are orthonormal. The manifold is said to be
_hypernormal_ if each almost contact metric structure
$(\varphi_{i},\xi_{i},\eta_{i},g)$ is normal. We denote an almost $3$-contact
metric manifold by $(M,\varphi_{i},\xi_{i},\eta_{i},g)$, understanding that
the index is running from $1$ to $3$.

One of the most interesting classes of almost $3$-contact metric manifolds is
given by _$3$ -$\alpha$-Sasaki manifolds_, for which each of the three
structures is $\alpha$-Sasaki. For $\alpha=1$, this is just the definition of
a _$3$ -Sasaki manifold_. As a comprehensive introduction to Sasaki and
$3$-Sasaki geometry, we refer to [BG08]. In the recent paper [AD20] the new
class of $3$-$(\alpha,\delta)$-Sasaki manifolds was introduced, generalizing
$3$-$\alpha$-Sasaki manifolds.

######  Definition 1.2.1.

An almost $3$-contact metric manifold $(M,\varphi_{i},\xi_{i},\eta_{i},g)$ is
called a _$3$ -$(\alpha,\delta)$-Sasaki manifold_ if it satisfies

$d\eta_{i}=2\alpha\Phi_{i}+2(\alpha-\delta)\eta_{j}\wedge\eta_{k}$ (1.1)

for every even permutation $(ijk)$ of $(123)$, where $\alpha\neq 0$ and
$\delta$ are real constants. A $3$-$(\alpha,\delta)$-Sasaki manifold is called
_degenerate_ if $\delta=0$ and _non-degenerate_ otherwise. Non-degenerate
$3$-$(\alpha,\delta)$-Sasaki manifolds will be distinguished into _positive_
and _negative_ ones, depending on whether $\alpha\delta>0$ or
$\alpha\delta<0$.

######  Remark 1.2.1.

Recall that the distinction into degenerate, positive, and negative
$3$-$(\alpha,\delta)$-Sasaki manifolds stems from their behaviour under
_$\mathcal{H}$ -homothetic deformations_ [AD20, Section 2.3]:

$\eta_{i}^{\prime}=c\eta_{i},\quad\xi_{i}^{\prime}=\frac{1}{c}\xi_{i},\quad\varphi_{i}^{\prime}=\varphi_{i},\quad
g^{\prime}=ag+b\sum_{i=1}^{3}\eta_{i}\otimes\eta_{i}\qquad a>0,\,c^{2}=a+b>0.$

The deformed structure
$(\varphi^{\prime},\xi_{i}^{\prime},\eta^{\prime},g^{\prime})$ turns out to be
$3$-$(\alpha^{\prime},\delta^{\prime})$-Sasaki with $\alpha^{\prime}=\alpha
c/a$, $\delta^{\prime}=\delta/c$. In particular, $\mathcal{H}$-homothetic
deformations preserve the class of degenerate $3$-$(\alpha,\delta)$-Sasaki
manifolds. In the non-degenerate case the sign of the product $\alpha\delta$
is also preserved, which justifies the distinction between the positive and
negative case stated in the definition above. In fact a
$3$-$(\alpha,\delta)$-Sasaki manifold is positive if and only if it is
$\mathcal{H}$-homothetic to a $3$-Sasaki manifold, and negative if and only if
it is $\mathcal{H}$-homothetic to a
$3$-$(\tilde{\alpha},\tilde{\delta})$-Sasaki manifold with
$\tilde{\alpha}=-\tilde{\delta}=1$.

We recall some basic properties of $3$-$(\alpha,\delta)$-Sasaki manifolds
whose proofs can be found in [AD20]. Any $3$-$(\alpha,\delta)$-Sasaki manifold
is shown to be hypernormal, thus generalizing Kashiwada’s theorem [Ka01].
Hence, for $\alpha=\delta$ one has a $3$-$\alpha$-Sasaki manifold. Each Reeb
vector field $\xi_{i}$ is Killing and it is an infinitesimal automorphism of
the horizontal distribution $\mathcal{H}$, i.e. $d\eta_{i}(X,\xi_{j})=0$ for
every $X\in\mathcal{H}$ and $i,j=1,2,3$. The vertical distribution
$\mathcal{V}$ is integrable with totally geodesic leaves. In particular, the
commutators of the Reeb vector fields are purely vertical and for every even
permutation $(ijk)$ of $(123)$ they are given by

$[\xi_{i},\xi_{j}]=2\delta\xi_{k}.$

Meanwhile, the vertical part of commutators of horizontal vector fields is
encoded by the fundamental form, as is shown in the following useful lemma:

######  Lemma 1.2.1.

For two horizontal vectors $X,Y\in\mathcal{H}$ we have

$[X,Y]_{\mathcal{V}}=-2\alpha\sum_{i=1}^{3}\Phi_{i}(X,Y)\xi_{i}.$

###### Proof.

Since the vertical distribution is spanned by the Reeb vector fields, we have

$[X,Y]_{\mathcal{V}}=\sum_{i=1}^{3}\eta_{i}([X,Y])\xi_{i}=-\sum_{i=1}^{3}d\eta_{i}(X,Y)\xi_{i}=-2\alpha\sum_{i=1}^{3}\Phi_{i}(X,Y)\xi_{i}.\qed$

By the same argument $[X,Y]_{\mathcal{V}}=0$ if $X\in\mathcal{H}$ and
$Y=\xi_{j}$, $j=1,2,3$, which is equivalent to the fact that
$\mathrm{d}\eta_{i}(X,\xi_{j})=0$, $i=1,2,3$.

A remarkable property of $3$-$(\alpha,\delta)$-Sasaki manifolds is that they
are _canonical almost $3$-contact metric manifolds_, in the sense of [AD20],
which is equivalent to the existence of a _canonical connection_.

We recall here some basic facts about connections with totally skew-symmetric
torsion—we refer to [Ag06] for further details. A metric connection $\nabla$
with torsion $T$ on a Riemannian manifold $(M,g)$ is said to have _totally
skew-symmetric torsion_ , or _skew torsion_ for short, if the $(0,3)$-tensor
field $T$ defined by

$T(X,Y,Z)=g(T(X,Y),Z)$

is a $3$-form. The relation between $\nabla$ and the Levi-Civita connection
$\nabla^{g}$ is then given by

$\nabla_{X}Y=\nabla^{g}_{X}Y+\frac{1}{2}T(X,Y).$

It is well-known that any Sasaki manifold $(M,\varphi,\xi,\eta,g)$ admits a
_characteristic connection_ , i. e. a unique metric connection $\nabla$ with
skew torsion such that $\nabla\eta=\nabla\varphi=0$. Its torsion is given by
$T=\eta\wedge d\eta$ [FI02]. As a consequence, a $3$-Sasaki manifold
$(M,\varphi_{i},\xi_{i},\eta_{i},g)$ cannot admit any metric connection with
skew torsion such that $\nabla\eta_{i}=\nabla\varphi_{i}=0$ for every
$i=1,2,3$. By relaxing the requirement on the parallelism of the structure
tensor fields in a suitable way, one can define a large class of almost
$3$-contact metric manifolds, called _canonical_ , including
$3$-$(\alpha,\delta)$-Sasaki manifolds, and thus $3$-Sasaki manifolds.

Any $3$-$(\alpha,\delta)$-Sasaki manifold $(M,\varphi_{i},\xi_{i},\eta_{i},g)$
is canonical, in the sense that it admits a unique metric connection $\nabla$
with skew torsion such that

$\nabla_{X}\varphi_{i}\,=\,\beta(\eta_{k}(X)\varphi_{j}-\eta_{j}(X)\varphi_{k})\quad\forall
X\in{\mathfrak{X}}(M)$ (1.2)

for every even permutation $(ijk)$ of $(123)$, where
$\beta=2(\delta-2\alpha)$. The covariant derivatives of the other structure
tensor fields are given by

$\nabla_{X}\xi_{i}=\beta(\eta_{k}(X)\xi_{j}-\eta_{j}(X)\xi_{k}),\qquad\nabla_{X}\eta_{i}=\beta(\eta_{k}(X)\eta_{j}-\eta_{j}(X)\eta_{k}).$

If $\delta=2\alpha$, then $\beta=0$ and the canonical connection parallelizes
all the structure tensor fields. Any $3$-$(\alpha,\delta)$-Sasaki manifold
with $\delta=2\alpha$ is called _parallel_. Notice that this is a positive
$3$-$(\alpha,\delta)$-Sasaki manifold.

The torsion $T$ of the canonical connection is given by

$T\ =\ 2\alpha\sum_{i=1}^{3}\eta_{i}\wedge\Phi_{i}-2(\alpha-\delta)\eta_{123}\
=\
2\alpha\sum_{i=1}^{3}\eta_{i}\wedge\Phi^{\mathcal{H}}_{i}+2(\delta-4\alpha)\,\eta_{123},$
(1.3)

where $\Phi^{\mathcal{H}}_{i}=\Phi_{i}+\eta_{jk}\in\Lambda^{2}({\mathcal{H}})$
is the horizontal part of the fundamental $2$-form $\Phi_{i}$. Here we put
$\eta_{jk}\coloneqq\eta_{j}\wedge\eta_{k}$ and
$\eta_{123}\coloneqq\eta_{1}\wedge\eta_{2}\wedge\eta_{3}$. In particular, for
every $X,Y\in\mathfrak{X}(M)$,

$T(X,Y)=2\alpha\sum_{i=1}^{3}\\{\eta_{i}(Y)\varphi_{i}X-\eta_{i}(X)\varphi_{i}Y+\Phi_{i}(X,Y)\xi_{i}\\}-2(\alpha-\delta)\stackrel{{\scriptstyle\scriptsize
i,j,k}}{{\mathfrak{S}}}\eta_{ij}(X,Y)\xi_{k}.$ (1.4)

The symbol $\stackrel{{\scriptstyle\scriptsize i,j,k}}{{\mathfrak{S}}}$ means
the sum over all even permutations of $(123)$. The torsion of the canonical
connection satisfies $\nabla T=0$. The curvature properties of
$3$-$(\alpha,\delta)$-Sasaki manifolds will be discussed in detail in a
separate publication [ADS21]. We cite from there without proof the following
special result that will be needed in the following section. It is a side
result of a lengthy and non-trivial, but otherwise straightforward
computation.

######  Proposition 1.2.1 ([ADS21]).

The curvature tensor $R$ of the canonical connection of a
$3$-$(\alpha,\delta)$-Sasaki manifold satisfies for any $X,Y,Z\in\mathcal{H}$
and $i,j,k,l=1,2,3$ the identities

$\displaystyle R(X,\xi_{i},Y,\xi_{j})$
$\displaystyle=R(X,Y,Z,\xi_{i})=R(\xi_{i},\xi_{j},\xi_{k},X)=0,$ (1.5)
$\displaystyle R(\xi_{i},\xi_{j},\xi_{k},\xi_{l})$
$\displaystyle=-4\alpha\beta(\delta_{ik}\delta_{jl}-\delta_{il}\delta_{jk}),$
(1.6) $\displaystyle R(\xi_{i},\xi_{j},X,Y)$
$\displaystyle=2\alpha\beta\Phi_{k}(X,Y),$ (1.7) $\displaystyle
R(X,Y,Z,\varphi_{i}Z)+R(X,Y,\varphi_{j}Z,\varphi_{k}Z)$
$\displaystyle=2\alpha\beta\Phi_{i}(X,Y)\|Z\|^{2},$ (1.8)

where in the last two identities $(ijk)$ is an even permutation of $(123)$.

##  2 The Riemannian submersion over a quaternionic Kähler base

###  2.1 The canonical submersion

In [CMS18] the authors discuss the geometry of Riemannian manifolds admitting
metric connections $\nabla^{\tau}$ with parallel skew torsion $\tau$ and
reducible holonomy. This applies, in particular, to the canonical connection
of $3$-$(\alpha,\delta)$-Sasaki manifolds. We shortly recall their notation.

Suppose the tangent space $TM$ decomposes under the action of the holonomy
group $\mathrm{Hol}$ of $\nabla^{\tau}$ into a sum of irreducible
representations
$\mathfrak{v}_{1},\dots,\mathfrak{v}_{r},\mathfrak{h}_{1},\dots,\mathfrak{h}_{s}$.
Here an irreducible submodule is called vertical, adequately denoted by
$\mathfrak{v}_{j}$, if the subspace of $\mathfrak{hol}$ acting purely on
$\mathfrak{v}_{j}$ is trivial. Conversely, a subspace $\mathfrak{h}_{a}$ is
called horizontal if the subspace
$\mathfrak{k}_{a}=\mathfrak{so}(\mathfrak{h}_{a})\cap\mathfrak{hol}\neq\\{0\\}$
of $\mathfrak{hol}$ acting purely on $\mathfrak{h}_{a}$ is non-trivial.

We need a slight generalization of the results obtained in [CMS18]. Suppose
the tangent space decomposes into
$TM=\mathfrak{v}_{1}\oplus\dots\oplus\mathfrak{v}_{r}\oplus\mathfrak{h}_{1}\oplus\dots\oplus\mathfrak{h}_{s}$
as before. Let $TM=\mathcal{V}_{\Gamma}\oplus\mathcal{H}_{\Gamma}$ be a
decomposition such that

$\mathcal{H}_{\Gamma}\coloneqq\bigoplus_{a=1}^{s}\mathfrak{h}_{a}\oplus\bigoplus_{j\in\Gamma_{0}\setminus\Gamma}\mathfrak{v}_{j},\qquad\qquad\mathcal{V}_{\Gamma}\coloneqq\bigoplus_{j\in\Gamma}\mathfrak{v}_{j},$
(2.1)

for some subset $\Gamma\subset\Gamma_{0}=\\{1,\dots,r\\}$. Suppose further
that for this decomposition the projection of $\tau$ onto the space
$\mathcal{H}_{\Gamma}\otimes\Lambda^{2}\mathcal{V}_{\Gamma}$ satisfies

$0=\mathrm{pr}_{\mathcal{H}_{\Gamma}\otimes\Lambda^{2}\mathcal{V}_{\Gamma}}\tau\in\mathcal{H}_{\Gamma}\otimes\Lambda^{2}\mathcal{V}_{\Gamma}\subset\Lambda^{3}(\mathcal{H}_{\Gamma}\oplus\mathcal{V}_{\Gamma}).$
(2.2)

This condition turns out to be sufficient to prove Lemma 3.7-3.10 and Remark
3.11 from [CMS18]. We obtain

######  Corollary 2.1.1.

Suppose the decomposition $TM=\mathcal{V}_{\Gamma}\oplus\mathcal{H}_{\Gamma}$
from (2.1) fulfills condition (2.2). Then

  1. a)

the distribution $\mathcal{V}_{\Gamma}$ is the vertical distribution of a
locally defined Riemannian submersion
$(M,g)\overset{\pi}{\longrightarrow}(N,g_{N})$ with totally geodesic leaves,

  2. b)

there exists a $3$-form $\sigma\in\Lambda^{3}N$ satisfying
$\pi^{*}\sigma=\mathrm{pr}_{\Lambda^{3}\mathcal{H}_{\Gamma}}\tau$,

  3. c)

$\nabla^{\sigma}\coloneqq\nabla^{g_{N}}+\frac{1}{2}\sigma$ defines a
connection with parallel skew torsion $\sigma$ on $N$. In particular, we have

$\nabla^{\sigma}_{X}Y=\pi_{*}(\nabla^{\tau}_{\overline{X}}\overline{Y}),$
(2.3)

for the horizontal lifts $\overline{X},\overline{Y}\in TM$ of the vectors
fields $X,Y\in TN$.

Equation (2.3) is not stated explicitely in [CMS18] but follows directly from
$\nabla^{g_{N}}_{X}{Y}=\pi_{*}(\nabla^{g}_{\overline{X}}\overline{Y})$ for
Riemannian submersions [Pe06, Prop. 13]. To a Riemannian submersion one
assigns the O’Neill tensors

$\mathcal{A}_{X}Y=(\nabla^{g}_{X_{\mathcal{H}}}Y_{\mathcal{H}})_{\mathcal{V}}+(\nabla^{g}_{X_{\mathcal{H}}}Y_{\mathcal{V}})_{\mathcal{H}},\qquad\mathcal{T}_{X}Y=(\nabla^{g}_{X_{\mathcal{V}}}Y_{\mathcal{H}})_{\mathcal{V}}+(\nabla^{g}_{X_{\mathcal{V}}}Y_{\mathcal{V}})_{\mathcal{H}}.$

Here the subscripts denote projection on the respective subspaces. For the
submersion above $\mathcal{A}$ and $\mathcal{T}$ simplify:

######  Lemma 2.1.1.

The O’Neill tensors $\mathcal{A}$ and $\mathcal{T}$ associated to the
submersion defined by $TM=\mathcal{V}_{\Gamma}\oplus\mathcal{H}_{\Gamma}$ are
given by

$g(\mathcal{A}_{X}Y,Z)=-\frac{1}{2}(\tau(X_{\mathcal{H}_{\Gamma}},Y_{\mathcal{H}_{\Gamma}},Z_{\mathcal{V}_{\Gamma}})+\tau(X_{\mathcal{H}_{\Gamma}},Y_{\mathcal{V}_{\Gamma}},Z_{\mathcal{H}_{\Gamma}})),\qquad\mathcal{T}=0.$

###### Proof.

Since $\mathcal{H}_{\Gamma}$ and $\mathcal{V}_{\Gamma}$ are
$\nabla^{\tau}$-holonomy invariant
$(\nabla^{\tau}_{X}Y_{\mathcal{H}_{\Gamma}})_{\mathcal{V}_{\Gamma}}=(\nabla^{\tau}_{X}Y_{\mathcal{V}_{\Gamma}})_{\mathcal{H}_{\Gamma}}=0$.
Thus,
$g(\nabla^{g}_{X}Y_{\mathcal{H}_{\Gamma}},Z_{\mathcal{V}_{\Gamma}})=-\frac{1}{2}\tau(X,Y_{\mathcal{H}_{\Gamma}},Z_{\mathcal{V}_{\Gamma}})$
and
$g(\nabla^{g}_{X}Y_{\mathcal{V}_{\Gamma}},Z_{\mathcal{H}_{\Gamma}})=-\frac{1}{2}\tau(X,Y_{\mathcal{V}_{\Gamma}},Z_{\mathcal{H}_{\Gamma}})$.
The first expression follows directly. The identity $\mathcal{T}=0$ is then an
immediate consequence of condition (2.2). ∎

The vanishing of $\mathcal{T}$ does not come as a surprise since it is
equivalent to the fibers being totally geodesic.

We now discuss the situation for $3$-$(\alpha,\delta)$-Sasaki manifolds. By
(1.2) the holonomy representation of the canonical connection $\nabla$ of a
$3$-$(\alpha,\delta)$-Sasaki manifold splits into the horizontal and vertical
subspaces $\mathcal{H}$ and $\mathcal{V}$. In the non-parallel case
$\mathcal{V}$ is irreducible, in the parallel case it decomposes into $3$
trivial $1$-dimensional representations. In either case the curvature
properties stated in Proposition 1.2.1 allow us to prove:

######  Lemma 2.1.1.

The vertical distribution $\mathcal{V}$ of a $3$-$(\alpha,\delta)$-Sasaki
manifold is vertical with respect to the above notation.

###### Proof.

By the Ambrose-Singer Theorem the holonomy algebra $\mathfrak{hol}$ of the
holonomy group $\mathrm{Hol}(p)$ at a point $p$ is given by

$\mathfrak{hol}=\\{P^{-1}_{\gamma}\circ R(P_{\gamma}X,P_{\gamma}Y)\circ
P_{\gamma}\ |\ \gamma\ \text{some path from $p$ to $q$,}\ X,Y\in
T_{p}M\\}\subset\mathfrak{so}(T_{p}M)$

where $P_{\gamma}$ denotes parallel transport along $\gamma$ and
$R(X,Y)\in\mathfrak{so}(T_{q}M)$ the curvature operator. The horizontal and
vertical distribution are invariant under parallel transport with respect to
the canonical connection. Thus, we may assume $\gamma$ to be trivial when
investigating the holonomy action on these distributions. By (1.5) we know
that the holonomy is only non-trivial if $X,Y\in\mathcal{V}$ or
$X,Y\in\mathcal{H}$. In the first case (1.6) and (1.7) show that every element
of $\mathfrak{hol}$ acting non-trivially on $\mathcal{V}$ must also act non-
trivially on $\mathcal{H}$. The action of an element $R(X,Y)$,
$X,Y\in\mathcal{H}$, on $\mathcal{V}$ is again given by (1.7). Any such
element of $\mathfrak{hol}$ acts non-trivially on $\mathcal{V}$ if $\beta\neq
0$ and $\Phi_{i}(X,Y)\neq 0$ for some $i=1,2,3$. In this case $R(X,Y)$ is also
a non-trivial operator on $\mathcal{H}$ by (1.8). ∎

######  Proposition 2.1.1.

The decomposition $TM=\mathcal{H}\oplus\mathcal{V}$ of a
$3$-$(\alpha,\delta)$-Sasaki manifold $M$ satisfies the conditions in
Corollary 2.1.1. In particular, there exists a locally defined Riemannian
submersion $\pi\colon M\to N$ such that

$\nabla^{g_{N}}_{X}Y=\pi_{*}(\nabla_{\overline{X}}\overline{Y}).$ (2.4)

######  Definition 2.1.1.

We will call $\pi\colon M\to N$ the _canonical submersion_ of a
$3$-$(\alpha,\delta)$-Sasaki manifold.

###### Proof (of Proposition 2.1.1).

By (1.2) and Lemma 2.1.1 the decomposition $TM=\mathcal{V}\oplus\mathcal{H}$
is of type (2.1) with respect to the canonical connection $\nabla$. By (1.3)
the projection of the torsion onto $\mathcal{H}\otimes\Lambda^{2}\mathcal{V}$
vanishes, satisfying (2.2). Therefore the conditions of Corollary 2.1.1 are
satisfied. Moreover, (1.3) shows that the projection of $\tau$ onto
$\Lambda^{3}\mathcal{H}$ vanishes so the connection $\nabla^{\sigma}$ in (2.3)
for the canonical submersion is the Levi-Civita connection $\nabla^{g_{N}}$ on
$N$. ∎

We observe that the canonical submersion is, indeed, an almost contact metric
$3$-submersion in the sense of [Wa84], although we never make explicity use of
this property (our formulas are much more detailed than the general results
obtained therein).

###  2.2 The quaternionic Kähler structure on the base

We give a preliminary lemma needed to prove that the base of the canonical
submersion admits a qK structure. Recall that a basic vector field on $M$ is a
horizontal vector field which is projectable, that is $\pi$-related to some
vector field defined on $N$. If $X\in TN$, the horizontal lift of $X$ is the
unique basic vector field $\overline{X}\in TM$ such that
$\pi_{*}\overline{X}=X$.

######  Lemma 2.2.1.

For any vertical vector field $X\in\mathcal{V}$ and for any basic vector field
$Y\in\mathcal{H}$ we have

$(\nabla_{X}Y)_{\mathcal{H}}=-2\alpha\sum_{i=1}^{3}\eta_{i}(X)\varphi_{i}Y.$

###### Proof.

We first use the identity $g(\nabla^{g}_{X}Y,Z)=-\frac{1}{2}g([Y,Z],X)$ for
any vector fields $X\in\mathcal{V},Y,Z\in\mathcal{H}$, with $Y$ and $Z$
projectable, of a Riemannian submersion [Pe06, Proposition $13$]. Note that
the horizontal and vertical distributions of the Riemannian submersion agree
with the same notion in the $3$-$(\alpha,\delta)$-Sasaki setting. Further, we
make use of Lemma 1.2.1 to obtain

$g(\nabla^{g}_{\xi_{i}}Y,Z)=-\frac{1}{2}g([Y,Z],\xi_{i})=\alpha\Phi_{i}(Y,Z).$

Therefore

$\displaystyle g(\nabla_{X}Y,Z)$
$\displaystyle=g(\nabla^{g}_{X}Y,Z)+\frac{1}{2}T(X,Y,Z)=\sum_{i=1}^{3}\eta_{i}(X)(\alpha\Phi_{i}(Y,Z)+\alpha\Phi_{i}(Y,Z))$
$\displaystyle=-2\alpha\sum_{i=1}^{3}\eta_{i}(X)g(\varphi_{i}Y,Z).\qed$

######  Theorem 2.2.1.

The base $N$ of the canonical submersion $\pi\colon M\to N$ of any
$3$-$(\alpha,\delta)$-Sasaki manifold $M$ carries a quaternionic Kähler
structure given by

$\check{\varphi_{i}}=\pi_{*}\circ\varphi_{i}\circ s_{*},\quad i=1,2,3,$

where $s\colon U\to M$ is any local smooth section of $\pi$. The covariant
derivatives of the almost complex structures $\check{\varphi}_{i}$ are given
by

$\nabla^{g_{N}}_{X}\check{\varphi}_{i}=2\delta(\check{\eta}_{k}(X)\check{\varphi}_{j}-\check{\eta}_{j}(X)\check{\varphi}_{k}),$

where $\check{\eta}_{i}(X)=\eta_{i}(s_{*}X)$ for $i=1,2,3$.

###### Proof.

Let $s$ be a local section of the canonical submersion $\pi\colon M\to N$,
hence $\pi_{*}\circ s_{*}=\mathrm{id}$ and
$\operatorname{Im}(s_{*}\circ\pi_{*}-\mathrm{id})\subset\mathcal{V}$ on the
image $s(N)\subset M$. Define

$\check{\varphi}_{i}=\pi_{*}\circ\varphi_{i}\circ s_{*}$

for $i=1,2,3$. The horizontal and vertical distributions, $\mathcal{H}$ and
$\mathcal{V}$, are invariant under $\varphi_{i}$. Thus,
$\pi_{*}\circ\varphi_{i}=\check{\varphi_{i}}\circ\pi_{*}$ on $s(N)$. This
yields

$\displaystyle\check{\varphi}_{i}\check{\varphi}_{j}$
$\displaystyle=\check{\varphi}_{i}\circ(\pi_{*}\circ\varphi_{j}\circ
s_{*})=\pi_{*}\circ(\varphi_{i}\varphi_{j})\circ s_{*}.$

Now use that $(\varphi_{i}|_{\mathcal{H}})^{2}=-\mathrm{id}|_{\mathcal{H}}$
and
$(\varphi_{i}|_{\mathcal{H}})(\varphi_{j}|_{\mathcal{H}})=\pm\varphi_{k}|_{\mathcal{H}}$
with sign $\pm$ depending on whether $(ijk)$ is an even or odd permutation of
$(123)$. This shows $\check{\varphi}_{i}^{2}=-\mathrm{id}$ and
$\check{\varphi}_{i}\check{\varphi}_{j}=\pm\check{\varphi}_{k}$.

Finally, by means of (2.4) and (1.2), we show that the quaternionic structure
is parallel. First

$\displaystyle(\nabla^{g_{N}}_{X}\check{\varphi}_{i})Y$
$\displaystyle=(\nabla^{g_{N}}_{X}(\check{\varphi}_{i}Y))-(\check{\varphi}_{i}(\nabla^{g_{N}}_{X}Y))=\pi_{*}\nabla_{\overline{X}}\overline{(\check{\varphi}_{i}Y)}-\check{\varphi}_{i}\left(\pi_{*}(\nabla_{\overline{X}}\overline{Y})\right)$
$\displaystyle=\pi_{*}\nabla_{\overline{X}}\overline{\left(\pi_{*}\left(\varphi_{i}(s_{*}Y)\right)\right)}-\pi_{*}\left(\varphi_{i}\left(s_{*}\left(\pi_{*}\left(\nabla_{\overline{X}}\overline{Y}\right)\right)\right)\right).$

By the properties of any Riemannian submersion we have that
$\overline{\left(\pi_{*}\left(\varphi_{i}(s_{*}Y)\right)\right)}=(\varphi_{i}(s_{*}Y))_{\mathcal{H}}$
wherever the right side is defined, that is on the image $s(N)\subset M$.
Thus, we take the covariant derivatives in the direction of $s_{*}X$ resulting
in a vertical correction term $\hat{X}=\overline{X}-s_{*}X\in\mathcal{V}$.
Recall that $\nabla$ and $\varphi_{i}$ preserve the horizontal and the
vertical distribution. Using Lemma 2.2.1, we obtain

$\displaystyle\nabla_{\overline{X}}\overline{\left(\pi_{*}\left(\varphi_{i}(s_{*}Y)\right)\right)}$
$\displaystyle=\nabla_{s_{*}X}(\varphi_{i}(s_{*}Y))_{\mathcal{H}}+\nabla_{\hat{X}}(\overline{\left(\pi_{*}\left(\varphi_{i}(s_{*}Y)\right)\right)})$
$\displaystyle=\left(\nabla_{s_{*}X}\left(\varphi_{i}(s_{*}Y)\right)\right)_{\mathcal{H}}-2\alpha\sum_{l=1}^{3}\eta_{l}(\hat{X})\varphi_{l}\left(\varphi_{i}(s_{*}Y)\right)_{\mathcal{H}}.$

For the second summand, the horizontal projection is given by

$\displaystyle\left(\varphi_{i}\left(s_{*}\left(\pi_{*}\left(\nabla_{\overline{X}}\overline{Y}\right)\right)\right)\right)_{\mathcal{H}}$
$\displaystyle=\varphi_{i}\left(s_{*}\left(\pi_{*}\left(\nabla_{\overline{X}}\overline{Y}\right)\right)\right)_{\mathcal{H}}$
$\displaystyle=\varphi_{i}\left(\nabla_{s_{*}X}(s_{*}Y)\right)_{\mathcal{H}}+\varphi_{i}\nabla_{\hat{X}}(\overline{Y})$
$\displaystyle=\left(\varphi_{i}\left(\nabla_{s_{*}X}(s_{*}Y)\right)\right)_{\mathcal{H}}-2\alpha\sum_{l=1}^{3}\eta_{l}(\hat{X})\varphi_{i}(\varphi_{l}(s_{*}Y))_{\mathcal{H}}.$

Recombining both identities we obtain

$\displaystyle(\nabla^{g_{N}}_{X}\check{\varphi}_{i})Y$
$\displaystyle=\pi_{*}\big{(}\nabla_{s_{*}X}\left(\varphi_{i}(s_{*}Y)\right)-\varphi_{i}\left(\nabla_{s_{*}X}(s_{*}Y)\right)+2\alpha\sum_{l=1}^{3}\eta_{l}(\hat{X})(\varphi_{i}\varphi_{l}(s_{*}Y)_{\mathcal{H}}-\varphi_{l}\varphi_{i}(s_{*}Y)_{\mathcal{H}})\big{)}$
$\displaystyle=\pi_{*}\big{(}(\nabla_{s_{*}X}\varphi_{i})s_{*}Y-2\alpha\sum_{l=1}^{3}\eta_{l}(s_{*}X)(\varphi_{i}\varphi_{l}(s_{*}Y)_{\mathcal{H}}-\varphi_{l}\varphi_{i}(s_{*}Y)_{\mathcal{H}})\big{)}$
$\displaystyle=(\beta+4\alpha)\big{(}(\eta_{k}(s_{*}X)\circ
s)\pi_{*}(\varphi_{j}(s_{*}Y))-(\eta_{j}(s_{*}X)\circ
s)\pi_{*}(\varphi_{k}(s_{*}Y))\big{)}$
$\displaystyle=2\delta(\check{\eta}_{k}(X)\check{\varphi}_{j}-\check{\eta}_{j}(X)\check{\varphi}_{k})Y.$

Here we used the defining identity (1.2) of the canonical connection for any
even permutation $(ijk)$ of $(123)$. Therefore, the quaternionic structure is
parallel and $N$ is quaternionic Kähler. ∎

######  Remark 2.2.1.

A priori the quaternionic structure may depend on the chosen section $s$.
Indeed, the individual almost complex structures $\check{\varphi}_{i}$ vary
with $s$. However, following the work of P. Piccinni and I. Vaisman [PV01],
one can see that the quaternionic structure is preserved under the Bott
connection $\mathring{D}\colon\mathcal{V}\times\mathcal{H}\to\mathcal{H}$
defined by $\mathring{D}_{V}X=[V,X]_{\mathcal{H}}$, since

$(\mathring{D}_{\xi_{i}}\varphi_{j})X=[\xi_{i},\varphi_{j}X]_{\mathcal{H}}-\varphi_{j}[\xi_{i},X]_{\mathcal{H}}=((\mathcal{L}_{\xi_{i}}\varphi_{j})X)_{\mathcal{H}}=2\delta\epsilon_{ijk}\varphi_{k}X.$

This implies that the quaternionic structure is projectable and, thus,
independent of choices.

######  Corollary 2.2.1.

A $3$-$(\alpha,\delta)$-Sasaki manifold fibers locally over a hyperkähler
manifold if it is degenerate.

######  Remark 2.2.2.

Apart from the degenerate case the induced quaternionic Kähler structure is
hyperkähler if and only if $s_{*}X\in\mathcal{H}$ for all $X\in TN$. Such a
section exists if and only if the horizontal distribution is tangent to $s(N)$
and, thus, integrable. This is in contrast to Lemma 1.2.1 for any
$3$-$(\alpha,\delta)$-Sasaki manifold.

We can now relate the curvature of $N$ with that of $M$.

######  Theorem 2.2.2.

Let $\pi\colon M\to N$ be the canonical submersion of a
$3$-$(\alpha,\delta)$-Sasaki manifold. Then

$\mathrm{scal}_{g_{N}}=16n(n+2)\alpha\delta.$

###### Proof.

By Lemma 2.1.1 and (1.3) for $X,Y\in\mathcal{H}$

$\mathcal{A}_{X}Y=-\frac{1}{2}T(X,Y)_{\mathcal{V}}=-\alpha\sum_{i=1}^{3}\Phi_{i}(X,Y)\xi_{i}.$

Let $e_{1},\dots,e_{4n}$ be a local adapted frame for $\mathcal{H}$, i.e. an
orthonormal frame such that $\varphi_{1}e_{4p+1}=e_{4p+2}$,
$\varphi_{2}e_{4p+1}=e_{4p+3}$ and $\varphi_{3}e_{4p+1}=e_{4p+4}$, $0\leq
p\leq n-1$. Then

$\sum_{i,j=1}^{4n}g(A_{e_{i}}e_{j},A_{e_{i}}e_{j})=\alpha^{2}\sum_{i,j=1}^{4n}\sum_{k=1}^{3}\Phi^{2}_{k}(e_{i},e_{j})=\alpha^{2}\cdot
3\cdot 4n=12n\alpha^{2}.$

From the O’Neill identities one obtains for submersions with $\mathcal{T}=0$
the Ricci curvature identity [Be87, Proposition 9.36]

$\mathrm{Ric}_{g}(X,Y)=\pi^{*}\mathrm{Ric}_{g_{N}}(X,Y)-2\sum_{j=1}^{4n}g(A_{X}e_{j},A_{Y}e_{j}).$

The Ricci curvature of $M$ is
$\mathrm{Ric}_{g}=2\alpha(2\delta(n+2)-3\alpha)g+2(\alpha-\delta)((2n+3)\alpha-\delta)g|_{\mathcal{V}}$
by [AD20, Proposition 2.3.3]. Combining both identities we have

$\displaystyle\mathrm{scal}_{g_{N}}$
$\displaystyle=\sum_{i=1}^{4n}\mathrm{Ric}_{g_{N}}(\pi_{*}e_{i},\pi_{*}e_{i})=\sum_{i=1}^{4n}\mathrm{Ric}_{g}(e_{i},e_{i})+2\sum_{i,j=1}^{4n}g(A_{e_{i}}e_{j},A_{e_{i}}e_{j})$
$\displaystyle=4n\cdot
2\alpha(2\delta(n+2)-3\alpha)+24n\alpha^{2}=16n(n+2)\alpha\delta.\qed$

######  Remark 2.2.3.

In particular, we recover the scalar curvature result
$\mathrm{scal}_{g_{N}}=16n(n+2)$ known in the $3$-Sasaki case [BG08, Theorem
13.3.13].

##  3 Construction of non-degenerate homogeneous $3$-$(\alpha,\delta)$-Sasaki
manifolds

For homogeneous $3$-$(\alpha,\delta)$-Sasaki manifolds the canonical
submersion is invariant. Hence, the base $N$ is a homogeneous qK space. In the
non-degenerate case Theorem 2.2.2 shows that $N$ is a homogeneous quaternionic
Kähler space of non-vanishing scalar curvature. There are two families of such
spaces known: Compact qK symmetric spaces, named Wolf spaces, their non-
compact duals and Alekseevsky spaces. The latter are homogeneous qK spaces
admitting a solvable transitive group action. D. Alekseevsky conjectured that
all homogeneous qK spaces with negative scalar curvature are Alekseevsky
spaces [Ale75]. In particular, the class of non-compact qK symmetric spaces is
included in the class of Alekseevsky. We will give independent constructions
of homogeneous $3$-$(\alpha,\delta)$-Sasaki manifolds over symmetric base
spaces and such fibering over Alekseevsky spaces.

###  3.1 Homogeneous $3$-$(\alpha,\delta)$-Sasaki manifolds over symmetric
quaternionic Kähler spaces

Let $G/G_{0}$ be a real symmetric space, i.e.
$\mathfrak{g}=\mathfrak{g}_{0}\oplus\mathfrak{g}_{1}$ with
$[\mathfrak{g}_{i},\mathfrak{g}_{j}]\subset\mathfrak{g}_{i+j}$ on the level of
Lie algebras. Suppose there exists a connected subgroup $H\subset G_{0}$ such
that $\mathfrak{g}_{0}$ splits into a direct sum of Lie algebras
$\mathfrak{g}_{0}=\mathfrak{h}\oplus\mathfrak{sp}(1)$. Finally, assume that
$\mathfrak{g}_{1}^{\mathbb{C}}=\mathbb{C}^{2}\otimes_{\mathbb{C}}W$, for some
$\mathfrak{h}^{\mathbb{C}}$-module $W$ of $\dim_{\mathbb{C}}W=2n$, and the
adjoint action of $\mathfrak{g}_{0}^{\mathbb{C}}$ is given by

$\mathfrak{h}^{\mathbb{C}}\oplus\mathfrak{sp}(1)^{\mathbb{C}}\ni(A,B)\cdot((z_{1},z_{2})\otimes
w)=B(z_{1},z_{2})\otimes w+(z_{1},z_{2})\otimes Aw,$

where
$\mathfrak{sp}(1)^{\mathbb{C}}=\mathfrak{su}(2)^{\mathbb{C}}=\mathfrak{sl}(2,\mathbb{C})$
acts by multiplication on $\mathbb{C}^{2}$. We will call $(G,G_{0},H)$
_generalized 3-Sasaki data_.

######  Remark 3.1.1.

  1. a)

For compact $G$ this is called $3$-Sasaki data in [DOP18, Definition 12, p.
12].

  2. b)

Consider the homogeneous space $M=G/H$. The assumptions above imply that
$\mathfrak{g}=\mathfrak{h}\oplus\mathfrak{m}$ with
$\mathfrak{m}=\mathfrak{sp}(1)\oplus\mathfrak{g}_{1}$ is a reductive
decomposition. We rename the spaces $\mathcal{V}=\mathfrak{sp}(1)$ and
$\mathcal{H}=\mathfrak{g}_{1}$ to express their role as vertical and
horizontal subspaces of a $3$-$(\alpha,\delta)$-Sasaki manifold via
$T_{p}M\cong\mathfrak{m}$. For clarity we restate the bracket relations
between all these spaces. We have
$\mathfrak{g}=\mathfrak{h}\oplus\mathcal{V}\oplus\mathcal{H}$, where
$\mathfrak{h}$ and $\mathcal{V}$ are commuting subalgebras. Thus they form the
joint subalgebra
$\mathfrak{h}\oplus\mathcal{V}=\mathfrak{g}_{0}\subset\mathfrak{g}$. The full
set of commutator relations is

$\displaystyle[\mathfrak{h},\mathfrak{h}]\subset\mathfrak{h}\qquad[\mathcal{V},\mathcal{V}]\subset\mathcal{V}\qquad[\mathfrak{h},\mathcal{V}]=0$
$\displaystyle[\mathfrak{h},\mathcal{H}]\subset\mathcal{H}\qquad[\mathcal{V},\mathcal{H}]\subset\mathcal{H}\qquad[\mathcal{H},\mathcal{H}]\subset\mathcal{V}\oplus\mathfrak{h}.$

In particular, both $\mathcal{V}$ and $\mathcal{H}$ are
$\mathfrak{h}$-invariant.

  3. c)

Since $G/G_{0}$ is a symmetric space there exists a dual symmetric space
$G^{*}/G_{0}$ for every generalized $3$-Sasaki data $(G,G_{0},H)$. The Lie
algebras can then be identified as

$\mathfrak{g}^{*}=\mathfrak{h}\oplus\mathcal{V}\oplus
i\mathcal{H}\subset\mathfrak{g}^{\mathbb{C}}.$ (3.1)

It is then clear that $(G^{*},G_{0},H)$ is generalized $3$-Sasaki data as
well. This yields pairs of compact and non-compact generalized $3$-Sasaki
data. For clarity we will denote the compact top Lie group by $G$ and the non-
compact one by $G^{*}$.

  4. d)

By [DOP18] any $3$-Sasaki data gives rise to a homogeneous $3$-Sasaki
manifold. They were completely determined in [BGM94] by the fact that they are
fiber-bundles over the quaternionic Kähler base space $G/G_{0}$. The non-
compact $G^{*}$ are thus given as the isometry group of the non-compact
quaternionic Kähler symmetric spaces [Be87, p. 409]. Alltogether, we obtain
Table 3.1.

$\begin{array}[]{cccccc}\hline\cr\hline\cr\boldsymbol{G}&\boldsymbol{G^{*}}&\boldsymbol{H}&\boldsymbol{G_{0}}&\boldsymbol{\dim}&\\\
\hline\cr\mathrm{Sp}(n+1)&\mathrm{Sp}(n,1)&\mathrm{Sp}(n)&\mathrm{Sp}(n)\mathrm{Sp}(1)&4n+3&n\geq
0\\\
\hline\cr\mathrm{SU}(n+2)&\mathrm{SU}(n,2)&S(\mathrm{U}(n)\times\mathrm{U}(1))&S(\mathrm{U}(n)\times\mathrm{U}(2))&4n+3&n\geq
1\\\
\hline\cr\mathrm{SO}(n+4)&\mathrm{SO}(n,4)&\mathrm{SO}(n)\times\mathrm{Sp}(1)&\mathrm{SO}(n)\mathrm{SO}(4)&4n+3&n\geq
3\\\
\hline\cr\mathrm{G}_{2}&\mathrm{G}_{2}^{2}&\mathrm{Sp}(1)&\mathrm{SO}(4)&11&\\\
\hline\cr\mathrm{F}_{4}&\mathrm{F}_{4}^{-20}&\mathrm{Sp}(3)&\mathrm{Sp}(3)\mathrm{Sp}(1)&31&\\\
\hline\cr\mathrm{E}_{6}&\mathrm{E}_{6}^{2}&\mathrm{SU}(6)&\mathrm{SU}(6)\mathrm{Sp}(1)&43&\\\
\hline\cr\mathrm{E}_{7}&\mathrm{E}_{7}^{-5}&\mathrm{Spin}(12)&\mathrm{Spin}(12)\mathrm{Sp}(1)&67&\\\
\hline\cr\mathrm{E}_{8}&\mathrm{E}_{8}^{-24}&\mathrm{E}_{7}&\mathrm{E}_{7}\mathrm{Sp}(1)&115&\\\
\hline\cr\hline\cr\end{array}$ Table 3.1: Complete table of generalized
$3$-Sasaki data

######  Theorem 3.1.1.

Consider some generalized $3$-Sasaki data $(G,G_{0},H)$ and
$0\neq\alpha,\delta\in\mathbb{R}$. Additionally suppose $\alpha\delta>0$ if
$G$ is compact and $\alpha\delta<0$ if $G$ is non-compact.

Let $\kappa(X,Y)=\mathrm{tr}(\mathrm{ad}(X)\circ\mathrm{ad}(Y))$ denote the
Killing form on $\mathfrak{g}$. Then define the inner product $g$ on the
tangent space $T_{p}M=T_{p}(G/H)\cong\mathfrak{m}$ by

$\displaystyle g|_{\mathcal{V}}$
$\displaystyle=\frac{-\kappa}{4\delta^{2}(n+2)},\qquad
g|_{\mathcal{H}}=\frac{-\kappa}{8\alpha\delta(n+2)},\qquad\mathcal{V}\perp\mathcal{H}.$

Let $\xi_{i}=\delta\sigma_{i}\in\mathcal{V}=\mathfrak{sp}(1)$, where the
$\sigma_{i}$ are the elements of $\mathfrak{sp}(1)=\mathfrak{su}(2)$ given by

$\sigma_{1}=\begin{pmatrix}i&0\\\
0&-i\end{pmatrix},\quad\sigma_{2}=\begin{pmatrix}0&-1\\\
1&0\end{pmatrix},\quad\sigma_{3}=\begin{pmatrix}0&-i\\\ -i&0\end{pmatrix}.$

Define endomorphisms $\varphi_{i}\in\mathrm{End}_{\mathfrak{h}}(\mathfrak{m})$
for $i=1,2,3$ by

$\displaystyle\varphi_{i}|_{\mathcal{V}}$
$\displaystyle=\frac{1}{2\delta}\mathrm{ad}\,\xi_{i},\qquad\varphi_{i}|_{\mathcal{H}}=\frac{1}{\delta}\mathrm{ad}\,\xi_{i}.$

Together with $\eta_{i}=g(\xi_{i},\cdot)$ the collection
$(G/H,\varphi_{i},\xi_{i},\eta_{i},g)$ defines a homogeneous
$3$-$(\alpha,\delta)$-Sasaki structure.

Before we proceed with the proof, we collect some observations.

######  Remark 3.1.2.

  1. a)

In case $\alpha\delta>0$ the given $3$-$(\alpha,\delta)$-Sasaki structure is
obtained via a $\mathcal{H}$-homothetic deformation with parameters
$a=\frac{1}{\alpha\delta}$, $b=\frac{1}{\delta^{2}}-\frac{1}{\alpha\delta}$,
$c=\frac{1}{\delta}$ from the $3$-Sasaki structure given in [DOP18].

  2. b)

Consider a homogeneous $3$-$(\alpha,\delta)$-Sasaki manifold
$(G/H,\varphi_{i},\xi_{i},\eta_{i},g)$ with $\alpha\delta>0$ such that the
isotropy group $H$ is connected, i.e. $G/H\neq\mathbb{RP}^{4n+3}$. Then a
$\mathcal{H}$-homothetic deformation with $a=\alpha\delta$ and $c=\delta$
induces a homogeneous $3$-Sasaki manifold with $G/H\neq\mathbb{RP}^{4n+3}$ and
thus is given by the model in [DOP18]. By definition of
$\mathcal{H}$-homothetic deformations the above inverse deformation will
restore the original objects. Thus, $(G/H,\varphi_{i},\xi_{i},\eta_{i},g)$ is
given by the construction in the theorem.

  3. c)

Usually the real representation $\mathfrak{g}_{1}$ of $\mathfrak{h}$ will be
irreducible and will only become reducible when complexified, thus we cannot
describe the action of $\mathcal{V}=\mathfrak{sp}(1)$ on $\mathcal{H}$ easily,
but from the complexified action we still find that the relations
$\mathrm{ad}\,\xi_{i}^{2}=-\delta^{2}\mathrm{id}$ and
$\mathrm{ad}\,\xi_{i}\circ\mathrm{ad}\,\xi_{j}=\pm\delta\mathrm{ad}\,\xi_{k}$
when $(ijk)$ is an even, resp. odd permutation of $(123)$ hold on
$\mathcal{H}$.

  4. d)

The Riemannian metric on $\mathcal{H}$ is a fixed multiple of the Killing form
on $\mathfrak{g}$ and thus the projection onto the symmetric orbit space

$G/H\rightarrow G/G_{0}$

is a Riemannian submersion. Indeed, this is the canonical submersion obtained
in Theorem 2.2.1.

  5. e)

The real projective space
$\mathbb{R}P^{4n+3}=\frac{\mathrm{Sp}(n+1)}{\mathrm{Sp}(n)\times\mathbb{Z}_{2}}$
and its non compact dual
$\frac{\mathrm{Sp}(n,1)}{\mathrm{Sp}(n)\times\mathbb{Z}_{2}}$ also admit
$3$-$(\alpha,\delta)$-Sasaki structures. They are obtained as the quotient of
$S^{4n+3}=\frac{\mathrm{Sp}(n+1)}{\mathrm{Sp}(n)}$, resp.
$\frac{\mathrm{Sp}(n,1)}{\mathrm{Sp}(n)}$ by the action of $\mathbb{Z}_{2}$
inside the fiber. Since the action is discrete these spaces cannot be
discerned in the Lie algebra picture. Note that all relevant tensors are
invariant under the $\mathbb{Z}_{2}$ action and thus local results obtained
for $S^{4n+3}=\frac{\mathrm{Sp}(n+1)}{\mathrm{Sp}(n)}$, resp.
$\frac{\mathrm{Sp}(n,1)}{\mathrm{Sp}(n)}$, remain true on $\mathbb{R}P^{4n+3}$
and its non compact dual.

  6. f)

Since the metric is a multiple of the Killing form and the Killing form is
$\mathrm{ad}$-invariant $[X,\,\cdot\;]$ will be metric if it preserves
$\mathcal{H}$ and $\mathcal{V}$. This is precisely the case if
$X\in\mathcal{V}$. For $X\in\mathcal{H}$, we compute with
$Y\in\mathcal{V},Z\in\mathcal{H}$

$g([X,Y],Z)=\frac{-1}{8\alpha\delta(n+2)}\kappa([X,Y],Z)=\frac{1}{8\alpha\delta(n+2)}\kappa(Y,[X,Z])=-\frac{\delta}{2\alpha}g(Y,[X,Z]).$

Thus $[X,\,\cdot\;]\in\mathfrak{so}(\mathfrak{m})$ if and only if
$\delta=2\alpha$, i.e. we are in the parallel case. This is exactly the
condition that our homogeneous space is naturally reductive. This can only
occur if $\alpha\delta>0$, i.e. we are in the positive case.

###### Proof (of Theorem 3.1.1).

If $G$ is compact, $\kappa<0$. If $G$ is of non-compact type, we have
$\kappa|_{\mathcal{V}}<0$ while $\kappa|_{\mathcal{H}}>0$ by (3.1) . Thus, in
both cases the given metric $g$ is indeed positive definite.

Remark 3.1.2 shows
$\mathrm{tr}(\mathrm{ad}^{2}\xi_{i}|_{\mathcal{H}})=\mathrm{tr}(-\delta^{2}\mathrm{id}|_{\mathcal{H}})=-4n\delta^{2}$.
On $\mathcal{V}$ we have

$[\xi_{i},[\xi_{i},\xi_{j}]]=\pm
2\delta[\xi_{i},\xi_{k}]=-4\delta^{2}\xi_{j},$

whenever $(ijk)$ is an even, respectively odd, permutation of $(123)$. Thus,
$\mathrm{tr}(\mathrm{ad}^{2}\xi_{i}|_{\mathcal{V}})=-8\delta^{2}$ and
therefore

$g(\xi_{i},\xi_{i})=\frac{-\mathrm{tr}(\mathrm{ad}^{2}\xi_{i})}{4\delta^{2}(n+2)}=\frac{-\mathrm{tr}(\mathrm{ad}^{2}\xi_{i}|_{\mathcal{H}})-\mathrm{tr}(\mathrm{ad}^{2}\xi_{i}|_{\mathcal{V}})}{4\delta^{2}(n+2)}=\frac{8\delta^{2}+4\delta^{2}n}{4\delta^{2}(n+2)}=1.$

On the contrary we have
$\mathrm{tr}(\mathrm{ad}\,\xi_{i}\circ\mathrm{ad}\,\xi_{j}|_{\mathcal{H}})=\mathrm{tr}(\pm\mathrm{ad}\,\xi_{k}|_{\mathcal{H}})=0$
as its trace on the complexification vanishes. And similar
$[\xi_{i},[\xi_{j},\xi_{k}]]=0$ if $(ijk)$ is any permutation of $(123)$ or
$[\xi_{i},[\xi_{j},\xi_{k}]]=4\delta^{2}\xi_{j}$ if $i=k\neq j$. In any case
$\mathrm{tr}(\mathrm{ad}\,\xi_{i}\circ\mathrm{ad}\,\xi_{j})=0$ and, hence,
$g(\xi_{i},\xi_{j})=0$ if $i\neq j$.

Next we check that the endomorphisms $\varphi_{i}$ are metric almost complex
structures on the complement to $\xi_{i}$. Note that they vanish on their
corresponding $\xi_{i}$. Furthermore,

$\displaystyle\varphi_{i}^{2}(\xi_{j})$
$\displaystyle=\frac{1}{4\delta^{2}}[\xi_{i},[\xi_{i},\xi_{j}]]=-\xi_{j},$
$\displaystyle\varphi_{i}^{2}|_{\mathcal{H}}$
$\displaystyle=\frac{1}{\delta^{2}}\mathrm{ad}\,\\!^{2}\xi_{i}|_{\mathcal{H}}=\frac{-\delta^{2}}{\delta^{2}}\mathrm{id}=-\mathrm{id}.$

Since $\mathcal{H}$ and $\mathcal{V}$ are invariant under $\varphi_{i}$ we
check orthogonality on each component individually. On $\mathcal{H}$ use the
associativity of $\kappa$ to find

$\kappa(\varphi_{i}X,\varphi_{i}Y)=-\kappa(X,\frac{1}{\delta^{2}}\mathrm{ad}\,\\!^{2}\xi_{i}Y)=\kappa(X,Y)$

and thus
$g(\varphi_{i}X,\varphi_{i}Y)=\frac{-\kappa(\varphi_{i}X,\varphi_{i}Y)}{8\alpha\delta(n+2)}=\frac{-\kappa(X,Y)}{8\alpha\delta^{2}(n+2)}=g(X,Y)$.
On $\mathcal{V}$ we have

$g(\varphi_{i}\xi_{j},\varphi_{i}\xi_{j^{\prime}})=g(\frac{1}{2\delta}\mathrm{ad}\,\xi_{i}(\xi_{j}),\frac{1}{2\delta}\mathrm{ad}\,\xi_{i}(\xi_{j^{\prime}}))=g(\pm\xi_{k},\pm\xi_{k^{\prime}})=g(\xi_{j},\xi_{j^{\prime}})$

if $(ijk)$, $(ij^{\prime}k^{\prime})$ are according permutations of $(123)$
and the left side vanishes whenever $j$ or $j^{\prime}$ equals $i$.

Next we check the compatibility conditions of the $3$ almost contact metric
structures. Suppose $(ijk)$ is an even permutation of $(123)$ then
$\varphi_{i}\xi_{j}=\xi_{k}$ and together with the invariance of $\mathcal{H}$
under $\varphi_{i}$ we conclude $\eta_{i}\circ\varphi_{j}=\eta_{k}$. Further,
$\varphi_{i}\varphi_{j}|_{\mathcal{H}}=\frac{1}{\delta^{2}}\mathrm{ad}\,\xi_{i}\circ\mathrm{ad}\,\xi_{j}|_{\mathcal{H}}=\frac{1}{\delta}\mathrm{ad}\,\xi_{k}|_{\mathcal{H}}=\varphi_{k}|_{\mathcal{H}}$
and on $\mathcal{V}$ we have

$\displaystyle\varphi_{i}\varphi_{j}\xi_{i}$
$\displaystyle=\frac{1}{4\delta^{2}}[\xi_{i},[\xi_{j},\xi_{i}]]=\xi_{j}=\varphi_{k}\xi_{i}=\varphi_{k}\xi_{i}+\eta_{j}(\xi_{i})\xi_{i},$
$\displaystyle\varphi_{i}\varphi_{j}\xi_{j}$
$\displaystyle=0=\xi_{i}-\xi_{i}=\varphi_{k}\xi_{j}-\eta_{j}(\xi_{j})\xi_{i},$
$\displaystyle\varphi_{i}\varphi_{j}\xi_{k}$
$\displaystyle=\frac{1}{4\delta^{2}}[\xi_{i},[\xi_{j},\xi_{k}]]=\frac{1}{2\delta}[\xi_{i},\xi_{i}]=0=\varphi_{k}\xi_{k}+\eta_{j}(\xi_{k})\xi_{i}.$

We have thus shown that the given structure is a homogeneous almost
$3$-contact metric structure. It remains to show the
$3$-$(\alpha,\delta)$-Sasaki condition
$\mathrm{d}\eta_{i}=2\alpha\Phi_{i}+2(\alpha-\delta)\eta_{j}\wedge\eta_{k}$,
for any even permutation $(ijk)$ of $(123)$. We show this case by case. Note
that the last summand vanishes whenever either entry is in $\mathcal{H}$. Let
$X\in\mathcal{H}$. Then, since $\mathrm{ad}\,\xi_{j}X\in\mathcal{H}$,

$\displaystyle\mathrm{d}\eta_{i}(\xi_{j},X)$
$\displaystyle=\xi_{j}(\eta_{i}(X))-X(\eta_{i}(\xi_{j}))-\eta_{i}(\mathrm{ad}\,\xi_{j}X)=-\eta_{i}(\mathrm{ad}\,\xi_{j}X)=0,$
$\displaystyle 2\alpha\Phi_{i}(\xi_{j},X)$ $\displaystyle=2\alpha
g(\xi_{j},\varphi_{i}X)=\frac{2\alpha}{\delta}g(\xi_{j},\mathrm{ad}\,\xi_{i}X)=0.$

For $X,Y\in\mathcal{H}$ we use associativity of $\kappa$

$\displaystyle\mathrm{d}\eta_{i}(X,Y)$
$\displaystyle=X(\eta_{i}(Y))-Y(\eta_{i}(X))-\eta_{i}([X,Y])=-g(\xi_{i},[X,Y])$
$\displaystyle=\frac{1}{4\delta^{2}(n+2)}\kappa(\xi_{i},[X,Y])=\frac{-1}{4\delta^{2}(n+2)}\kappa(\mathrm{ad}\,\xi_{i}Y,X),$
$\displaystyle 2\alpha\Phi_{i}(X,Y)$ $\displaystyle=2\alpha
g(X,\varphi_{i}Y)=\frac{2\alpha}{\delta}g(X,\mathrm{ad}\,\xi_{i}Y)=\frac{-2\alpha}{8\alpha\delta^{2}(n+2)}\kappa(X,\mathrm{ad}\,\xi_{i}Y)$
$\displaystyle=\frac{-1}{4\delta^{2}(n+2)}\kappa(X,\mathrm{ad}\,\xi_{i}Y).$

Finally, we have

$\displaystyle\mathrm{d}\eta_{i}(\xi_{j},\xi_{k})$
$\displaystyle=\xi_{j}(\eta_{i}(\xi_{k}))-\xi_{k}(\eta_{i}(\xi_{j}))-\eta_{i}([\xi_{j},\xi_{k}])=-\eta_{i}(2\delta\xi_{i})=-2\delta,$
(3.2) $\displaystyle 2\alpha\Phi_{i}(\xi_{j},\xi_{k})$ $\displaystyle=2\alpha
g(\xi_{j},\varphi_{i}\xi_{k})=-2\alpha g(\xi_{j},\xi_{j})=-2\alpha,$
$\displaystyle 2(\alpha-\delta)\eta_{i+1}\wedge\eta_{i+2}(\xi_{j},\xi_{k})$
$\displaystyle=2(\alpha-\delta)=2\alpha-2\delta.\qed$

###  3.2 Negative homogeneous $3$-$(\alpha,\delta)$-Sasaki manifolds over
Alekseevsky spaces

In order to construct homogeneous $3$-$(\alpha,\delta)$-Sasaki manifolds we
recall the setup in the unified construction of Alekseevsky spaces due to V.
Cortés [Co00]. Let $q\in\mathbb{N}$. Set $V=\mathbb{R}^{3,q}$ the real vector
space with signature $(3,q)$. Let $C\\!\ell^{0}(V)$ denote the even Clifford
algebra over $V$. Depending on $q\mod 4$ there exist exactly one or two
inequivalent irreducible $C\\!\ell^{0}(V)$-modules. Accordingly, let
$l\in\mathbb{N}$, if $q\not\equiv 3\mod 4$, or $l^{+},l^{-}\in\mathbb{N}$, if
$q\equiv 3\mod 4$. Then set

$\mathfrak{g}=\mathfrak{so}(V)\oplus V\oplus\mathbb{R}D\oplus W,$

where $W$ is the sum of $l$ equivalent irreducible $C\\!\ell^{0}(V)$-modules
(or the sum of $l^{+}$, $l^{-}$ irreducible $C\\!\ell^{0}(V)$-modules if there
are two inequivalent ones) and $D$ a derivation with eigenvalue decomposition
$\mathfrak{so}(V)\oplus V\oplus W$ and respective eigenvalues $(0,1,1/2)$. The
action of $\mathfrak{so}(V)$ on $V$ is given by the standard representation
and $\mathfrak{so}(V)$ acts on $W$ via the isomorphism
$\mathfrak{so}(V)\cong\mathfrak{spin}(V)\subset C\\!\ell^{0}(V)$ $e\wedge
e^{\prime}\mapsto-\frac{1}{2}ee^{\prime}$ if $e,e^{\prime}$ are orthogonal.
$V$ commutes with itself and $W$. Finally the commutators $[W,W]$ are given by
some non-degenerate $\mathfrak{so}(V)$-equivariant map
$\Pi\colon\Lambda^{2}W\to V$ where $\mathfrak{so}(V)$ acts on $W$ as
$\mathfrak{spin}(V)$.

######  Remark 3.2.1.

Note that $\Pi$ is unique up to rescaling along the irreducible summands of
$W$ [Co00, Theorem 5]. This rescaling leads to an isomorphism of the Lie
algebras $\mathfrak{g}(\Pi)$ and $\mathfrak{g}(\Pi^{\prime})$ corresponding to
two such maps $\Pi$ and $\Pi^{\prime}$. The isomorphism extends to an
isomorphism of the $3$-$(\alpha,\delta)$-Sasaki structures defined later on.
Thus, we will ignore the ambiguity in $\Pi$ from here on.

######  Notation.

On $V=\mathbb{R}^{3,q}$ fix an ONB
$\hat{e}_{1},\hat{e}_{2},\hat{e}_{3},e_{1},\dots,e_{q}$ with signature
$(+,+,+,-,\dots,-)$. Then with the identification
$\mathfrak{so}(V)\cong\Lambda^{2}V$ we also obtain a standard basis of the
space $\mathfrak{so}(V)$ given by
$\\{\hat{e}_{i}\wedge\hat{e}_{j},\hat{e}_{i}\wedge e_{k},e_{k}\wedge
e_{l}\\}_{{}^{i,j=1,2,3}_{k,l=1,\dots,3}}$.

Denote $\sigma_{i}=2\hat{e}_{k}\wedge\hat{e}_{j}$ for any even permutation
$(ijk)$ of $(123)$. Using the identification $\mathrm{End}(V)=V\otimes V^{*}$
this implies $[\sigma_{i},\hat{e}_{j}]=2\hat{e}_{k}$ and
$[\sigma_{i},\sigma_{j}]=2\sigma_{k}$ where again $(ijk)$ is an even
permutation of $(123)$.

We further set $\mathcal{V}=\mathfrak{so}(3)\subset\mathfrak{so}(3,q)$,
$\mathcal{H}_{0}$ the subspace generated by the elements $D$ and
$\hat{e}_{i}+\sigma_{i}$ and $\mathcal{H}_{1}$ the subspace generated by
$e_{1},\dots,e_{q}\in V$ and $e_{i}\wedge\hat{e}_{j}\in\mathfrak{so}(3,q)$.

The $4$-dimensional spaces $\mathcal{H}_{0}$ and $\langle
e_{l},e_{l}\wedge\hat{e}_{j}\rangle\subset\mathcal{H}_{1}$ will form the
quaternionic subspaces inside $\mathfrak{so}(V)\oplus
V\oplus\mathbb{R}D\subset\mathfrak{g}$. Accordingly, we show that they have
the only commutators with non-trivial $\mathcal{V}$-part.

######  Lemma 3.2.1.

The only non-trivial projections on $\mathcal{V}$ of commutators are

$\displaystyle\pi_{\mathcal{V}}([\sigma_{i},\sigma_{j}])=\pm
2\sigma_{k},\quad\pi_{\mathcal{V}}([D,\hat{e}_{i}+\sigma_{i}])=-\sigma_{i},\quad\pi_{\mathcal{V}}([\hat{e}_{i}+\sigma_{i},\hat{e}_{j}+\sigma_{j}])=\mp
2\sigma_{k},$ $\displaystyle\pi_{\mathcal{V}}([e_{l},\hat{e}_{i}\wedge
e_{l}])=-\sigma_{i},\quad\pi_{\mathcal{V}}([\hat{e}_{i}\wedge
e_{l},\hat{e}_{j}\wedge
e_{l}])=\pm\frac{1}{2}\sigma_{k},\quad\pi_{\mathcal{V}}([w_{1},w_{2}])=\pi_{\mathcal{V}}(\Pi(w_{1},w_{2}))$

for all permutations $(ijk)$ of $(123)$ with $\pm$ indicating the sign of the
permutation, $l=1,\dots,q$ and $w_{1},w_{2}\in W$.

###### Proof.

The full list of commutators of basis vectors is

$\displaystyle[\sigma_{i},\sigma_{j}]$ $\displaystyle=\pm 2\sigma_{k},$
$\displaystyle[\sigma_{i},D]$ $\displaystyle=0,$
$\displaystyle[\hat{e}_{i}+\sigma_{i},\hat{e}_{i}\wedge e_{l}]$
$\displaystyle=e_{l},$ $\displaystyle[\sigma_{i},\hat{e}_{j}+\sigma_{j}]$
$\displaystyle=\pm 2(\hat{e}_{k}+\sigma_{k}),$
$\displaystyle[\sigma_{i},e_{l}]$ $\displaystyle=0,$
$\displaystyle[\sigma_{i},\hat{e}_{i}\wedge e_{l}]$ $\displaystyle=0,$
$\displaystyle[\hat{e}_{i}\wedge e_{l},W]$ $\displaystyle=W,$
$\displaystyle[\sigma_{i},\hat{e}_{j}\wedge e_{l}]$ $\displaystyle=\pm
2\hat{e}_{k}\wedge e_{l},$ $\displaystyle[D,e_{l}]$ $\displaystyle=e_{l},$
$\displaystyle[D,\hat{e}_{i}\wedge e_{l}]$ $\displaystyle=0,$
$\displaystyle[\hat{e}_{i}\wedge e_{l},\hat{e}_{j}\wedge e_{m}]$
$\displaystyle=0,$ $\displaystyle[D,\hat{e}_{i}+\sigma_{i}]$
$\displaystyle=\hat{e}_{i}=(\hat{e}_{i}+\sigma_{i})-\sigma_{i},$
$\displaystyle[D,W]$ $\displaystyle=W,$
$\displaystyle[\sigma_{i},\hat{e}_{i}+\sigma_{i}]$ $\displaystyle=0,$
$\displaystyle[\hat{e}_{i}+\sigma_{i},W]$ $\displaystyle=W,$
$\displaystyle[\hat{e}_{i}+\sigma_{i},\hat{e}_{j}\wedge e_{l}]$
$\displaystyle=\pm 2\hat{e}_{k}\wedge e_{l},$ $\displaystyle[\sigma_{i},W]$
$\displaystyle=W,$ $\displaystyle[\hat{e}_{i}+\sigma_{i},e_{l}]$
$\displaystyle=0,$ $\displaystyle[\hat{e}_{i}\wedge e_{l},\hat{e}_{i}\wedge
e_{m}]$ $\displaystyle=-e_{l}\wedge e_{m},$ $\displaystyle[\hat{e}_{i}\wedge
e_{l},\hat{e}_{j}\wedge e_{l}]$
$\displaystyle=-\hat{e}_{i}\wedge\hat{e}_{j}=\pm\frac{1}{2}\sigma_{k}$
$\displaystyle[e_{l},e_{m}]$ $\displaystyle=0,$ $\displaystyle[e_{l},W]$
$\displaystyle=W,$ $\displaystyle[e_{l},\hat{e_{i}}\wedge e_{m}]$
$\displaystyle=0,$ $\displaystyle[e_{l},\hat{e_{i}}\wedge e_{l}]$
$\displaystyle=\hat{e_{i}}=(\hat{e_{i}}+\sigma_{i})-\sigma_{i}$

and finally

$[\hat{e}_{i}+\sigma_{i},\hat{e}_{j}+\sigma_{j}]\ =\
[\sigma_{i},\hat{e}_{j}]-[\sigma_{j},\hat{e}_{i}]+[\sigma_{i},\sigma_{j}]\ =\
\pm 4\hat{e}_{k}\pm 2\sigma_{k}=\pm 4(\hat{e}_{k}+\sigma_{k})\mp 2\sigma_{k},$

where $(ijk)$ is a permutation of $(123)$ with $\pm$ indicating the sign of
the permutation and $l,m=1,\dots,q$ with $l\neq m$. For the commutator $[W,W]$
we have $[w_{1},w_{2}]=\Pi(w_{1},w_{2})\in
V\subset\mathcal{H}_{0}\oplus\mathcal{H}_{1}\oplus\mathcal{V}$. ∎

By [Co00, Proposition 3] the adjoint action
$\mathfrak{g}\curvearrowright\mathfrak{r}=\mathbb{R}D\oplus V\oplus
W\subset\mathfrak{g}$ is faithful. Thus, $\mathfrak{g}$ is a subalgebra
$\mathfrak{g}\subset\mathrm{der}(\mathfrak{r})$. Set $G$ the subgroup
$G\subset\mathrm{Aut}(\mathfrak{r})$ with Lie Algebra $\mathfrak{g}$. Let
$\mathfrak{h}=\mathfrak{so}(q)\subset\mathfrak{so}(V)\subset\mathfrak{g}$ and
$H\subset G$ the corresponding connected subgroup. Then both $G$ and $H$ are
closed subgroups of $\mathrm{Aut}(\mathfrak{r})$. This follows from [Co00,
Corollary 3] and the fact that $H$ is closed in $\mathrm{Spin}_{0}(V)\subset
G$. In particular, $G/H$ is a homogeneous space. We now define the desired
negative $3$-$(\alpha,\delta)$-Sasaki structure on $M=G/H$.

######  Theorem 3.2.1.

Let $\alpha,\delta\in\mathbb{R}$ with $\alpha\delta<0$. Let $G,H$ with Lie
algebras $\mathfrak{g},\mathfrak{h}$ as above. Then
$\mathfrak{m}=\mathcal{V}\oplus\mathcal{H}_{0}\oplus\mathcal{H}_{1}\oplus W$
is a reductive complement to $\mathfrak{h}$ in $\mathfrak{g}$. Set

$\xi_{1}=\delta\sigma_{1},\qquad\xi_{2}=\delta\sigma_{2},\qquad\xi_{3}=\delta\sigma_{3}.$

Define the almost complex structures
$\varphi_{i}\colon\mathfrak{m}\to\mathfrak{m}$ on $\mathcal{V}$,
$\mathcal{H}_{0}$, $\mathcal{H}_{1}$ and $W$ individually. For any permutation
$(ijk)$ of $(123)$ with signature $\pm$ we set

$\displaystyle\varphi_{i}(\sigma_{j})=\pm\sigma_{k},\qquad\varphi_{i}(\sigma_{i})=0,$
($\mathcal{V}$)
$\displaystyle\varphi_{i}(2D)=\hat{e}_{i}+\sigma_{i},\qquad\varphi_{i}(\hat{e}_{i}+\sigma_{i})=-2D,\qquad\varphi_{i}(\hat{e}_{j}+\sigma_{j})=\pm(\hat{e}_{k}+\sigma_{k}),$
($\mathcal{H}_{0}$) $\displaystyle\varphi_{i}(e_{l})=2\hat{e}_{i}\wedge
e_{l},\qquad\varphi_{i}(2\hat{e}_{i}\wedge
e_{l})=-e_{l},\qquad\varphi_{i}(\hat{e}_{j}\wedge e_{l})=\pm\hat{e}_{k}\wedge
e_{l},$ ($\mathcal{H}_{1}$) $\displaystyle\varphi_{i}|_{W}=\rho(\sigma_{i}),$
($W$)

where $\rho$ is the Clifford-multiplication on $W$.

Define a scalar product $g_{[e]}$ by declaring the following vectors to be an
orthonormal basis of $\mathcal{V}\oplus\mathcal{H}_{0}\oplus\mathcal{H}_{1}$:

$\displaystyle\delta\sigma_{i},\sqrt{-4\alpha\delta}D,\sqrt{-\alpha\delta}(\hat{e}_{i}+\sigma_{i}),\sqrt{-4\alpha\delta}\,\hat{e}_{i}\wedge
e_{l},\sqrt{-\alpha\delta}\,e_{l}.$

On $W$ we set the scalar product

$g_{[e]}|_{W\times
W}(s,t)=(-2\alpha\delta)^{-1}b(s,t)\coloneqq(-2\alpha\delta)^{-1}\langle\hat{e}_{i},\Pi(\rho(\hat{e}_{j}\hat{e}_{k})s,t)\rangle,$

where $\langle,\rangle$ is the scalar product on $V$ and $(ijk)$ is any even
permutation of $(123)$. We set $W$ orthogonal to
$\mathcal{V}\oplus\mathcal{H}_{0}\oplus\mathcal{H}_{1}$. Set
$\eta_{i}=g(\xi_{i},\,\cdot\,)$ the dual to $\xi_{i}$.

Then $(G/H,g,\xi_{i},\eta_{i},\varphi_{i})$ defines a homogeneous
$3$-$(\alpha,\delta)$-Sasaki manifold.

###### Proof.

We first note that the defined scalar product is positive definite and
$\mathrm{Spin}(q)$-invariant. This is clear on
$\mathcal{V}\oplus\mathcal{H}_{0}\oplus\mathcal{H}_{1}$ and it is shown for
$b$ in [Co00, Theorem 1 and Proposition 9]. Thus, the scalar product extends
to an invariant Riemannian metric on $G/H$. The invariance under $H$ of the
$\xi_{i}$ is obvious. For an invariant $3$-a.c.m. structure, it remains to
check that the $\varphi_{i}$ are invariant as well. $\mathrm{Spin}(q)$ acts
trivial on $\mathcal{V}\oplus\mathcal{H}_{0}$ and on $\mathcal{H}_{1}$ by its
adjoint action on $e_{l}\in\mathbb{R}^{q}\subset V$. On $W$ it acts by
Clifford multiplication with vectors in $\mathbb{R}^{q}$ twice, thus commuting
with the Clifford multiplication defining the almost complex structures on
$W$.

The endomorphisms $\varphi_{i}$ are compatible with the metric by definition
on $\mathcal{V}\oplus\mathcal{H}_{0}\oplus\mathcal{H}_{1}$ and by
$\mathrm{Spin}(q)\cdot\mathrm{Spin}(3)$-invariance of $b$ on $W$. Next we
check the compatibility conditions of the $3$ almost contact structures. Again
on $\mathcal{V}\oplus\mathcal{H}_{0}\oplus\mathcal{H}_{1}$ this is a direct
consequence of the definition and on $W$ we have

$\rho(\sigma_{i})\rho(\sigma_{j})w=\hat{e}_{k}\cdot\hat{e}_{j}\cdot\hat{e}_{i}\cdot\hat{e}_{k}\cdot
w=(-1)^{2}\hat{e}_{j}\cdot\hat{e}_{k}\cdot\hat{e}_{k}\cdot\hat{e}_{i}\cdot
w=-\hat{e}_{j}\cdot\hat{e}_{i}\cdot w=\rho(\sigma_{k})w.$

Finally we need to check the defining condition
$\mathrm{d}\eta_{i}=2\alpha\Phi_{i}+2(\alpha-\delta)\eta_{j}\wedge\eta_{k}$.
By bilinearity it suffices to check it for any pair of two basis vectors
individually. On $\mathcal{V}\times\mathcal{V}$ this is exactly the same
computation as in the $3$-$(\alpha,\delta)$-Sasaki structure over symmetric
bases (compare (3.2)). Apart from $\mathcal{V}\times\mathcal{V}$ the equation
reduces to $\mathrm{d}\eta_{i}=2\alpha\Phi_{i}$. Note that the left hand side
reduces to checking the commutators. From Lemma 3.2.1 and the definition of
the $\varphi_{i}$ we see that both sides vanish for all mixed terms regarding
the decomposition $\mathcal{V}\oplus\mathcal{H}_{0}\oplus\mathcal{H}_{1}\oplus
W$ of the tangent space. Similarly on $\mathcal{H}_{1}$ if the index $l$ of
$\hat{e}_{i}\wedge e_{l}$, respectively $e_{l}$, is not the same both sides
vanish. On $\mathcal{H}_{0}\times\mathcal{H}_{0}$ we compute

$\displaystyle\mathrm{d}\eta_{i}(D,\hat{e}_{i}+\sigma_{i})$
$\displaystyle=-\eta_{i}([D,\hat{e}_{i}+\sigma_{i}])=-\eta_{i}(-\sigma_{i})=\frac{1}{\delta}g(\delta\sigma_{i},\delta\sigma_{i})=\frac{1}{\delta},$
$\displaystyle 2\alpha\Phi_{i}(D,\hat{e}_{i}+\sigma_{i})$
$\displaystyle=2\alpha
g(D,\varphi_{i}(\hat{e}_{i}+\sigma_{i}))=\frac{2\alpha}{-2\alpha\delta}g(\sqrt{-4\alpha\delta}\;D,-\sqrt{-\alpha\delta}\;2D)=\frac{1}{\delta}.$

In similar fashion for the remaining pairs in
$\mathcal{H}_{0}\times\mathcal{H}_{0}$ and on
$\mathcal{H}_{1}\times\mathcal{H}_{1}$ we have

$\displaystyle\frac{2}{\delta}=-\eta_{k}(2\sigma_{k})=\mathrm{d}\eta_{k}(\hat{e}_{i}+\sigma_{i},\hat{e}_{j}+\sigma_{j})$
$\displaystyle=2\alpha\Phi_{k}(\hat{e}_{i}+\sigma_{i},\hat{e}_{j}+\sigma_{j})=\frac{2}{\delta},$
$\displaystyle\frac{1}{\delta}=-\eta_{i}(-\sigma_{i})=\mathrm{d}\eta_{i}(e_{l},\hat{e}_{i}\wedge
e_{l})$ $\displaystyle=2\alpha\Phi_{i}(e_{l},\hat{e}_{i}\wedge
e_{l})=\frac{1}{\delta},$
$\displaystyle\frac{1}{2\delta}=-\eta_{k}(-\frac{1}{2}\sigma_{k})=\mathrm{d}\eta_{k}(\hat{e}_{i}\wedge
e_{l},\hat{e}_{j}\wedge e_{l})$
$\displaystyle=2\alpha\Phi_{k}(\hat{e}_{i}\wedge e_{l},\hat{e}_{j}\wedge
e_{l})=\frac{1}{2\delta}$

for any even permutation $(ijk)$ of $(123)$. Finally, we look at $W\times W$.
Let $w_{1},w_{2}\in W$ and suppose
$\Pi(w_{1},w_{2})=\sum_{r=1}^{q}a_{r}e_{r}+\sum_{s=1}^{3}\hat{a}_{s}\hat{e}_{s}$.
Then

$\displaystyle\mathrm{d}\eta_{i}([w_{1},w_{2}])$
$\displaystyle=-\eta_{i}(\Pi(w_{1},w_{2}))=-\eta_{i}\left(\sum_{r=1}^{q}a_{r}e_{r}+\sum_{s=1}^{3}\hat{a}_{s}\hat{e}_{s}\right)=-\eta_{i}\left(\sum_{s=1}^{3}\hat{a}_{s}((\hat{e}_{s}+\sigma_{s})-\sigma_{s})\right)=\frac{\hat{a}_{i}}{\delta}$

and

$\displaystyle 2\alpha\Phi_{i}(w_{1},w_{2})$ $\displaystyle=2\alpha
g(w_{1},\varphi_{i}w_{2})=\frac{2\alpha}{-2\alpha\delta}\left\langle\hat{e}_{i},\Pi\left(w_{1},\hat{e}_{j}\hat{e}_{k}\hat{e}_{j}\hat{e}_{k}w_{2}\right)\right\rangle=\frac{(-1)^{3}2\alpha}{-2\alpha\delta}\langle\hat{e_{i}},\Pi(w_{1},w_{2})\rangle$
$\displaystyle=\frac{1}{\delta}\left\langle\hat{e}_{i},\sum_{r=1}^{q}a_{r}e_{r}+\sum_{s=1}^{3}\hat{a}_{s}\hat{e}_{s}\right\rangle=\frac{\hat{a}_{i}}{\delta}.$

This concludes the proof. ∎

######  Remark.

We try to motivate the definition. Recall that in [Co00] Cortés shows that
$\mathfrak{so}(V)\oplus V\oplus\mathbb{R}D$ is isomorphic to a subalgebra of
$\mathfrak{so}(4,q+1)=\Lambda^{2}(V\oplus\langle e^{+},e^{-}\rangle)$,
$e^{+},e^{-}$ unit length vectors of corresponding signature, given by the
inclusion

$\mathfrak{so}(V)\mapsto\Lambda^{2}V,\qquad V\mapsto
V\wedge(e^{+}-e^{-}),\qquad D\mapsto e^{+}\wedge e^{-}.$

Now $\varphi_{i}$ is modeled on $\mathfrak{so}(V)\oplus V\oplus\mathbb{R}D$
after the adjoint action with $\hat{e}_{j}\wedge\hat{e}_{k}+\hat{e}_{i}\wedge
e^{+}\in\mathfrak{so}(3)_{+}\subset\mathfrak{so}(3)_{+}\oplus\mathfrak{so}(3)_{-}=\mathfrak{so}(4)$
in the known $\mathrm{SO}(4,q+1)/\mathrm{SO}(q+1)\mathrm{SO}(3)$ setting.
However, this does not exist as an inner derivative in $\mathfrak{g}$ unlike
int the (semi-) simple case.

###  3.3 Examples

We begin with an example of the construction over a symmetric Wolf space.

######  Example 3.3.1.

Our first example is the Aloff-Wallach space
$W^{1,1}=\mathrm{SU(3)}/S^{1}=G/H$. In this case the isotropy algebra
$\mathfrak{h}$ inside $\mathfrak{g}=\mathfrak{su}(3)$ is the 1-dimensional
space generated by

$h=\begin{bmatrix}-i&0&0\\\ 0&-i&0\\\ 0&0&2i\end{bmatrix}.$

We locate the space $\mathfrak{sp}(1)=\mathfrak{su}(2)\subset\mathfrak{su}(3)$
as the upper left 2-by-2 block. One checks that this is a splitting of
$\mathfrak{su}(3)$ as necessary. Then for $\alpha,\delta>0$ the Reeb vector
fields are given by

$\displaystyle\xi_{1}=\delta\begin{bmatrix}i&0&0\\\ 0&-i&0\\\
0&0&0\end{bmatrix},\qquad\xi_{2}=\delta\begin{bmatrix}0&-1&0\\\ 1&0&0\\\
0&0&0\end{bmatrix},\qquad\xi_{3}=\delta\begin{bmatrix}0&-i&0\\\ -i&0&0\\\
0&0&0\end{bmatrix}.$

On the horizontal subspace we choose a basis vector

$\displaystyle\tilde{e}_{1}=\begin{bmatrix}0&0&1\\\ 0&0&0\\\
-1&0&0\end{bmatrix}.$

Then we normalize it
$g(\tilde{e}_{1},\tilde{e}_{1})=\frac{-6\mathrm{tr}(\tilde{e}_{1}\cdot\tilde{e}_{1})}{24\alpha\delta}=\frac{1}{2\alpha\delta}$,
i.e. $e_{1}=\sqrt{2\alpha\delta}\cdot\tilde{e}_{1}$ and generate an adapted
basis:

$\displaystyle e_{2}=\sqrt{2\alpha\delta}\begin{bmatrix}0&0&i\\\ 0&0&0\\\
i&0&0\end{bmatrix},\qquad e_{3}=\sqrt{2\alpha\delta}\begin{bmatrix}0&0&0\\\
0&0&1\\\ 0&-1&0\end{bmatrix},\qquad
e_{4}=\sqrt{2\alpha\delta}\begin{bmatrix}0&0&0\\\ 0&0&-i\\\
0&-i&0\end{bmatrix}.$

######  Example 3.3.2.

Next consider the dual negative $3$-$(\alpha,\delta)$-Sasaki space
$\mathrm{SU}(2,1)/S^{1}$. We realize the Lie algebra

$\mathfrak{su}(2,1)=\mathfrak{g}^{*}=\mathfrak{h}\oplus\mathfrak{sp}(1)\oplus
i\mathcal{H}\subset\mathfrak{su}(3)^{\mathbb{C}}$

as described in (3.1). Then as for the Aloff-Wallach space we identify the
$1$-dimensional isotropy $\mathfrak{h}$ generated by

$h=\begin{bmatrix}-i&0&0\\\ 0&-i&0\\\ 0&0&2i\end{bmatrix}.$

Analogously the Reeb vector fields are given by

$\displaystyle\xi_{1}=\delta\begin{bmatrix}i&0&0\\\ 0&-i&0\\\
0&0&0\end{bmatrix},\qquad\xi_{2}=\delta\begin{bmatrix}0&-1&0\\\ 1&0&0\\\
0&0&0\end{bmatrix},\qquad\xi_{3}=\delta\begin{bmatrix}0&-i&0\\\ -i&0&0\\\
0&0&0\end{bmatrix}.$

On the horizontal subspace we choose

$\tilde{e}^{*}_{1}=i\tilde{e}_{1}=\begin{bmatrix}0&0&i\\\ 0&0&0\\\
-i&0&0\end{bmatrix}\subset i\mathcal{H}.$

We have

$g(\tilde{e}^{*}_{1},\tilde{e}^{*}_{1})=\frac{-i^{2}}{24\alpha\delta}\kappa(\tilde{e}_{1},\tilde{e}_{1})=-\frac{1}{2\alpha\delta}.$

Thus we find an adapted base of $\mathrm{SU}(2,1)/S^{1}$ by
$e_{1}^{*}=i\sqrt{-2\alpha\delta}\tilde{e}_{1}$ and

$\displaystyle e^{*}_{2}=\sqrt{-2\alpha\delta}\begin{bmatrix}0&0&-1\\\
0&0&0\\\
-1&0&0\end{bmatrix},\;e^{*}_{3}=\sqrt{-2\alpha\delta}\begin{bmatrix}0&0&0\\\
0&0&i\\\
0&-i&0\end{bmatrix},\;e^{*}_{4}=\sqrt{-2\alpha\delta}\begin{bmatrix}0&0&0\\\
0&0&1\\\ 0&1&0\end{bmatrix}.$

We now discuss the lowest dimensional example $\hat{\mathcal{T}}(1)$ of a
negative homogeneous $3$-$(\alpha,\delta)$-Sasaki manifold fibering over
Alekseevsky space $\mathcal{T}(1)$ not obtained by the construction over
symmetric spaces.

$\begin{array}[]{cclll}\hline\cr\hline\cr\text{Dimension}&\text{parameters}&\mathfrak{g}&\mathfrak{h}&\text{alternative
description}\\\ \hline\cr
7&q=0,\;l=0&\mathfrak{so}(3)\oplus\mathbb{R}^{3}\oplus\mathbb{R}D&0&\mathrm{Sp}(1,1)/\mathrm{Sp}(1)\\\
\hline\cr\hbox{\multirowsetup$11$}&q=0,\;l=1&\mathfrak{so}(3)\oplus\mathbb{R}^{3}\oplus\mathbb{R}D\oplus
W_{0}&0&\mathrm{Sp}(2,1)/\mathrm{Sp}(2)\\\
&q=1,\;l=0&\mathfrak{so}(3,1)\oplus\mathbb{R}^{3,1}\oplus\mathbb{R}D&0&\mathrm{SU}(2,2)/\mathrm{S}(\mathrm{U}(2)\times\mathrm{U}(1))\\\
\hline\cr\hbox{\multirowsetup$15$}&q=0,\;l=2&\mathfrak{so}(3)\oplus\mathbb{R}^{3}\oplus\mathbb{R}D\oplus
2W_{0}&0&\mathrm{Sp}(3,1)/\mathrm{Sp}(3)\\\
&q=1,\;l=1&\mathfrak{so}(3,1)\oplus\mathbb{R}^{3,1}\oplus\mathbb{R}D\oplus
W_{1}&0&\mathrm{SU}(3,2)/\mathrm{S}(\mathrm{U}(3)\times\mathrm{U}(1))\\\
&q=2,\;l=0&\mathfrak{so}(3,2)\oplus\mathbb{R}^{3,2}\oplus\mathbb{R}D&\mathfrak{so}(2)&\mathrm{SO}_{0}(3,4)/\mathrm{SO}(3)\times\mathrm{SO}(3)\\\
\hline\cr\hbox{\multirowsetup$19$}&q=0,\;l=3&\mathfrak{so}(3)\oplus\mathbb{R}^{3}\oplus\mathbb{R}D\oplus
3W_{0}&0&\mathrm{Sp}(4,1)/\mathrm{Sp}(4)\\\
&q=1,\;l=2&\mathfrak{so}(3,1)\oplus\mathbb{R}^{3,1}\oplus\mathbb{R}D\oplus
2W_{1}&0&\mathrm{SU}(4,2)/\mathrm{S}(\mathrm{U}(4)\times\mathrm{U}(1))\\\
&q=2,\;l=1&\mathfrak{so}(3,2)\oplus\mathbb{R}^{3,2}\oplus\mathbb{R}D\oplus
W_{2}&\mathfrak{so}(2)&\hat{\mathcal{T}}(1)\ \text{non symmetric base}\\\
&q=3,\;l=0&\mathfrak{so}(3,3)\oplus\mathbb{R}^{3,3}\oplus\mathbb{R}D&\mathfrak{so}(3)&\mathrm{SO}_{0}(4,4)/\mathrm{SO}(4)\times\mathrm{SO}(3)\\\
\hline\cr\hline\cr\end{array}$ Table 3.2: $3$-$(\alpha,\delta)$-Sasaki
manifolds over Alekseevsky spaces of $\dim\leq 19$, see Remark 3.3.1.

######  Remark 3.3.1.

The first new example arising from the construction over Alekseevsky spaces
appears only in dimension $19$. Table 3.2 lists all homogeneous
$3$-$(\alpha,\delta)$-Sasaki manifolds obtained by Theorem 3.2.1 up to
dimension $19$ and, if existing, the isomorphic ones appearing in Table 3.1,
i.e. obtained by Theorem 3.1.1 over non-compact symmetric spaces. The list
gets more intricate with higher dimension, in particular, there appear two
inequivalent even Clifford modules for $q=3$ beginning in $\dim 27$ and for
$q\geq 4$ we have $\dim W_{q}>4$. Further, observe that the symmetric base
cases $\mathrm{SU}(2,1)/\mathrm{U}(1)$, $\mathrm{G}_{2}^{(2)}/\mathrm{SO}(3)$
are not obtained by this construction.

We now give more concrete descriptions of the $C\\!\ell^{0}(3,q)$-modules
$W_{q}$ for $q=0,1,2$. Note that there are choices to be made though these
lead to isomorphisms of the modules since all these modules are unique. Let
$\mathbb{R}^{3,q}=\langle
e_{\hat{1}},e_{\hat{2}},e_{\hat{3}},e_{1},\dots,e_{q}\rangle$, where
$e_{\hat{i}}$ have signature $+1$ while $e_{i}$ have signature $-1$. Then we
have $C\\!\ell^{0}(3,0)=\mathbb{H}$,
$C\\!\ell^{0}(3,1)=\mathcal{M}_{2}(\mathbb{C})$,
$C\\!\ell^{0}(3,2)=\mathcal{M}_{4}(\mathbb{R})$ realized as follows. Table 3.3
lists the cases $q=0$ and $q=1$, while Table 3.4 is devoted to the case $q=2$.

$\begin{array}[]{@{}lllll@{}}\hline\cr\hline\cr q=0&\deg
0:&\begin{bmatrix}1\end{bmatrix}&\phantom{x}&+1\\\ \cline{2-5}\cr&\deg
2:&e_{\hat{1}\hat{2}}=\begin{bmatrix}i\end{bmatrix},\quad
e_{\hat{2}\hat{3}}=\begin{bmatrix}j\end{bmatrix},\quad
e_{\hat{3}\hat{1}}=\begin{bmatrix}k\end{bmatrix}&&-1\\\ \hline\cr q=1&\deg
0:&\begin{bmatrix}1&\\\ &1\end{bmatrix}&&+1\\\ \cline{2-5}\cr&\deg
2:&e_{\hat{1}\hat{2}}=\begin{bmatrix}i&\\\ &-i\end{bmatrix},\quad
e_{\hat{2}\hat{3}}=\begin{bmatrix}&i\\\ i&\end{bmatrix},\quad
e_{\hat{3}\hat{1}}=\begin{bmatrix}&-1\\\ 1&\end{bmatrix}&&-1\\\
&&e_{\hat{3}1}=\begin{bmatrix}1&\\\ &-1\end{bmatrix},\quad
e_{\hat{1}1}=\begin{bmatrix}&1\\\ 1&\end{bmatrix},\quad
e_{\hat{2}1}=\begin{bmatrix}&i\\\ -i&\end{bmatrix}&&+1\\\ \cline{2-5}\cr&\deg
4:&e_{\hat{1}\hat{2}\hat{3}1}=\begin{bmatrix}i&\\\ &i\end{bmatrix}&&-1\\\
\hline\cr\hline\cr\end{array}$ Table 3.3: Choice of
$C\\!\ell^{0}(3,q)$-representations for $q=0$ and $q=1$

The notation is as follow: We denote elements $e_{ij}=e_{i}e_{j}\in
C\\!\ell^{0}(V)$ and analogous for the action of elements in $C\\!\ell^{0}(V)$
of higher degree. The last line denotes the square of elements in the
respective row, which are invariant of choices unlike the matrices itself.

$\begin{array}[]{@{}lll@{}}\hline\cr\hline\cr\deg 0:&\begin{bmatrix}1&&&\\\
&1&&\\\ &&1&\\\ &&&1\end{bmatrix}&+1\\\ \hline\cr\deg
2:&e_{\hat{1}\hat{2}}=\begin{bmatrix}&-1&&\\\ 1&&&\\\ &&&1\\\
&&-1&\end{bmatrix},\ e_{\hat{2}\hat{3}}=\begin{bmatrix}&&&-1\\\ &&1&\\\
&-1&&\\\ 1&&&\end{bmatrix},\ e_{\hat{3}\hat{1}}=\begin{bmatrix}&&-1&\\\
&&&-1\\\ 1&&&\\\ &1&&\end{bmatrix},\ e_{12}=\begin{bmatrix}&&1&\\\ &&&-1\\\
-1&&&\\\ &1&&\end{bmatrix}&-1\\\
\cline{2-3}\cr&e_{\hat{3}1}=\begin{bmatrix}1&&&\\\ &1&&\\\ &&-1&\\\
&&&-1\end{bmatrix},\ e_{\hat{1}1}=\begin{bmatrix}&&1&\\\ &&&1\\\ 1&&&\\\
&1&&\end{bmatrix},\ e_{\hat{2}1}=\begin{bmatrix}&&&-1\\\ &&1&\\\ &1&&\\\
-1&&&\end{bmatrix},\ e_{\hat{3}2}=\begin{bmatrix}&&1&\\\ &&&-1\\\ 1&&&\\\
&-1&&\end{bmatrix}&+1\\\ &e_{\hat{1}2}=\begin{bmatrix}-1&&&\\\ &1&&\\\ &&1&\\\
&&&-1\end{bmatrix},\ e_{\hat{2}2}=\begin{bmatrix}&-1&&\\\ -1&&&\\\ &&&-1\\\
&&-1&\end{bmatrix}&\\\ \hline\cr\deg
4:&e_{\hat{1}\hat{2}\hat{3}1}=\begin{bmatrix}&-1&&\\\ 1&&&\\\ &&&-1\\\
&&1&\end{bmatrix},\ e_{\hat{1}\hat{2}\hat{3}2}=\begin{bmatrix}&&&1\\\ &&1&\\\
&-1&&\\\ -1&&&\end{bmatrix}&-1\\\
\cline{2-3}\cr&e_{\hat{1}\hat{2}12}=\begin{bmatrix}&&&1\\\ &&1&\\\ &1&&\\\
1&&&\end{bmatrix},\ e_{\hat{2}\hat{3}12}=\begin{bmatrix}&-1&&\\\ -1&&&\\\
&&&1\\\ &&1&\end{bmatrix},\ e_{\hat{3}\hat{1}12}=\begin{bmatrix}1&&&\\\
&-1&&\\\ &&1&\\\ &&&-1\end{bmatrix}&+1\\\ \hline\cr\hline\cr\end{array}$
Table 3.4: Choice of $C\\!\ell^{0}(3,q)$-representations for $q=2$

With this we can find the map $\Pi\colon\Lambda^{2}W_{2}\to\mathbb{R}^{3,2}$.

######  Theorem 3.3.1.

Setting $W_{2}\cong\mathbb{R}^{4}=\langle E_{1},E_{2},E_{3},E_{4}\rangle$ with
the $\mathfrak{spin}(3,2)$-module structure above the map
$\Pi\colon\Lambda^{2}W_{2}\to\mathbb{R}^{3,2}$ given by

$\displaystyle\Pi(E_{1}\wedge E_{2})=-\hat{e}_{3}-e_{1},\quad\Pi(E_{1}\wedge
E_{3})=-\hat{e}_{2},\quad\Pi(E_{1}\wedge E_{4})=-\hat{e}_{1}+e_{2},$
$\displaystyle\Pi(E_{2}\wedge E_{3})=\hat{e}_{1}+e_{2},\quad\Pi(E_{4}\wedge
E_{2})=\hat{e}_{2},\quad\Pi(E_{3}\wedge E_{4})=\hat{e}_{3}-e_{1}$

is $\mathfrak{spin}(3,2)$-invariant and non-degenerate.

Recall that the action of $\mathfrak{so}(3,q)$ on the
$C\\!\ell^{0}(3,q)$-module $W$, and thereby $W\wedge W$, is given by the
isomorphism
$\mathrm{ad}\,^{-1}\colon\mathfrak{so}(3,q)\to\mathfrak{spin}(3,q)=C\\!\ell^{0}(3,q)$,
$e_{i}\wedge e_{j}\mapsto-\frac{1}{2}e_{ij}$, where
$i,j\in\\{\hat{1},\hat{2},\hat{3},1,\dots,q\\}$.

###### Proof.

Non-degeneracy is clear. It suffices to check the invariance on a generating
set of $C\\!\ell^{0}(3,2)$. One such set is given by
$e_{\hat{1}\hat{2}},e_{\hat{2}\hat{3}},e_{\hat{1}1},e_{12}$. Each of these map
certain subspaces of $W_{2}$ onto one another, hence their action on the
exterior product of these subspaces vanishes. This yields the identities

$\displaystyle-2\hat{e}_{1}\wedge\hat{e}_{2}(\hat{e}_{3}\pm e_{1})$
$\displaystyle=0=\Pi(e_{\hat{1}\hat{2}}(E_{1}\wedge
E_{2}))=\Pi(e_{\hat{1}\hat{2}}(E_{3}\wedge E_{4})),$
$\displaystyle-2\hat{e}_{2}\wedge\hat{e}_{3}(\hat{e}_{1}\pm e_{2})$
$\displaystyle=0=\Pi(e_{\hat{2}\hat{3}}(E_{1}\wedge
E_{4}))=\Pi(e_{\hat{2}\hat{3}}(E_{2}\wedge E_{3})),$
$\displaystyle-2\hat{e}_{1}\wedge e_{1}(\hat{e}_{2})$
$\displaystyle=0=\Pi(e_{\hat{1}1}(E_{1}\wedge
E_{3}))=\Pi(e_{\hat{1}1}(E_{4}\wedge E_{2})),$ $\displaystyle-2e_{1}\wedge
e_{2}(\hat{e}_{2})$ $\displaystyle=0=\Pi(e_{12}(E_{1}\wedge
E_{3}))=\Pi(e_{12}(E_{4}\wedge E_{2})).$

The rest is just more computations. We start with $e_{\hat{1}\hat{2}}$:

$\displaystyle\Pi(e_{\hat{1}\hat{2}}(E_{1}\wedge E_{3}))$
$\displaystyle=\Pi(E_{2}\wedge E_{3})+\Pi(E_{1}\wedge-
E_{4})=\hat{e}_{1}+e_{2}+\hat{e}_{1}-e_{2}=2\hat{e}_{1}$
$\displaystyle=-2\hat{e}_{1}\wedge\hat{e}_{2}(-\hat{e}_{2})=-2\hat{e}_{1}\wedge\hat{e}_{2}(\Pi(E_{1}\wedge
E_{3})),$ $\displaystyle\Pi(e_{\hat{1}\hat{2}}(E_{1}\wedge E_{4}))$
$\displaystyle=\Pi(E_{2}\wedge E_{4})+\Pi(E_{1}\wedge
E_{3})=-\hat{e}_{2}-\hat{e}_{2}=-2\hat{e}_{2}$
$\displaystyle=-2\hat{e}_{1}\wedge\hat{e}_{2}(-\hat{e}_{1}+e_{2})=-2\hat{e}_{1}\wedge\hat{e}_{2}(\Pi(E_{1}\wedge
E_{4})),$ $\displaystyle\Pi(e_{\hat{1}\hat{2}}(E_{2}\wedge E_{3}))$
$\displaystyle=\Pi(-E_{1}\wedge E_{3})+\Pi(E_{2}\wedge-
E_{4})=\hat{e}_{2}+\hat{e}_{2}=2\hat{e}_{2}$
$\displaystyle=-2\hat{e}_{1}\wedge\hat{e}_{2}(\hat{e}_{1}+e_{2})=-2\hat{e}_{1}\wedge\hat{e}_{2}(\Pi(E_{2}\wedge
E_{3})),$ $\displaystyle\Pi(e_{\hat{1}\hat{2}}(E_{4}\wedge E_{2}))$
$\displaystyle=\Pi(E_{3}\wedge E_{2})+\Pi(E_{4}\wedge-
E_{1})=-\hat{e}_{1}-e_{2}-\hat{e}_{1}+e_{2}=-2\hat{e}_{1}$
$\displaystyle=-2\hat{e}_{1}\wedge\hat{e}_{2}(\hat{e}_{2})=-2\hat{e}_{1}\wedge\hat{e}_{2}(\Pi(E_{4}\wedge
E_{2})).$

For $e_{\hat{2}\hat{3}}$:

$\displaystyle\Pi(e_{\hat{2}\hat{3}}(E_{1}\wedge E_{2}))$
$\displaystyle=\Pi(E_{4}\wedge E_{2})+\Pi(E_{1}\wedge-
E_{3})=\hat{e}_{2}+\hat{e}_{2}=2\hat{e}_{2}$
$\displaystyle=-2\hat{e}_{2}\wedge\hat{e}_{3}(-\hat{e}_{3}-e_{1})=-2\hat{e}_{2}\wedge\hat{e}_{3}(\Pi(E_{1}\wedge
E_{2})),$ $\displaystyle\Pi(e_{\hat{2}\hat{3}}(E_{1}\wedge E_{3}))$
$\displaystyle=\Pi(E_{4}\wedge E_{3})+\Pi(E_{1}\wedge
E_{2})=-\hat{e}_{3}+e_{1}-\hat{e}_{3}-e_{1}=-2\hat{e}_{3}$
$\displaystyle=-2\hat{e}_{2}\wedge\hat{e}_{3}(-\hat{e}_{2})=-2\hat{e}_{2}\wedge\hat{e}_{3}(\Pi(E_{1}\wedge
E_{3})),$ $\displaystyle\Pi(e_{\hat{2}\hat{3}}(E_{4}\wedge E_{2}))$
$\displaystyle=\Pi(-E_{1}\wedge E_{2})+\Pi(E_{4}\wedge-
E_{3})=\hat{e}_{3}+e_{1}+\hat{e}_{3}-e_{1}=2\hat{e}_{3}$
$\displaystyle=-2\hat{e}_{2}\wedge\hat{e}_{3}(\hat{e}_{2})=-2\hat{e}_{2}\wedge\hat{e}_{3}(\Pi(E_{4}\wedge
E_{2})),$ $\displaystyle\Pi(e_{\hat{2}\hat{3}}(E_{2}\wedge E_{4}))$
$\displaystyle=\Pi(E_{3}\wedge E_{2})+\Pi(E_{3}\wedge-
E_{1})=-\hat{e}_{2}-\hat{e}_{2}=-2\hat{e}_{2}$
$\displaystyle=-2\hat{e}_{2}\wedge\hat{e}_{3}(\hat{e}_{3}-e_{1})=-2\hat{e}_{2}\wedge\hat{e}_{3}(\Pi(E_{3}\wedge
E_{4})).$

For $e_{\hat{1}1}$:

$\displaystyle\Pi(e_{\hat{1}1}(E_{1}\wedge E_{2}))$
$\displaystyle=\Pi(E_{3}\wedge E_{2})+\Pi(E_{1}\wedge
E_{4})=-\hat{e}_{1}-e_{2}-\hat{e}_{1}+e_{2}=-2\hat{e}_{1}$
$\displaystyle=-2\hat{e}_{1}\wedge
e_{1}(-\hat{e}_{3}-e_{1})=-2\hat{e}_{1}\wedge e_{1}(\Pi(E_{1}\wedge E_{2})),$
$\displaystyle\Pi(e_{\hat{1}1}(E_{1}\wedge E_{4}))$
$\displaystyle=\Pi(E_{3}\wedge E_{4})+\Pi(E_{1}\wedge
E_{2})=\hat{e}_{3}-e_{1}-\hat{e}_{3}-e_{1}=-2e_{1}$
$\displaystyle=-2\hat{e}_{1}\wedge
e_{1}(-\hat{e}_{1}+e_{2})=-2\hat{e}_{1}\wedge e_{1}(\Pi(E_{1}\wedge E_{4})),$
$\displaystyle\Pi(e_{\hat{1}1}(E_{2}\wedge E_{3}))$
$\displaystyle=\Pi(E_{4}\wedge E_{3})+\Pi(E_{2}\wedge
E_{1})=-\hat{e}_{3}+e_{1}+\hat{e}_{3}+e_{1}=2e_{1}$
$\displaystyle=-2\hat{e}_{1}\wedge
e_{1}(\hat{e}_{1}+e_{2})=-2\hat{e}_{1}\wedge e_{1}(\Pi(E_{2}\wedge E_{3})),$
$\displaystyle\Pi(e_{\hat{1}1}(E_{3}\wedge E_{4}))$
$\displaystyle=\Pi(E_{1}\wedge E_{4})+\Pi(E_{3}\wedge
E_{2})=-\hat{e}_{1}+e_{2}-\hat{e}_{1}-e_{2}=-2\hat{e}_{1}$
$\displaystyle=-2\hat{e}_{1}\wedge
e_{1}(\hat{e}_{3}-e_{1})=-2\hat{e}_{1}\wedge e_{1}(\Pi(E_{3}\wedge E_{4})).$

And finally for $e_{12}$:

$\displaystyle\Pi(e_{12}(E_{1}\wedge E_{2}))$ $\displaystyle=\Pi(-E_{3}\wedge
E_{2})+\Pi(E_{1}\wedge E_{4})=\hat{e}_{1}+e_{2}-\hat{e}_{1}+e_{2}=2e_{2}$
$\displaystyle=-2e_{1}\wedge e_{2}(-\hat{e}_{3}-e_{1})=-2e_{1}\wedge
e_{2}(\Pi(E_{1}\wedge E_{2})),$ $\displaystyle\Pi(e_{12}(E_{1}\wedge E_{4}))$
$\displaystyle=\Pi(-E_{3}\wedge E_{4})+\Pi(E_{1}\wedge-
E_{2})=-\hat{e}_{3}+e_{1}+\hat{e}_{3}+e_{1}=2e_{1}$
$\displaystyle=-2e_{1}\wedge e_{2}(-\hat{e}_{1}+e_{2})=-2e_{1}\wedge
e_{2}(\Pi(E_{1}\wedge E_{4})),$ $\displaystyle\Pi(e_{12}(E_{2}\wedge E_{3}))$
$\displaystyle=\Pi(E_{4}\wedge E_{3})+\Pi(E_{2}\wedge
E_{1})=-\hat{e}_{3}+e_{1}+\hat{e}_{3}+e_{1}=2e_{1}$
$\displaystyle=-2e_{1}\wedge e_{2}(\hat{e}_{1}+e_{2})=-2e_{1}\wedge
e_{2}(\Pi(E_{2}\wedge E_{3})),$ $\displaystyle\Pi(e_{12}(E_{3}\wedge E_{4}))$
$\displaystyle=\Pi(E_{1}\wedge E_{4})+\Pi(E_{3}\wedge-
E_{2})=-\hat{e}_{1}+e_{2}+\hat{e}_{1}+e_{2}=2e_{2}$
$\displaystyle=-2e_{1}\wedge e_{2}(\hat{e}_{3}-e_{1})=-2e_{1}\wedge
e_{2}(\Pi(E_{3}\wedge E_{4})).\qed$

To display the algebra $\mathfrak{g}$ with $q=2,\ l=1$ corresponding to
$\hat{\mathcal{T}}(1)=G/H$ we use the inclusion
$\mathfrak{g}\subset\mathfrak{der}(\mathfrak{r})\subset\mathfrak{gl}(\mathbb{R}^{3,2}\oplus\mathbb{R}D\oplus
W_{2})$. We give these elements as matrices with respect to the basis
$\hat{e}_{1},\hat{e}_{2},\hat{e}_{3},e_{1},e_{2},D,E_{1},E_{2},E_{3},E_{4}$ of
$\mathfrak{r}$.

Recall
$\mathfrak{g}=\mathfrak{so}(3,2)\oplus\mathbb{R}^{3,2}\oplus\mathbb{R}D\oplus
W_{2}$. We begin with $\mathfrak{so}(3,2)$:

$\displaystyle 2\hat{e}_{1}\wedge\hat{e}_{2}=\begin{bmatrix}0&2&&&&0&&&&\\\
-2&0&&&&0&&&&\\\ &&0&&&0&&&&\\\ &&&0&&0&&&&\\\ &&&&0&0&&&&\\\
0&0&0&0&0&0&0&0&0&0\\\ &&&&&0&&&0&1\\\ &&&&&0&&&-1&0\\\ &&&&&0&0&1&&\\\
&&&&&0&-1&0&&\\\ \end{bmatrix},\quad
2\hat{e}_{3}\wedge\hat{e}_{1}=\begin{bmatrix}0&&-2&&&0&&&&\\\ &0&&&&0&&&&\\\
2&&0&&&0&&&&\\\ &&&0&&0&&&&\\\ &&&&0&0&&&&\\\ 0&0&0&0&0&0&0&0&0&0\\\
&&&&&0&&&1&0\\\ &&&&&0&&&0&1\\\ &&&&&0&-1&0&&\\\ &&&&&0&0&-1&&\\\
\end{bmatrix},$ $\displaystyle
2\hat{e}_{2}\wedge\hat{e}_{3}=\begin{bmatrix}0&&&&&0&&&&\\\ &0&2&&&0&&&&\\\
&-2&0&&&0&&&&\\\ &&&0&&0&&&&\\\ &&&&0&0&&&&\\\ 0&0&0&0&0&0&0&0&0&0\\\
&&&&&0&&&0&1\\\ &&&&&0&&&-1&0\\\ &&&&&0&0&1&&\\\ &&&&&0&-1&0&&\\\
\end{bmatrix},$

$\displaystyle 2\hat{e}_{1}\wedge e_{1}=\begin{bmatrix}0&&&-2&&0&&&&\\\
&0&&&&0&&&&\\\ &&0&&&0&&&&\\\ -2&&&0&&0&&&&\\\ &&&&0&0&&&&\\\
0&0&0&0&0&0&0&0&0&0\\\ &&&&&0&&&-1&0\\\ &&&&&0&&&0&-1\\\ &&&&&0&-1&0&&\\\
&&&&&0&0&-1&&\\\ \end{bmatrix},\qquad 2\hat{e}_{1}\wedge
e_{2}=\begin{bmatrix}0&&&&-2&0&&&&\\\ &0&&&&0&&&&\\\ &&0&&&0&&&&\\\
&&&0&&0&&&&\\\ -2&&&&0&0&&&&\\\ 0&0&0&0&0&0&0&0&0&0\\\ &&&&&0&1&0&&\\\
&&&&&0&0&-1&&\\\ &&&&&0&&&-1&0\\\ &&&&&0&&&0&1\\\ \end{bmatrix},$
$\displaystyle 2\hat{e}_{2}\wedge e_{1}=\begin{bmatrix}0&&&&&0&&&&\\\
&0&&-2&&0&&&&\\\ &&0&&&0&&&&\\\ &-2&&0&&0&&&&\\\ &&&&0&0&&&&\\\
0&0&0&0&0&0&0&0&0&0\\\ &&&&&0&&&0&1\\\ &&&&&0&&&-1&0\\\ &&&&&0&0&-1&&\\\
&&&&&0&1&0&&\\\ \end{bmatrix},\qquad 2\hat{e}_{2}\wedge
e_{2}=\begin{bmatrix}0&&&&&0&&&&\\\ &0&&&-2&0&&&&\\\ &&0&&&0&&&&\\\
&&&0&&0&&&&\\\ &-2&&&0&0&&&&\\\ 0&0&0&0&0&0&0&0&0&0\\\ &&&&&0&0&1&&\\\
&&&&&0&1&0&&\\\ &&&&&0&&&0&1\\\ &&&&&0&&&1&0\\\ \end{bmatrix},$ $\displaystyle
2\hat{e}_{3}\wedge e_{1}=\begin{bmatrix}0&&&&&0&&&&\\\ &0&&&&0&&&&\\\
&&0&-2&&0&&&&\\\ &&-2&0&&0&&&&\\\ &&&&0&0&&&&\\\ 0&0&0&0&0&0&0&0&0&0\\\
&&&&&0&-1&0&&\\\ &&&&&0&0&-1&&\\\ &&&&&0&&&1&0\\\ &&&&&0&&&0&1\\\
\end{bmatrix},\qquad 2\hat{e}_{3}\wedge e_{2}=\begin{bmatrix}0&&&&&0&&&&\\\
&0&&&&0&&&&\\\ &&0&&-2&0&&&&\\\ &&&0&&0&&&&\\\ &&-2&&0&0&&&&\\\
0&0&0&0&0&0&0&0&0&0\\\ &&&&&0&&&-1&0\\\ &&&&&0&&&0&1\\\ &&&&&0&-1&0&&\\\
&&&&&0&0&1&&\\\ \end{bmatrix},$

$\displaystyle 2e_{1}\wedge e_{2}=\begin{bmatrix}0&&&&&0&&&&\\\ &0&&&&0&&&&\\\
&&0&&&0&&&&\\\ &&&0&-2&0&&&&\\\ &&&2&0&0&&&&\\\ 0&0&0&0&0&0&0&0&0&0\\\
&&&&&0&&&-1&0\\\ &&&&&0&&&0&1\\\ &&&&&0&1&0&&\\\ &&&&&0&0&-1&&\\\
\end{bmatrix}.$

Recall that $2e_{1}\wedge e_{2}$ generates the isotropy algebra
$\mathfrak{h}$.

Next we describe the element $2D$:
$2D=\begin{bmatrix}2&&&&&0&&&&\\\ &2&&&&0&&&&\\\ &&2&&&0&&&&\\\ &&&2&&0&&&&\\\
&&&&2&0&&&&\\\ 0&0&0&0&0&0&0&0&0&0\\\ &&&&&0&1&&&\\\ &&&&&0&&1&&\\\
&&&&&0&&&1&\\\ &&&&&0&&&&1\\\ \end{bmatrix}.$

The generators of $V=\mathbb{R}^{3,2}$ are:

$\displaystyle\hat{e}_{1}=\begin{bmatrix}&&&&&-1&&&&\\\ &&&&&0&&&&\\\
&&&&&0&&&&\\\ &&&&&0&&&&\\\ &&&&&0&&&&\\\ 0&0&0&0&0&0&0&0&0&0\\\ &&&&&0&&&&\\\
&&&&&0&&&&\\\ &&&&&0&&&&\\\ &&&&&0&&&&\\\
\end{bmatrix},\qquad\hat{e}_{2}=\begin{bmatrix}&&&&&0&&&&\\\ &&&&&-1&&&&\\\
&&&&&0&&&&\\\ &&&&&0&&&&\\\ &&&&&0&&&&\\\ 0&0&0&0&0&0&0&0&0&0\\\ &&&&&0&&&&\\\
&&&&&0&&&&\\\ &&&&&0&&&&\\\ &&&&&0&&&&\\\
\end{bmatrix},\qquad\hat{e}_{3}=\begin{bmatrix}&&&&&0&&&&\\\ &&&&&0&&&&\\\
&&&&&-1&&&&\\\ &&&&&0&&&&\\\ &&&&&0&&&&\\\ 0&0&0&0&0&0&0&0&0&0\\\
&&&&&0&&&&\\\ &&&&&0&&&&\\\ &&&&&0&&&&\\\ &&&&&0&&&&\\\ \end{bmatrix},$
$\displaystyle e_{1}=\begin{bmatrix}&&&&&0&&&&\\\ &&&&&0&&&&\\\ &&&&&0&&&&\\\
&&&&&-1&&&&\\\ &&&&&0&&&&\\\ 0&0&0&0&0&0&0&0&0&0\\\ &&&&&0&&&&\\\
&&&&&0&&&&\\\ &&&&&0&&&&\\\ &&&&&0&&&&\\\ \end{bmatrix},\qquad
e_{2}=\begin{bmatrix}&&&&&0&&&&\\\ &&&&&0&&&&\\\ &&&&&0&&&&\\\ &&&&&0&&&&\\\
&&&&&-1&&&&\\\ 0&0&0&0&0&0&0&0&0&0\\\ &&&&&0&&&&\\\ &&&&&0&&&&\\\
&&&&&0&&&&\\\ &&&&&0&&&&\\\ \end{bmatrix}.$

Finally we have the $4$ basis elements of $W_{2}$:
$\displaystyle 2E_{1}=\begin{bmatrix}&&&&&0&&&&-2\\\ &&&&&0&&&-2&\\\
&&&&&0&&-2&&\\\ &&&&&0&&2&&\\\ &&&&&0&&&&2\\\ 0&0&0&0&0&0&0&0&0&0\\\
&&&&&-1&&&&\\\ &&&&&0&&&&\\\ &&&&&0&&&&\\\ &&&&&0&&&&\\\ \end{bmatrix},\quad$
$\displaystyle 2E_{2}=\begin{bmatrix}&&&&&0&&&2&\\\ &&&&&0&&&&-2\\\
&&&&&0&2&&&\\\ &&&&&0&2&&&\\\ &&&&&0&&&2&\\\ 0&0&0&0&0&0&0&0&0&0\\\
&&&&&0&&&&\\\ &&&&&-1&&&&\\\ &&&&&0&&&&\\\ &&&&&&&&&\\\ \end{bmatrix}$
$\displaystyle 2E_{3}=\begin{bmatrix}&&&&&0&&-2&&\\\ &&&&&0&2&&&\\\
&&&&&0&&&&2\\\ &&&&&0&&&&-2\\\ &&&&&0&&-2&&\\\ 0&0&0&0&0&0&0&0&0&0\\\
&&&&&0&&&&\\\ &&&&&0&&&&\\\ &&&&&-1&&&&\\\ &&&&&0&&&&\\\ \end{bmatrix},\quad$
$\displaystyle 2E_{4}=\begin{bmatrix}&&&&&0&2&&&\\\ &&&&&0&&2&&\\\
&&&&&0&&&-2&\\\ &&&&&0&&&2&\\\ &&&&&0&-2&&&\\\ 0&0&0&0&0&0&0&0&0&0\\\
&&&&&0&&&&\\\ &&&&&0&&&&\\\ &&&&&0&&&&\\\ &&&&&-1&&&&\\\ \end{bmatrix}$

##  4 Nomizu maps of homogeneous $3$-$(\alpha,\delta)$-Sasaki manifolds

###  4.1 Nomizu map of the canonical connection

By the Nomizu theorem invariant connections on reductive homogeneous spaces
$M=G/H$ are in bijective correspondence with isotropy equivariant maps
$\Lambda\colon\mathfrak{m}\times\mathfrak{m}\to\mathfrak{m}$, where
$\mathfrak{m}$ is a reductive complement to the isotropy algebra
$\mathfrak{h}\subset\mathfrak{g}$. For fundamental vector fields $X,Y$
considered as elements in $\mathfrak{m}$ this correspondence is given by
$\Lambda^{\nabla}_{X}Y=\nabla_{X_{0}}Y-[X,Y]_{0}$, compare [KN69, Corollary
2.2, p.191]. By [KN69, Proposition X.2.3, p. 191] the torsion $T^{\nabla}$ of
the connection corresponding to $\Lambda^{\nabla}$ is given by

$T^{\nabla}(X,Y)=\Lambda^{\nabla}_{X}Y-\Lambda^{\nabla}_{Y}X-[X,Y]_{\mathfrak{m}}.$
(4.1)

For the following theorem we need a similar statement to Lemma 1.2.1 for two
fundamental vector fields. Note that even in the case when
$X,Y\in\mathfrak{g}$ are horizontal in the origin they fail to be horizontal
in other points. Yet we have

######  Lemma 4.1.1.

Let $G\curvearrowright M$ act by automorphisms on a
$3$-$(\alpha,\delta)$-Sasaki manifold $M$ from the left, $X,Y\in\mathfrak{g}$
with fundamental vector fields $\hat{X},\hat{Y}\in\mathfrak{X}(M)$. Then

$\widehat{[X,Y]}_{\mathcal{V}}=-\sum_{i=1}^{3}\big{(}2\alpha\Phi_{i}(\hat{X},\hat{Y})+2(\alpha-\delta)\eta_{j}\wedge\eta_{k}(\hat{X},\hat{Y})\big{)}\xi_{i}.$

###### Proof.

Since $G$ is a group of automorphisms we have $L_{\hat{X}}\eta_{i}=0$ so

$\displaystyle d\eta_{i}(\hat{X},\hat{Y})$
$\displaystyle=\hat{X}(\eta_{i}(\hat{Y}))-\hat{Y}(\eta_{i}(\hat{X}))-\eta_{i}([\hat{X},\hat{Y}])$
$\displaystyle=\eta_{i}([\hat{X},\hat{Y}])-\eta_{i}([\hat{Y},\hat{X}])-\eta_{i}([\hat{X},\hat{Y}])=\eta_{i}([\hat{X},\hat{Y}])=-\eta_{i}(\widehat{[X,Y]}).$

Using
$\widehat{[X,Y]}_{\mathcal{V}}=\sum_{i=1}^{3}\eta_{i}(\widehat{[X,Y]})\xi_{i}$
and (1.1) yields the result. ∎

######  Theorem 4.1.1.

The Nomizu map for the canonical connection
$\Lambda^{\nabla}\colon\mathfrak{m}\times\mathfrak{m}\to\mathfrak{m}$ of a
homogeneous $3$-$(\alpha,\delta)$-Sasaki manifold with
$\mathfrak{m}=\mathcal{V}\oplus\mathcal{H}$ is given by

$\Lambda^{\nabla}_{X}Y=\begin{cases}\Lambda^{g_{N}}_{X}Y&X,Y\in\mathcal{H}\\\
\frac{\beta}{2\delta}[X,Y]&X,Y\in\mathcal{V}\\\
[X,Y]-2\alpha\sum_{i=1}^{3}\eta_{i}(X)\varphi_{i}Y&X\in\mathcal{V},Y\in\mathcal{H}\\\
0&X\in\mathcal{H},Y\in\mathcal{V},\end{cases}$

where $\Lambda^{g_{N}}\colon\mathcal{H}\times\mathcal{H}\to\mathcal{H}$ is the
Nomizu map of the Levi-Civita connection on the homogeneous base of the
canonical submersion.

###### Proof.

We first prove that the torsion of $\Lambda^{\nabla}$ given by (4.1) agrees
with the canonical torsion

$T^{\nabla}=2\alpha\sum_{i=1}^{3}\eta_{i}\wedge\Phi_{i}^{\mathcal{H}}+2(\delta-4\alpha)\eta_{123}.$

We begin with the case $X,Y\in\mathcal{H}$. Evaluating Lemma 4.1.1 in the
origin $T_{0}M\cong\mathfrak{m}$ we obtain
$[X,Y]_{\mathcal{V}}=-2\alpha\sum_{i=1}^{3}\Phi_{i}(X,Y)\xi_{i}$. Thus,

$\displaystyle\Lambda^{\nabla}_{X}Y-\Lambda^{\nabla}_{Y}X-[X,Y]$
$\displaystyle=\Lambda^{g_{N}}_{X}Y-\Lambda^{g_{N}}_{Y}X-[X,Y]_{\mathcal{H}}-[X,Y]_{\mathcal{V}}$
$\displaystyle=T^{g_{N}}(X,Y)+2\alpha\sum_{i=1}^{3}\Phi_{i}(X,Y)\xi_{i}=0+2\alpha\sum_{i=1}^{3}\Phi_{i}(X,Y)\xi_{i}=T^{\nabla}(X,Y)$

as the torsion $T^{g_{N}}=0$ of the Levi-Civita connection on the base
vanishes. Suppose now that $X=\xi_{i}$ and $Y\in\mathcal{H}$. Then

$\Lambda^{\nabla}_{X}Y-\Lambda^{\nabla}_{Y}X-[X,Y]=[X,Y]-2\alpha\varphi_{i}Y-[X,Y]=-2\alpha\varphi_{i}Y=T^{\nabla}(X,Y).$

Finally if both $X=\xi_{i},Y=\xi_{j}\in\mathcal{V}$ and $(ijk)$ an even
permutation of $(123)$ we have

$\displaystyle\Lambda^{\nabla}_{X}Y-\Lambda^{\nabla}_{Y}X-[X,Y]$
$\displaystyle=\frac{\beta}{2\delta}[X,Y]-\frac{\beta}{2\delta}[Y,X]-[X,Y]=(2-\frac{4\alpha}{\delta}-1)[\xi_{i},\xi_{j}]$
$\displaystyle=(1-\frac{4\alpha}{\delta})2\delta\xi_{k}=T^{\nabla}(X,Y).$

We further need to verify that
$\Lambda^{\nabla}_{X}\in\mathfrak{so}(\mathfrak{m})$ for all
$X\in\mathfrak{m}$, that is
$g(\Lambda^{\nabla}_{X}Y,Z)+g(Y,\Lambda_{X}^{\nabla}Z)=0$ for all
$Y,Z\in\mathfrak{m}$. Suppose $X\in\mathcal{H}$ and $Y,Z\in\mathcal{H}$. Then
$\Lambda^{\nabla}_{X}=\Lambda^{g_{N}}_{X}$ and $g|_{\mathcal{H}}=g_{N}$ thus
$\Lambda^{\nabla}_{X}$ is metric as $\Lambda^{g_{N}}_{X}$ is. If
$Y\in\mathcal{V}$ and $Z\in\mathcal{H}$ we find that $\Lambda^{\nabla}_{X}Y=0$
by definition while $\Lambda^{\nabla}_{X}Z\in\mathcal{H}$ is orthogonal to
$Y$. Analogously, if $Y,Z\in\mathcal{V}$ the Nomizu map $\Lambda^{\nabla}_{X}$
acts trivially on both sides. Now suppose $X\in\mathcal{V}$. Then by [AD20,
Corollary 2.3.1] $X$ is Killing as a linear combination of the $\xi_{i}$ and
thus the Lie derivative $L_{X}\in\mathfrak{so}(\mathfrak{m})$. In particular,
if $Y,Z\in\mathcal{V}$ then
$\Lambda^{\nabla}_{X}Y=\frac{\beta}{2\delta}L_{X}Y$ is metric. If
$Y\in\mathcal{H}$ we have

$g(\Lambda^{\nabla}_{X}Y,Z)=g(L_{X}Y-2\alpha\sum\eta_{i}(X)\varphi_{i}Y,Z)=-g(Y,L_{X}Z-2\alpha\sum\eta_{i}(X)\varphi_{i}Z).$

If $Z\in\mathcal{H}$ the right hand side is just $-g(Y,\Lambda^{\nabla}_{X}Z)$
by definition while for $Z\in\mathcal{V}$

$L_{X}Z-2\alpha\sum\eta_{i}(X)\varphi_{i}Z\in\mathcal{V}$

is perpendicular to $Y$. Hence,
$g(\Lambda^{\nabla}_{X}Y,Z)=0=-g(Y,\Lambda^{\nabla}_{X}Z)$. ∎

######  Remark 4.1.1.

For symmetric spaces the Levi-Civita connection corresponds to the trivial
Nomizu map $\Lambda^{g_{N}}=0$. The Nomizu map of the Alekseevsky base is
given in [Co00, Lemma 5, p. 35].

###  4.2 Nomizu maps in the symmetric base case

In the case of a positive homogeneous $3$-$(\alpha,\delta)$-Sasaki manifold or
its non-compact sibling the Nomizu map $\Lambda^{\nabla}$ simplifies
drastically.

######  Proposition 4.2.1.

The canonical connection $\nabla$ of a homogeneous
$3$-$(\alpha,\delta)$-Sasaki manifold over a Wolf space or its non-compact
dual corresponds to the map

$\Lambda^{\nabla}_{X}=\begin{cases}0,&X\in\mathcal{H}\\\
\frac{\beta}{2\delta}\mathrm{ad}\,X,&X\in\mathcal{V}.\end{cases}$

###### Proof.

In the case of a Riemannian symmetric space the Levi–Civita connection agrees
with the Ambrose–Singer connection. Thus, $\Lambda^{g_{N}}\equiv 0$. Now let
$X\in\mathcal{V}$ and $Y\in\mathcal{H}$. Then

$2\alpha\sum_{i=1}^{3}\eta_{i}(X)\varphi_{i}|_{\mathcal{H}}=\frac{2\alpha}{\delta}\sum_{i=1}^{3}\eta_{i}(X)\mathrm{ad}\,{\xi_{i}}=\frac{2\alpha}{\delta}\mathrm{ad}\,X.$

It follows
$\Lambda^{\nabla}_{X}Y=(1-\frac{2\alpha}{\delta})[X,Y]=\frac{\beta}{2\delta}[X,Y]$.
∎

######  Remark 4.2.1.

As noted in Remark 3.1.2 the homogeneous $3$-$(\alpha,\delta)$-Sasaki space is
naturally reductive if and only if $\beta=0$. In this case the Ambrose-Singer
connection is metric. In fact, Proposition 4.2.1 shows that in this case the
canonical and Ambrose-Singer connections agree.

######  Proposition 4.2.2.

The Levi–Civita connection corresponds to the map

$\Lambda^{g}_{X}Y=\begin{cases}\frac{1}{2}[X,Y]_{\mathfrak{m}}&X,Y\in\mathcal{V}\text{
or }X,Y\in\mathcal{H}\\\
(1-\frac{\alpha}{\delta})[X,Y]&X\in\mathcal{V},Y\in\mathcal{H}\\\
\frac{\alpha}{\delta}[X,Y]&X\in\mathcal{H},Y\in\mathcal{V}.\end{cases}$

###### Proof.

Note that the correspondence is $\Lambda^{\nabla}_{X}Y=\nabla_{X}Y-[X,Y]$, for
$X,Y\in\mathfrak{m}$, and the canonical connection is given by
$\nabla=\nabla^{g}+\frac{1}{2}T$ where the canonical torsion is given by
(1.3), or equivalently (1.4). Thus we have
$\Lambda^{g}_{X}Y=\nabla^{g}_{X}Y-[X,Y]=\nabla_{X}Y-[X,Y]-\frac{1}{2}T(X,Y)=\Lambda^{\nabla}_{X}Y-\frac{1}{2}T(X,Y)$.
Again we look at each case individually. Let $X,Y\in\mathcal{H}$. Then

$\displaystyle\frac{1}{2}T(X,Y)$
$\displaystyle=\alpha\sum_{i=1}^{3}\Phi_{i}(X,Y)\xi_{i}=\alpha\sum_{i=1}^{3}g(X,\varphi_{i}Y)\xi_{i}$
$\displaystyle=\frac{\alpha}{\delta}\sum_{i=1}^{3}g(X,[\xi_{i},Y])\xi_{i}=\frac{\alpha}{8\alpha\delta^{2}(n+2)}\sum_{i=1}^{3}\kappa(X,[Y,\xi_{i}])\xi_{i}$
$\displaystyle=\frac{1}{8\delta^{2}(n+2)}\sum_{i=1}^{3}\kappa([X,Y],\xi_{i})\xi_{i}$
$\displaystyle=-\frac{1}{2}\sum_{i=1}^{3}g([X,Y]_{\mathfrak{m}},\xi_{i})\xi_{i}=-\frac{1}{2}[X,Y]_{\mathfrak{m}},$

where we have used that $\kappa(\mathfrak{h},\mathcal{V})=0$ and
$[X,Y]_{\mathfrak{m}}\in\mathcal{V}=\mathrm{span}\\{\xi_{1},\xi_{2},\xi_{3}\\}$.
Thus

$\Lambda^{g}_{X}Y=\Lambda^{\nabla}_{X}Y-\frac{1}{2}T(X,Y)=\frac{1}{2}[X,Y]_{\mathfrak{m}}$

For vertical vectors $X=\xi_{i},Y=\xi_{j}$, $(ijk)$ an even permutation of
$(123)$, we find

$\displaystyle\Lambda^{g}_{\xi_{i}}\xi_{j}$
$\displaystyle=\Lambda^{\nabla}_{\xi_{i}}\xi_{j}-(\delta-4\alpha)\eta_{123}(\xi_{i},\xi_{j},\cdot)=\frac{\beta}{2\delta}[\xi_{i},\xi_{j}]-(\delta-4\alpha)\xi_{k}$
$\displaystyle=\beta\xi_{k}+(\delta-4\alpha)\xi_{k}=\delta\xi_{k}=[\xi_{i},\xi_{j}]$

and by linearity for arbitrary $X,Y\in\mathcal{V}$. Let
$X\in\mathcal{V},Y\in\mathcal{H}$ then

$\displaystyle T(X,Y)$
$\displaystyle=2\alpha\sum_{i=1}^{3}\eta_{i}\wedge\Phi_{i}^{\mathcal{H}}(X,Y,\cdot\,)=2\alpha\sum_{i=1}^{3}\eta_{i}(X)\Phi_{i}^{\mathcal{H}}(Y,\cdot\,)$
$\displaystyle=-2\alpha\sum_{i=1}^{3}\eta_{i}(X)\varphi_{i}Y=-\frac{2\alpha}{\delta}\left[\sum_{i=1}^{3}\eta_{i}(X)\xi_{i},Y\right]=-\frac{2\alpha}{\delta}[X,Y]$

and thus

$\displaystyle\Lambda^{g}_{X}Y$
$\displaystyle=\Lambda^{\nabla}_{X}Y-\frac{1}{2}T(X,Y)=(1-\frac{2\alpha}{\delta})[X,Y]+\frac{\alpha}{\delta}[X,Y]=(1-\frac{\alpha}{\delta})[X,Y].$

For the final expression $X\in\mathcal{H},Y\in\mathcal{V}$ use the above
identity for $T$ with $X\leftrightarrow Y$. Then

$\displaystyle\Lambda^{g}_{X}Y=\Lambda^{\nabla}_{X}Y-\frac{1}{2}T(X,Y)$
$\displaystyle=\frac{1}{2}T(Y,X)=\frac{\alpha}{\delta}[X,Y].\qed$

The Nomizu maps allow for a detailed investigation of homogeneous
$3$-$(\alpha,\delta)$-Sasaki manifolds. A thorough discussion of curvature
operators and properties will be carried out in an upcoming publication
[ADS21].

### Declarations

No funding was received to assist with the preparation of this manuscript. The
authors have no relevant financial or non-financial interests to disclose.
Data sharing is not applicable to this article as no datasets were generated
or analysed during the current study.

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Ilka Agricola and Leander Stecker, Fachbereich Mathematik und Informatik,
Philipps-Universität Marburg, Campus Lahnberge, 35032 Marburg, Germany.
<EMAIL_ADDRESS><EMAIL_ADDRESS>

Giulia Dileo, Dipartimento di Matematica, Università degli Studi di Bari Aldo
Moro, Via E. Orabona 4, 70125 Bari, Italy<EMAIL_ADDRESS>