Homogeneous non-degenerate 3-(α,δ)-Sasaki manifolds and submersions over quaternionic Kähler spaces

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.

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 [9]. 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 [16]. For qK manifolds with negative scalar curvature one does not obtain a 3-Sasaki manifold but a so-called pseudo 3-Sasaki structure [19]. 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 [2]. 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 Sect. 1. Using results by Cleyton et al.  [11] 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 Sect. 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 [12] (see also [6, 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 Cortés [10]. 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

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

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 $N_\varphi :=[\varphi ,\varphi ]+d\eta \otimes \xi$ vanishes, where $[\varphi ,\varphi ]$ is the Nijenhuis torsion of $\varphi$ [7].

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 $\eta \wedge (d\eta )^n\ne 0$ 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

$$\begin{aligned} \varphi _k=\varphi _i\varphi _j-\eta _j\otimes \xi _i=-\varphi _j\varphi _i+\eta _i\otimes \xi _j,\quad \xi _k=\varphi _i\xi _j=-\varphi _j\xi _i, \quad \eta _k=\eta _i\circ \varphi _j=-\eta _j\circ \varphi _i \end{aligned}$$

for any even permutation (ijk) of (123) [7]. 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

$$\begin{aligned} {\mathcal {H}}\, :=\, \bigcap _{i=1}^{3}\ker \eta _i,\quad {\mathcal {V}}\, :=\, \langle \xi _1,\xi _2,\xi _3\rangle . \end{aligned}$$

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 [8]. In the recent paper [2] 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

$$\begin{aligned} d\eta _i=2\alpha \Phi _i+2(\alpha -\delta )\eta _j\wedge \eta _k \end{aligned}$$

(1)

for every even permutation (ijk) of (123), where $\alpha \ne 0$ 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 [2, Section 2.3]:

$$\begin{aligned}&\eta _i'=c\eta _i,\quad \xi _i'=\frac{1}{c}\xi _i,\quad \varphi _i'=\varphi _i,\quad g'=ag+b\sum _{i=1}^3\eta _i\otimes \eta _i, \qquad a>0, \,c^2=a+b>0. \end{aligned}$$

The deformed structure $(\varphi ',\xi _i',\eta ',g')$ turns out to be 3-$(\alpha ',\delta ')$-Sasaki with $\alpha '=\alpha c/a$, $\delta '=\delta /c$. 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 [2]. Any 3-$(\alpha ,\delta )$-Sasaki manifold is shown to be hypernormal, thus generalizing Kashiwada’s theorem [14]. 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

$$\begin{aligned}{}[\xi _i,\xi _j]=2\delta \xi _k. \end{aligned}$$

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

$$\begin{aligned} {[X,Y]_{\mathcal {V}}}=-2\alpha {\sum _{i=1}^3} \Phi _i (X,Y) {\xi _i}. \end{aligned}$$

Proof

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

$$\begin{aligned}{}[X,Y]_{\mathcal V}=\sum _{i=1}^3\eta _i([X,Y])\xi _i=-\sum _{i=1}^3d\eta _i(X,Y)\xi _i=-2\alpha \sum _{i=1}^3\Phi _i(X,Y)\xi _i. \end{aligned}$$

$\square$

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 [2], 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 [1] 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

$$\begin{aligned} T(X,Y,Z)=g(T(X,Y),Z) \end{aligned}$$

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

$$\begin{aligned} \nabla _XY=\nabla ^g_XY+\frac{1}{2}T(X,Y). \end{aligned}$$

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 $T=\eta \wedge d\eta$ [13]. 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

$$\begin{aligned} \nabla _X\varphi _i\, =\, \beta (\eta _k(X)\varphi _j -\eta _j(X)\varphi _k) \quad \forall X\in {{\mathfrak {X}}}(M) \end{aligned}$$

(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

$$\begin{aligned} \nabla _X\xi _i=\beta (\eta _k(X)\xi _j-\eta _j(X)\xi _k),\qquad \nabla _X\eta _i=\beta (\eta _k(X)\eta _j-\eta _j(X)\eta _k). \end{aligned}$$

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

$$\begin{aligned} T\ =\ 2\alpha \sum _{i=1}^3\eta _i\wedge \Phi _i-2(\alpha -\delta )\eta _{123}\ =\ 2\alpha \sum _{i=1}^3\eta _i\wedge \Phi ^{\mathcal H}_i+2(\delta -4 \alpha )\,\eta _{123}, \end{aligned}$$

(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 $\eta _{jk}:=\eta _j\wedge \eta _k$ and $\eta _{123}:=\eta _1\wedge \eta _2\wedge \eta _3$. In particular, for every $X,Y\in {\mathfrak {X}}(M)$,

$$\begin{aligned} T(X,Y)=2\alpha \sum _{i=1}^3\{\eta _i(Y)\varphi _iX-\eta _i(X)\varphi _iY+\Phi _i(X,Y)\xi _i\}-2(\alpha -\delta ) {\mathop {\mathfrak {S}}\limits ^{i,j,k}}\eta _{ij}(X,Y)\xi _k. \end{aligned}$$

(4)

The symbol ${\mathop {\mathfrak {S}}\limits ^{i,j,k}}$ 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 [3]. 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

[3] 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

$$\begin{aligned} R(X,\xi _i,Y,\xi _j)&=R(X,Y,Z,\xi _i)=R(\xi _i,\xi _j,\xi _k,X)=0, \end{aligned}$$

(5)

$$\begin{aligned} R(\xi _i,\xi _j,\xi _k,\xi _l)&=-4\alpha \beta (\delta _{ik}\delta _{jl}-\delta _{il}\delta _{jk}), \end{aligned}$$

(6)

$$\begin{aligned} R(\xi _i,\xi _j,X,Y)&=2\alpha \beta \Phi _k(X,Y), \end{aligned}$$

(7)

$$\begin{aligned} R(X,Y,Z,\varphi _iZ)+R(X,Y,\varphi _jZ,\varphi _kZ)&=2\alpha \beta \Phi _i(X,Y)\Vert Z\Vert ^2, \end{aligned}$$

(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 [11] 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 $\mathfrak {k}_a=\mathfrak {so}(\mathfrak {h}_a)\cap \mathfrak {hol}\ne \{0\}$ of $\mathfrak {hol}$ acting purely on $\mathfrak {h}_a$ is non-trivial.

We need a slight generalization of the results obtained in [11]. 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

$$\begin{aligned} {\mathcal {H}}_{\Gamma }:=\bigoplus _{a=1}^s\mathfrak {h}_a\oplus \bigoplus _{j\in \Gamma _0\setminus \Gamma }{\mathfrak {v}}_j, \quad {\mathcal {V}}_{\Gamma }:=\bigoplus _{j\in \Gamma }{\mathfrak {v}}_j, \end{aligned}$$

(9)

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

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

(10)

This condition turns out to be sufficient to prove Lemmas 3.7-3.10 and Remark 3.11 from [11]. We obtain

Corollary 2.1.1

Suppose the decomposition $TM={\mathcal {V}}_\Gamma \oplus {\mathcal {H}}_\Gamma$ from (9) fulfills condition (10). Then

• (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,

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

• (c) $\nabla ^\sigma :=\nabla ^{g_N}+\frac{1}{2}\sigma$ defines a connection with parallel skew torsion $\sigma$ on N. In particular, we have

  $$\begin{aligned} \nabla ^\sigma _XY=\pi _*(\nabla ^\tau _{\overline{X}}\overline{Y}), \end{aligned}$$

  (11)

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

Equation (11) is not stated explicitely in [11] but follows directly from $\nabla ^{g_N}_{X}{Y}=\pi _*(\nabla ^{g}_{\overline{X}}\overline{Y})$ for Riemannian submersions [17, Prop. 13]. To a Riemannian submersion one assigns the O’Neill tensors

$$\begin{aligned} \mathcal {A}_XY=(\nabla ^g_{X_{{\mathcal {H}}}}Y_{{\mathcal {H}}})_{\mathcal {V}}+(\nabla ^g_{X_{{\mathcal {H}}}}Y_{\mathcal {V}})_{{\mathcal {H}}}, \qquad \mathcal {T}_XY=(\nabla ^g_{X_{\mathcal {V}}}Y_{{\mathcal {H}}})_{\mathcal {V}}+(\nabla ^g_{X_{\mathcal {V}}}Y_{\mathcal {V}})_{{\mathcal {H}}}. \end{aligned}$$

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

$$\begin{aligned} g(\mathcal {A}_XY,Z)=-\frac{1}{2}(\tau (X_{{\mathcal {H}}_\Gamma },Y_{{\mathcal {H}}_\Gamma },Z_{{\mathcal {V}}_\Gamma })+\tau (X_{{\mathcal {H}}_\Gamma },Y_{{\mathcal {V}}_\Gamma },Z_{{\mathcal {H}}_\Gamma })),\quad \mathcal {T}=0. \end{aligned}$$

Proof

Since ${\mathcal {H}}_\Gamma$ and ${\mathcal {V}}_\Gamma$ are $\nabla ^\tau$-holonomy invariant $(\nabla ^\tau _XY_{{\mathcal {H}}_\Gamma })_{{\mathcal {V}}_\Gamma }=(\nabla ^\tau _XY_{{\mathcal {V}}_\Gamma })_{{\mathcal {H}}_\Gamma }=0$. 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 (10). $\square$

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 (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 2.1.2

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

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

where $P_{\gamma }$ denotes parallel transport along $\gamma$ and $R(X,Y)\in \mathfrak {so}(T_qM)$ 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 (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 (6) and (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 (7). Any such element of $\mathfrak {hol}$ acts non-trivially on ${\mathcal {V}}$ if $\beta \ne 0$ and $\Phi _i(X,Y)\ne 0$ for some $i=1,2,3$. In this case R(X, Y) is also a non-trivial operator on ${\mathcal {H}}$ by (8). $\square$

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 $\pi :M\rightarrow N$ such that

$$\begin{aligned} \nabla ^{g_N}_XY=\pi _*(\nabla _{\overline{X}}\overline{Y}). \end{aligned}$$

(12)

Definition 2.1.1

We will call $\pi :M\rightarrow N$ the canonical submersion of a 3-$(\alpha ,\delta )$-Sasaki manifold.

Proof (of Proposition 2.1.1)

By (2) and Lemma 2.1.2 the decomposition $TM={\mathcal {V}}\oplus {\mathcal {H}}$ is of type (9) with respect to the canonical connection $\nabla$. By (3) the projection of the torsion onto ${\mathcal {H}}\otimes \Lambda ^2{\mathcal {V}}$ vanishes, satisfying (10). Therefore the conditions of Corollary 2.1.1 are satisfied. Moreover, (3) shows that the projection of $\tau$ onto $\Lambda ^3{\mathcal {H}}$ vanishes so the connection $\nabla ^\sigma$ in (11) for the canonical submersion is the Levi-Civita connection $\nabla ^{g_N}$ on N. $\square$

We observe that the canonical submersion is, indeed, an almost contact metric 3-submersion in the sense of [20], 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

$$\begin{aligned} (\nabla _{X}Y)_{{\mathcal {H}}}=-2\alpha \sum _{i=1}^3\eta _i(X)\varphi _i Y. \end{aligned}$$

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 [17, 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

$$\begin{aligned} g(\nabla ^g_{\xi _i}Y,Z)=-\frac{1}{2} g([Y,Z],\xi _i)=\alpha \Phi _i(Y,Z). \end{aligned}$$

Therefore

$$\begin{aligned} g(\nabla _{X}Y,Z)&=g(\nabla ^g_{X}Y,Z)+\frac{1}{2} T(X,Y,Z)=\sum _{i=1}^3\eta _i(X)(\alpha \Phi _i(Y,Z)+\alpha \Phi _i(Y,Z))\\&=-2\alpha \sum _{i=1}^3\eta _i(X)g(\varphi _i Y,Z). \end{aligned}$$

$\square$

Theorem 2.2.1

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

$$\begin{aligned} \check{\varphi _i}=\pi _*\circ \varphi _i\circ s_*,\quad i=1,2,3, \end{aligned}$$

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

$$\begin{aligned} \nabla ^{g_N}_X\check{\varphi }_i=2\delta (\check{\eta }_k(X)\check{\varphi }_j-\check{\eta }_j(X)\check{\varphi }_k), \end{aligned}$$

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 $\pi :M\rightarrow N$, hence $\pi _*\circ s_*={\mathrm {id}}$ and ${\text {Im}}(s_*\circ \pi _*-{\mathrm {id}})\subset {\mathcal {V}}$ on the image $s(N)\subset M$. Define

$$\begin{aligned} \check{\varphi }_i=\pi _*\circ \varphi _i\circ s_* \end{aligned}$$

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

$$\begin{aligned} \check{\varphi }_i\check{\varphi }_j&=\check{\varphi }_i\circ (\pi _*\circ \varphi _j\circ s_*) =\pi _*\circ (\varphi _i \varphi _j) \circ s_*. \end{aligned}$$

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 ± 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 (12) and (2), we show that the quaternionic structure is parallel. First

$$\begin{aligned} (\nabla ^{g_N}_X\check{\varphi }_i)Y&=(\nabla ^{g_N}_X(\check{\varphi }_iY))-(\check{\varphi }_i(\nabla ^{g_N}_XY)) =\pi _*\nabla _{\overline{X}}\overline{(\check{\varphi }_iY)}-\check{\varphi }_i\left( \pi _*(\nabla _{\overline{X}}\overline{Y})\right) \\&=\pi _*\nabla _{\overline{X}}\overline{\left( \pi _*\left( \varphi _i( s_*Y)\right) \right) }-\pi _*\left( \varphi _i\left( s_*\left( \pi _*\left( \nabla _{\overline{X}}\overline{Y}\right) \right) \right) \right) . \end{aligned}$$

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

$$\begin{aligned} \nabla _{\overline{X}}\overline{\left( \pi _*\left( \varphi _i( s_*Y)\right) \right) }&=\nabla _{s_*X}(\varphi _i( s_*Y))_{{\mathcal {H}}}+\nabla _{\hat{X}}(\overline{\left( \pi _*\left( \varphi _i( s_*Y)\right) \right) })\\&=\left( \nabla _{s_*X}\left( \varphi _i( s_*Y)\right) \right) _{{\mathcal {H}}}-2\alpha \sum _{l=1}^3\eta _l(\hat{X})\varphi _l \left( \varphi _i( s_*Y)\right) _{{\mathcal {H}}}. \end{aligned}$$

For the second summand, the horizontal projection is given by

$$\begin{aligned} \left( \varphi _i\left( s_*\left( \pi _*\left( \nabla _{\overline{X}}\overline{Y}\right) \right) \right) \right) _{{\mathcal {H}}}&=\varphi _i\left( s_*\left( \pi _*\left( \nabla _{\overline{X}}\overline{Y}\right) \right) \right) _{{\mathcal {H}}}\\&=\varphi _i\left( \nabla _{s_*X}(s_*Y)\right) _{{\mathcal {H}}}+\varphi _i\nabla _{\hat{X}}(\overline{Y})\\&=\left( \varphi _i\left( \nabla _{s_*X}(s_*Y)\right) \right) _{{\mathcal {H}}}-2\alpha \sum _{l=1}^3\eta _l(\hat{X})\varphi _i(\varphi _l (s_*Y))_{{\mathcal {H}}}. \end{aligned}$$

Recombining both identities we obtain

$$\begin{aligned} (\nabla ^{g_N}_X\check{\varphi }_i)Y&=\pi _*\big (\nabla _{s_*X}\left( \varphi _i( s_*Y)\right) -\varphi _i\left( \nabla _{s_*X}(s_*Y)\right) \\&\quad +2\alpha \sum _{l=1}^3\eta _l(\hat{X})(\varphi _i\varphi _l (s_*Y)_{{\mathcal {H}}}-\varphi _l\varphi _i(s_*Y)_{{\mathcal {H}}})\big )\\&=\pi _*\big ((\nabla _{s_*X}\varphi _i)s_*Y-2\alpha \sum _{l=1}^3\eta _l(s_*X)(\varphi _i\varphi _l (s_*Y)_{{\mathcal {H}}}-\varphi _l\varphi _i(s_*Y)_{{\mathcal {H}}})\big )\\&=(\beta +4\alpha )\big ((\eta _k(s_*X)\circ s)\pi _*(\varphi _j(s_*Y))-(\eta _j(s_*X)\circ s)\pi _*(\varphi _k(s_*Y))\big )\\&=2\delta (\check{\eta }_k(X)\check{\varphi }_j-\check{\eta }_j(X)\check{\varphi }_k)Y. \end{aligned}$$

Here we used the defining identity (2) of the canonical connection for any even permutation (ijk) of (123). Therefore, the quaternionic structure is parallel and N is quaternionic Kähler. $\square$

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 Piccinni and Vaisman [18], one can see that the quaternionic structure is preserved under the Bott connection $\mathring{D}:{\mathcal {V}}\times {\mathcal {H}}\rightarrow {\mathcal {H}}$ defined by $\mathring{D}_VX=[V,X]_{{\mathcal {H}}}$, since

$$\begin{aligned} (\mathring{D}_{\xi _i}\varphi _j)X=[\xi _i,\varphi _j X]_{{\mathcal {H}}}-\varphi _j[\xi _i,X]_{\mathcal {H}}=((\mathcal {L}_{\xi _i}\varphi _j)X)_{\mathcal {H}}=2\delta \epsilon _{ijk}\varphi _kX. \end{aligned}$$

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

$$\begin{aligned} {\mathrm {scal}}_{g_N}=16n(n+2)\alpha \delta . \end{aligned}$$

Proof

By Lemma 2.1.1 and (3) for $X,Y\in {\mathcal {H}}$

$$\begin{aligned} \mathcal {A}_XY=-\frac{1}{2} T(X,Y)_{\mathcal {V}}=-\alpha \sum _{i=1}^3 \Phi _i(X,Y)\xi _i. \end{aligned}$$

Let $e_1,\dots , e_{4n}$ be a local adapted frame for ${\mathcal {H}}$, i.e. an orthonormal frame such that $\varphi _1e_{4p+1}=e_{4p+2}$, $\varphi _2e_{4p+1}=e_{4p+3}$ and $\varphi _3e_{4p+1}=e_{4p+4}$, $0\le p\le n-1$. Then

$$\begin{aligned} \sum _{i,j=1}^{4n}g(A_{e_i}e_j,A_{e_i}e_j)=\alpha ^2\sum _{i,j=1}^{4n}\sum _{k=1}^3\Phi ^2_k(e_i,e_j)=\alpha ^2\cdot 3\cdot 4n=12n\alpha ^2. \end{aligned}$$

From the O’Neill identities one obtains for submersions with $\mathcal {T}=0$ the Ricci curvature identity [5, Proposition 9.36]

$$\begin{aligned} {\mathrm {Ric}}_g(X,Y)=\pi ^*{\mathrm {Ric}}_{g_N}(X,Y)-2\sum _{j=1}^{4n}g(A_Xe_j,A_Ye_j). \end{aligned}$$

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 [2, Proposition 2.3.3]. Combining both identities we have

$$\begin{aligned} {\mathrm {scal}}_{g_N}&=\sum _{i=1}^{4n}{\mathrm {Ric}}_{g_N}(\pi _*e_i,\pi _*e_i)=\sum _{i=1}^{4n}{\mathrm {Ric}}_g(e_i,e_i) +2\sum _{i,j=1}^{4n}g(A_{e_i}e_j,A_{e_i}e_j)\\&=4n\cdot 2\alpha (2\delta (n+2)-3\alpha )+24n\alpha ^2=16n(n+2)\alpha \delta . \end{aligned}$$

$\square$

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 [8, 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. Alekseevsky conjectured that all homogeneous qK spaces with negative scalar curvature are Alekseevsky spaces [4]. 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

$$\begin{aligned} \mathfrak h^{\mathbb {C}} \oplus \mathfrak {sp}(1)^{\mathbb {C}} \ni (A,B)\cdot ((z_1,z_2)\otimes w) =B(z_1,z_2)\otimes w + (z_1,z_2)\otimes Aw, \end{aligned}$$

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

• (a) For compact G this is called 3-Sasaki data in [12, Definition 12, p. 12].

• (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 $T_pM\cong \mathfrak {m}$. 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

  $$\begin{aligned} {[}\mathfrak {h},\mathfrak {h}]\subset \mathfrak {h}\quad [{\mathcal {V}},{\mathcal {V}}]\subset {\mathcal {V}}\quad [\mathfrak {h},{\mathcal {V}}]=0\\ {[}\mathfrak {h},{\mathcal {H}}]\subset {\mathcal {H}}\quad [{\mathcal {V}},{\mathcal {H}}]\subset {\mathcal {H}}\quad [{\mathcal {H}},{\mathcal {H}}]\subset {\mathcal {V}}\oplus \mathfrak {h}. \end{aligned}$$

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

• (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

  $$\begin{aligned} \mathfrak {g}^*=\mathfrak {h}\oplus {\mathcal {V}}\oplus i{\mathcal {H}}\subset \mathfrak {g}^\mathbb {C}. \end{aligned}$$

  (13)

  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 $G^*$.

• (d) By [12] any 3-Sasaki data gives rise to a homogeneous 3-Sasaki manifold. They were completely determined in [9] 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 [5, p. 409]. Alltogether, we obtain Table 1.

Table 1 Complete table of generalized 3-Sasaki data

Theorem 3.1.1

Consider some generalized 3-Sasaki data $(G,G_0,H)$ and $0\ne \alpha ,\delta \in \mathbb {R}$. 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 $T_pM=T_p(G/H)\cong \mathfrak {m}$ by

$$\begin{aligned} g\vert _{\mathcal {V}}&=\frac{-\kappa }{4\delta ^2(n+2)},\quad g\vert _{{\mathcal {H}}}=\frac{-\kappa }{8\alpha \delta (n+2)},\quad {\mathcal {V}}\perp {\mathcal {H}}. \end{aligned}$$

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

$$\begin{aligned} \sigma _1= \begin{bmatrix} i &{}0\\ 0&{}-i \end{bmatrix},\quad \sigma _2= \begin{bmatrix} 0 &{}-1\\ 1&{}0 \end{bmatrix},\quad \sigma _3= \begin{bmatrix} 0&{}-i\\ -i &{}0 \end{bmatrix}. \end{aligned}$$

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

$$\begin{aligned} \varphi _i\vert _{\mathcal {V}}&=\frac{1}{2\delta } {\mathrm {ad}}\,\xi _i,\quad \varphi _i\vert _{{\mathcal {H}}}=\frac{1}{\delta }{\mathrm {ad}}\,\xi _i. \end{aligned}$$

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

• (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 [12].

• (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. $G/H\ne \mathbb {RP}^{4n+3}$. Then a ${\mathcal {H}}$-homothetic deformation with $a=\alpha \delta$ and $c=\delta$ induces a homogeneous 3-Sasaki manifold with $G/H\ne \mathbb {RP}^{4n+3}$ and thus is given by the model in [12]. 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.

• (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 ${\mathcal {H}}$.

• (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

  $$\begin{aligned} G/H\rightarrow G/G_0 \end{aligned}$$

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

• (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.

• (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 $Y\in {\mathcal {V}}, Z\in {\mathcal {H}}$

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

  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 (13) . Thus, in both cases the given metric g is indeed positive definite.

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

$$\begin{aligned}{}[\xi _i,[\xi _i,\xi _j]]=\pm 2\delta [\xi _i,\xi _k] =-4\delta ^2\xi _j, \end{aligned}$$

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

$$\begin{aligned} g(\xi _i,\xi _i)= \frac{-{\mathrm {tr}}({\mathrm {ad}}^2\xi _i)}{4\delta ^2(n+2)} =\frac{-{\mathrm {tr}}({\mathrm {ad}}^2\xi _i\vert _{{\mathcal {H}}})-{\mathrm {tr}}({\mathrm {ad}}^2\xi _i\vert _{\mathcal {V}})}{4\delta ^2(n+2)} =\frac{8\delta ^2+4\delta ^2n}{4\delta ^2(n+2)}=1. \end{aligned}$$

On the contrary we have ${\mathrm {tr}}({\mathrm {ad}}\,\xi _i\circ {\mathrm {ad}}\,\xi _j\vert _{{\mathcal {H}}})={\mathrm {tr}}(\pm {\mathrm {ad}}\,\xi _k\vert _{{\mathcal {H}}})=0$ 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 $i=k\ne j$. In any case ${\mathrm {tr}}({\mathrm {ad}}\,\xi _i\circ {\mathrm {ad}}\,\xi _j)=0$ and, hence, $g(\xi _i,\xi _j)=0$ if $i\ne j$.

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,

$$\begin{aligned} \varphi _i^2(\xi _j)&=\frac{1}{4\delta ^2}[\xi _i,[\xi _i,\xi _j]]=-\xi _j,\\ \varphi _i^2\vert _{{\mathcal {H}}}&=\frac{1}{\delta ^2}{\mathrm {ad}}\,\!^2\xi _i\vert _{{\mathcal {H}}}=\frac{-\delta ^2}{\delta ^2}{\mathrm {id}}=-{\mathrm {id}}. \end{aligned}$$

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

$$\begin{aligned} \kappa (\varphi _iX,\varphi _iY)=-\kappa \left( X,\frac{1}{\delta ^2} {\mathrm {ad}}\,\!^2\xi _i Y\right) =\kappa (X,Y) \end{aligned}$$

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

$$\begin{aligned} g(\varphi _i\xi _j,\varphi _i\xi _{j^\prime })=g\left( \frac{1}{2\delta }{\mathrm {ad}}\,\xi _i(\xi _j),\frac{1}{2\delta }{\mathrm {ad}}\,\xi _i(\xi _{j^\prime })\right) =g(\pm \xi _k,\pm \xi _{k^\prime })=g(\xi _j,\xi _{j^\prime }) \end{aligned}$$

if (ijk), $(ij^\prime k^\prime )$ are according permutations of (123) and the left side vanishes whenever j or j' 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, $\varphi _i\varphi _j\vert _{{\mathcal {H}}}=\frac{1}{\delta ^2} {\mathrm {ad}}\,\xi _i\circ {\mathrm {ad}}\,\xi _j\vert _{{\mathcal {H}}}=\frac{1}{\delta }{\mathrm {ad}}\,\xi _k|_{{\mathcal {H}}}=\varphi _k|_{{\mathcal {H}}}$ and on ${\mathcal {V}}$ we have

$$\begin{aligned} \varphi _i\varphi _j\xi _i&=\frac{1}{4\delta ^2}[\xi _i,[\xi _j,\xi _i]]=\xi _j=\varphi _k\xi _i=\varphi _k\xi _i+\eta _j(\xi _i)\xi _i,\\ \varphi _i\varphi _j\xi _j&=0=\xi _i-\xi _i=\varphi _k\xi _j-\eta _j(\xi _j)\xi _i,\\ \varphi _i\varphi _j\xi _k&=\frac{1}{4\delta ^2}[\xi _i,[\xi _j,\xi _k]]=\frac{1}{2\delta }[\xi _i,\xi _i]=0=\varphi _k\xi _k+\eta _j(\xi _k)\xi _i. \end{aligned}$$

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 ${\mathrm {ad}}\,\xi _j X\in {\mathcal {H}}$,

$$\begin{aligned} {\mathrm {d}}\eta _i(\xi _j,X)&=\xi _j(\eta _i(X))-X(\eta _i(\xi _j))-\eta _i({\mathrm {ad}}\,\xi _j X)=-\eta _i({\mathrm {ad}}\,\xi _j X)=0,\\ 2\alpha \Phi _i(\xi _j,X)&=2\alpha g(\xi _j,\varphi _i X)=\frac{2\alpha }{\delta }g(\xi _j,{\mathrm {ad}}\,\xi _i X)=0. \end{aligned}$$

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

$$\begin{aligned} {\mathrm {d}}\eta _i(X,Y)&=X(\eta _i(Y))-Y(\eta _i(X))-\eta _i([X,Y])=-g(\xi _i,[X,Y])\\&=\frac{1}{4\delta ^2(n+2)}\kappa (\xi _i,[X,Y])=\frac{-1}{4\delta ^2(n+2)}\kappa ({\mathrm {ad}}\,\xi _iY,X),\\ 2\alpha \Phi _i(X,Y)&=2\alpha g(X,\varphi _i Y)=\frac{2\alpha }{\delta }g(X,{\mathrm {ad}}\,\xi _i Y)=\frac{-2\alpha }{8\alpha \delta ^2(n+2)}\kappa (X,{\mathrm {ad}}\,\xi _i Y)\\&=\frac{-1}{4\delta ^2(n+2)}\kappa (X,{\mathrm {ad}}\,\xi _i Y). \end{aligned}$$

Finally, we have

$$\begin{aligned} \begin{aligned} {\mathrm {d}}\eta _i(\xi _j,\xi _k)&=\xi _j(\eta _i(\xi _k))-\xi _k(\eta _i(\xi _j))-\eta _i([\xi _j,\xi _k])=-\eta _i(2\delta \xi _i)=-2\delta ,\\ 2\alpha \Phi _i(\xi _j,\xi _k)&=2\alpha g(\xi _j,\varphi _i\xi _k)=- 2\alpha g(\xi _j,\xi _j)=- 2\alpha ,\\ 2(\alpha -\delta )\eta _{i+1}\wedge \eta _{i+2}(\xi _j,\xi _k)&=2(\alpha -\delta )=2\alpha -2\delta . \end{aligned} \end{aligned}$$

(14)

$\square$

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 Cortés [10]. 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

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

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 $\Pi :\Lambda ^2W\rightarrow V$ 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 [10, 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 $(+,+,+,-,\cdots , -)$. 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

$$\begin{aligned} \pi _{\mathcal {V}}([\sigma _i,\sigma _j]) &=\pm 2\sigma _k,\quad \pi _{\mathcal {V}}([D,\hat{e}_i+\sigma _i])=-\sigma _i, \quad \pi _{\mathcal {V}}([\hat{e}_i+\sigma _i,\hat{e}_j+\sigma _j])=\mp 2\sigma _k, \\ \pi _{\mathcal {V}}([e_l,\hat{e}_i\wedge e_l])&=-\sigma _i,\quad \pi _{\mathcal {V}}([\hat{e}_i\wedge e_l,\hat{e}_j\wedge e_l])=\pm \frac{1}{2}\sigma _k,\quad \pi _{\mathcal {V}}([w_1,w_2])=\pi _{\mathcal {V}}(\Pi (w_1,w_2)) \end{aligned}$$

for all permutations (ijk) of (123) with ± 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

$$\begin{aligned}{}[\sigma _i,\sigma _j]&=\pm 2\sigma _k,&[\sigma _i,D]&=0,&[\hat{e}_i+\sigma _i,\hat{e}_i\wedge e_l]&=e_l,&[\sigma _i,\hat{e}_j+\sigma _j]&=\pm 2(\hat{e}_k+\sigma _k),\\ [\sigma _i,e_l]&=0,&[\sigma _i,\hat{e}_i\wedge e_l]&=0,&[\hat{e}_i\wedge e_l, W]&=W,&[\sigma _i,\hat{e}_j\wedge e_l]&=\pm 2\hat{e}_k\wedge e_l,\\ [D,e_l]&=e_l,&[D,\hat{e}_i\wedge e_l]&=0,&[\hat{e}_i\wedge e_l,\hat{e}_j\wedge e_m]&=0,&[D,\hat{e}_i+\sigma _i]&=\hat{e}_i=(\hat{e}_i+\sigma _i)-\sigma _i,\\ [D,W]&=W,&[\sigma _i,\hat{e}_i+\sigma _i]&=0,&[\hat{e}_i+\sigma _i,W]&=W,&[\hat{e}_i+\sigma _i,\hat{e}_j\wedge e_l]&=\pm 2\hat{e}_k\wedge e_l,\\ [\sigma _i,W]&= W,&[\hat{e}_i+\sigma _i,e_l]&=0,&[\hat{e}_i\wedge e_l,\hat{e}_i\wedge e_m]&=-e_l\wedge e_m,&[\hat{e}_i\wedge e_l,\hat{e}_j\wedge e_l]&=-\hat{e}_i\wedge \hat{e}_j=\pm \frac{1}{2}\sigma _k\\ [e_l,e_m]&=0,&[e_l, W]&=W,&[e_l,\hat{e_i}\wedge e_m]&=0,&[e_l,\hat{e_i}\wedge e_l]&=\hat{e_i}=(\hat{e_i}+\sigma _i)-\sigma _i \end{aligned}$$

and finally

$$\begin{aligned}{}[\hat{e}_i+\sigma _i,\hat{e}_j+\sigma _j]\ = \ [\sigma _i,\hat{e}_j]-[\sigma _j,\hat{e}_i]+[\sigma _i,\sigma _j] \ = \ \pm 4\hat{e}_k\pm 2\sigma _k=\pm 4(\hat{e}_k+\sigma _k)\mp 2\sigma _k, \end{aligned}$$

where (ijk) is a permutation of (123) with ± indicating the sign of the permutation and $l,m=1,\dots ,q$ with $l\ne m$. For the commutator [W, W] we have $[w_1,w_2]=\Pi (w_1,w_2)\in V\subset {\mathcal {H}}_0\oplus {\mathcal {H}}_1\oplus {\mathcal {V}}$. $\square$

By [10, 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 [10, 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

$$\begin{aligned} \xi _1=\delta \sigma _1, \quad \xi _2=\delta \sigma _2, \quad \xi _3=\delta \sigma _3. \end{aligned}$$

Define the almost complex structures $\varphi _i:\mathfrak {m}\rightarrow \mathfrak {m}$ on ${\mathcal {V}}$, ${\mathcal {H}}_0$, ${\mathcal {H}}_1$ and W individually. For any permutation (ijk) of (123) with signature ± we set

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 ${\mathcal {V}}\oplus {\mathcal {H}}_0\oplus {\mathcal {H}}_1$:

$$\begin{aligned} \delta \sigma _i, \sqrt{-4\alpha \delta }D, \sqrt{-\alpha \delta }(\hat{e}_i+\sigma _i), \sqrt{-4\alpha \delta }\,\hat{e}_i\wedge e_l, \sqrt{-\alpha \delta }\,e_l. \end{aligned}$$

On W we set the scalar product

$$\begin{aligned} g_{[e]}|_{W\times W}(s,t)=(-2\alpha \delta )^{-1}b(s,t):=(-2\alpha \delta )^{-1}\langle \hat{e}_i,\Pi (\rho (\hat{e}_j\hat{e}_k)s,t)\rangle , \end{aligned}$$

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 [10, 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

$$\begin{aligned} \rho (\sigma _i)\rho (\sigma _j)w=\hat{e}_k\cdot \hat{e}_j\cdot \hat{e}_i\cdot \hat{e}_k\cdot w=(-1)^2\hat{e}_j\cdot \hat{e}_k\cdot \hat{e}_k\cdot \hat{e}_i\cdot w=-\hat{e}_j\cdot \hat{e}_i\cdot w=\rho (\sigma _k)w. \end{aligned}$$

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 (14)). 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

$$\begin{aligned} {\mathrm {d}}\eta _i(D,\hat{e}_i+\sigma _i)&=-\eta _i([D,\hat{e}_i+\sigma _i])=-\eta _i(-\sigma _i)=\frac{1}{\delta }g(\delta \sigma _i,\delta \sigma _i)=\frac{1}{\delta },\\ 2\alpha \Phi _i(D,\hat{e}_i+\sigma _i)&=2\alpha g(D,\varphi _i(\hat{e}_i+\sigma _i))=\frac{2\alpha }{-2\alpha \delta } g(\sqrt{-4\alpha \delta }\;D,-\sqrt{-\alpha \delta }\;2D)=\frac{1}{\delta }. \end{aligned}$$

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

$$\begin{aligned} \frac{2}{\delta }=-\eta _k(2\sigma _k)={\mathrm {d}}\eta _k(\hat{e}_i+\sigma _i,\hat{e}_j+\sigma _j)&=2\alpha \Phi _k(\hat{e}_i+\sigma _i,\hat{e}_j+\sigma _j)=\frac{2}{\delta },\\ \frac{1}{\delta }=-\eta _i(-\sigma _i)={\mathrm {d}}\eta _i(e_l,\hat{e}_i\wedge e_l)&=2\alpha \Phi _i(e_l,\hat{e}_i\wedge e_l)=\frac{1}{\delta },\\ \frac{1}{2\delta }=-\eta _k(-\frac{1}{2}\sigma _k)={\mathrm {d}}\eta _k(\hat{e}_i\wedge e_l,\hat{e}_j\wedge e_l)&=2\alpha \Phi _k(\hat{e}_i\wedge e_l,\hat{e}_j\wedge e_l)=\frac{1}{2\delta } \end{aligned}$$

for any even permutation (ijk) of (123). Finally, we look at $W\times W$. Let $w_1,w_2\in W$ and suppose $\Pi (w_1,w_2)=\sum _{r=1}^q a_re_r+\sum _{s=1}^3\hat{a}_s\hat{e}_s$. Then

$$\begin{aligned} {\mathrm {d}}\eta _i([w_1,w_2])&=-\eta _i(\Pi (w_1,w_2))=-\eta _i\left( \sum _{r=1}^q a_re_r+\sum _{s=1}^3\hat{a}_s\hat{e}_s\right) \\&=-\eta _i\left( \sum _{s=1}^3\hat{a}_s((\hat{e}_s+\sigma _s)-\sigma _s)\right) =\frac{\hat{a}_i}{\delta } \end{aligned}$$

and

$$\begin{aligned} 2\alpha \Phi _i(w_1,w_2)&=2\alpha g(w_1,\varphi _i w_2)=\frac{2\alpha }{-2\alpha \delta }\left\langle \hat{e}_i,\Pi \left( w_1,\hat{e}_j\hat{e}_k\hat{e}_j\hat{e}_k w_2\right) \right\rangle \\&=\frac{(-1)^32\alpha }{-2\alpha \delta }\langle \hat{e_i},\Pi (w_1,w_2)\rangle \\&=\frac{1}{\delta }\left\langle \hat{e}_i,\sum _{r=1}^q a_re_r+\sum _{s=1}^3\hat{a}_s\hat{e}_s\right\rangle =\frac{\hat{a}_i}{\delta }. \end{aligned}$$

This concludes the proof. $\square$

Remark

We try to motivate the definition. Recall that in [10] 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

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

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 in 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

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

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

$$\begin{aligned} \xi _1=\delta \begin{bmatrix} i&{}0&{}0\\ 0&{}-i&{}0\\ 0&{}0&{}0 \end{bmatrix},\qquad \xi _2=\delta \begin{bmatrix} 0&{}-1&{}0\\ 1&{}0&{}0\\ 0&{}0&{}0 \end{bmatrix},\qquad \xi _3=\delta \begin{bmatrix} 0&{}-i&{}0\\ -i&{}0&{}0\\ 0&{}0&{}0 \end{bmatrix}. \end{aligned}$$

On the horizontal subspace we choose a basis vector

$$\begin{aligned} \tilde{e}_1=\begin{bmatrix} 0&{}0&{}1\\ 0&{}0&{}0\\ -1&{}0&{}0 \end{bmatrix}. \end{aligned}$$

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:

$$\begin{aligned} e_2=\sqrt{2\alpha \delta } \begin{bmatrix} 0&{}0&{}i\\ 0&{}0&{}0\\ i&{}0&{}0 \end{bmatrix},\qquad e_3=\sqrt{2\alpha \delta } \begin{bmatrix} 0&{}0&{}0\\ 0&{}0&{}1\\ 0&{}-1&{}0 \end{bmatrix},\qquad e_4=\sqrt{2\alpha \delta } \begin{bmatrix} 0&{}0&{}0\\ 0&{}0&{}-i\\ 0&{}-i&{}0 \end{bmatrix}. \end{aligned}$$

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

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

as described in (13). Then as for the Aloff–Wallach space we identify the 1-dimensional isotropy $\mathfrak {h}$ generated by

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

Analogously the Reeb vector fields are given by

$$\begin{aligned} \xi _1=\delta \begin{bmatrix} i&{}0&{}0\\ 0&{}-i&{}0\\ 0&{}0&{}0 \end{bmatrix},\qquad \xi _2=\delta \begin{bmatrix} 0&{}-1&{}0\\ 1&{}0&{}0\\ 0&{}0&{}0 \end{bmatrix},\qquad \xi _3=\delta \begin{bmatrix} 0&{}-i&{}0\\ -i&{}0&{}0\\ 0&{}0&{}0 \end{bmatrix}. \end{aligned}$$

On the horizontal subspace we choose

$$\begin{aligned} \tilde{e}^*_1=i\tilde{e}_1=\begin{bmatrix} 0&{}0&{}i\\ 0&{}0&{}0\\ -i&{}0&{}0 \end{bmatrix}\subset i{\mathcal {H}}. \end{aligned}$$

We have

$$\begin{aligned} g(\tilde{e}^*_1,\tilde{e}^*_1)=\frac{-i^2}{24\alpha \delta }\kappa (\tilde{e}_1,\tilde{e}_1)=-\frac{1}{2\alpha \delta }. \end{aligned}$$

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

$$\begin{aligned} e^*_2=\sqrt{-2\alpha \delta } \begin{bmatrix} 0&{}0&{}-1\\ 0&{}0&{}0\\ -1&{}0&{}0 \end{bmatrix},\; e^*_3=\sqrt{-2\alpha \delta } \begin{bmatrix} 0&{}0&{}0\\ 0&{}0&{}i\\ 0&{}-i&{}0 \end{bmatrix},\; e^*_4=\sqrt{-2\alpha \delta } \begin{bmatrix} 0&{}0&{}0\\ 0&{}0&{}1\\ 0&{}1&{}0 \end{bmatrix}. \end{aligned}$$

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.

Table 2 3-$(\alpha ,\delta )$-Sasaki manifolds over Alekseevsky spaces of $\dim \le 19$, 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 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 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 $q\ge 4$ 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 lists the cases $q=0$ and $q=1$, while Table 4 is devoted to the case $q=2$.

Table 3 Choice of $C\!\ell ^0(3,q)$-representations for $q=0$ and $q=1$

The notation is as follow: We denote elements $e_{ij}=e_ie_j\in C\!\ell ^0(V)$ 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.

Table 4 Choice of $C\!\ell ^0(3,q)$-representations for $q=2$

With this we can find the map $\Pi :\Lambda ^2W_2\rightarrow \mathbb {R}^{3,2}$.

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 $\Pi :\Lambda ^2W_2\rightarrow \mathbb {R}^{3,2}$ given by

$$\begin{aligned} \Pi (E_1\wedge E_2)=-\hat{e}_3-e_1,\quad \Pi (E_1\wedge E_3)=-\hat{e}_2,\quad \Pi (E_1\wedge E_4)=-\hat{e}_1+e_2,\\ \Pi (E_2\wedge E_3)=\hat{e}_1+e_2,\quad \Pi (E_4\wedge E_2)=\hat{e}_2,\quad \Pi (E_3\wedge E_4)=\hat{e}_3-e_1 \end{aligned}$$

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 ${\mathrm {ad}}\,^{-1}:\mathfrak {so}(3,q)\rightarrow \mathfrak {spin}(3,q)=C\!\ell ^0(3,q)$, $e_i\wedge e_j\mapsto -\frac{1}{2} e_{ij}$, 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

$$\begin{aligned} -2\hat{e}_1\wedge \hat{e}_2(\hat{e}_3\pm e_1)&=0= \Pi (e_{\hat{1}\hat{2}}(E_1\wedge E_2))=\Pi (e_{\hat{1}\hat{2}}(E_3\wedge E_4)),\\ -2\hat{e}_2\wedge \hat{e}_3(\hat{e}_1\pm e_2)&=0= \Pi (e_{\hat{2}\hat{3}}(E_1\wedge E_4))=\Pi (e_{\hat{2}\hat{3}}(E_2\wedge E_3)),\\ -2\hat{e}_1\wedge e_1(\hat{e}_2)&=0= \Pi (e_{\hat{1}1}(E_1\wedge E_3))=\Pi (e_{\hat{1}1}(E_4\wedge E_2)),\\ -2e_1\wedge e_2(\hat{e}_2)&=0= \Pi (e_{12}(E_1\wedge E_3))=\Pi (e_{12}(E_4\wedge E_2)). \end{aligned}$$

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

$$\begin{aligned} \Pi (e_{\hat{1}\hat{2}}(E_1\wedge E_3))&=\Pi (E_2\wedge E_3)+\Pi (E_1\wedge -E_4)=\hat{e}_1+e_2+\hat{e}_1-e_2=2\hat{e}_1\\&=-2\hat{e}_1\wedge \hat{e}_2(-\hat{e}_2)=-2\hat{e}_1\wedge \hat{e}_2(\Pi (E_1\wedge E_3)),\\ \Pi (e_{\hat{1}\hat{2}}(E_1\wedge E_4))&=\Pi (E_2\wedge E_4)+\Pi (E_1\wedge E_3)=-\hat{e}_2-\hat{e}_2=-2\hat{e}_2\\&=-2\hat{e}_1\wedge \hat{e}_2(-\hat{e}_1+e_2)=-2\hat{e}_1\wedge \hat{e}_2(\Pi (E_1\wedge E_4)),\\ \Pi (e_{\hat{1}\hat{2}}(E_2\wedge E_3))&=\Pi (-E_1\wedge E_3)+\Pi (E_2\wedge -E_4)=\hat{e}_2+\hat{e}_2=2\hat{e}_2\\&=-2\hat{e}_1\wedge \hat{e}_2(\hat{e}_1+e_2)=-2\hat{e}_1\wedge \hat{e}_2(\Pi (E_2\wedge E_3)),\\ \Pi (e_{\hat{1}\hat{2}}(E_4\wedge E_2))&=\Pi (E_3\wedge E_2)+\Pi (E_4\wedge -E_1)=-\hat{e}_1-e_2-\hat{e}_1+e_2=-2\hat{e}_1\\&=-2\hat{e}_1\wedge \hat{e}_2(\hat{e}_2)=-2\hat{e}_1\wedge \hat{e}_2(\Pi (E_4\wedge E_2)). \end{aligned}$$

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

$$\begin{aligned} \Pi (e_{\hat{2}\hat{3}}(E_1\wedge E_2))&=\Pi (E_4\wedge E_2)+\Pi (E_1\wedge -E_3)=\hat{e}_2+\hat{e}_2=2\hat{e}_2\\&=-2\hat{e}_2\wedge \hat{e}_3(-\hat{e}_3-e_1)=-2\hat{e}_2\wedge \hat{e}_3(\Pi (E_1\wedge E_2)),\\ \Pi (e_{\hat{2}\hat{3}}(E_1\wedge E_3))&=\Pi (E_4\wedge E_3)+\Pi (E_1\wedge E_2)=-\hat{e}_3+e_1-\hat{e}_3-e_1=-2\hat{e}_3\\&=-2\hat{e}_2\wedge \hat{e}_3(-\hat{e}_2)=-2\hat{e}_2\wedge \hat{e}_3(\Pi (E_1\wedge E_3)),\\ \Pi (e_{\hat{2}\hat{3}}(E_4\wedge E_2))&=\Pi (-E_1\wedge E_2)+\Pi (E_4\wedge -E_3)=\hat{e}_3+e_1+\hat{e}_3-e_1=2\hat{e}_3\\&=-2\hat{e}_2\wedge \hat{e}_3(\hat{e}_2)=-2\hat{e}_2\wedge \hat{e}_3(\Pi (E_4\wedge E_2)),\\ \Pi (e_{\hat{2}\hat{3}}(E_2\wedge E_4))&=\Pi (E_3\wedge E_2)+\Pi (E_3\wedge -E_1)=-\hat{e}_2-\hat{e}_2=-2\hat{e}_2\\&=-2\hat{e}_2\wedge \hat{e}_3(\hat{e}_3-e_1)=-2\hat{e}_2\wedge \hat{e}_3(\Pi (E_3\wedge E_4)). \end{aligned}$$

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

$$\begin{aligned} \Pi (e_{\hat{1}1}(E_1\wedge E_2))&=\Pi (E_3\wedge E_2)+\Pi (E_1\wedge E_4)=-\hat{e}_1-e_2-\hat{e}_1+e_2=-2\hat{e}_1\\&=-2\hat{e}_1\wedge e_1(-\hat{e}_3-e_1)=-2\hat{e}_1\wedge e_1(\Pi (E_1\wedge E_2)),\\ \Pi (e_{\hat{1}1}(E_1\wedge E_4))&=\Pi (E_3\wedge E_4)+\Pi (E_1\wedge E_2)=\hat{e}_3-e_1-\hat{e}_3-e_1=-2e_1\\&=-2\hat{e}_ 1\wedge e_1(-\hat{e}_1+e_2)=-2\hat{e}_1\wedge e_1(\Pi (E_1\wedge E_4)),\\ \Pi (e_{\hat{1}1}(E_2\wedge E_3))&=\Pi (E_4\wedge E_3)+\Pi (E_2\wedge E_1)=-\hat{e}_3+e_1+\hat{e}_3+e_1=2e_1\\&=-2\hat{e}_1\wedge e_1(\hat{e}_1+e_2)=-2\hat{e}_1\wedge e_1(\Pi (E_2\wedge E_3)),\\ \Pi (e_{\hat{1}1}(E_3\wedge E_4))&=\Pi (E_1\wedge E_4)+\Pi (E_3\wedge E_2)=-\hat{e}_1+e_2-\hat{e}_1-e_2=-2\hat{e}_1\\&=-2\hat{e}_1\wedge e_1(\hat{e}_3-e_1)=-2\hat{e}_1\wedge e_1(\Pi (E_3\wedge E_4)). \end{aligned}$$

And finally for $e_{12}$:

$$\begin{aligned} \Pi (e_{12}(E_1\wedge E_2))&=\Pi (-E_3\wedge E_2)+\Pi (E_1\wedge E_4)=\hat{e}_1+e_2-\hat{e}_1+e_2=2e_2\\&=-2 e_1\wedge e_2(-\hat{e}_3-e_1)=-2 e_1\wedge e_2(\Pi (E_1\wedge E_2)),\\ \Pi (e_{12}(E_1\wedge E_4))&=\Pi (-E_3\wedge E_4)+\Pi (E_1\wedge -E_2)=-\hat{e}_3+e_1+\hat{e}_3+e_1=2e_1\\&=-2 e_1\wedge e_2(-\hat{e}_1+e_2)=-2 e_1\wedge e_2(\Pi (E_1\wedge E_4)),\\ \Pi (e_{12}(E_2\wedge E_3))&=\Pi (E_4\wedge E_3)+\Pi (E_2\wedge E_1)=-\hat{e}_3+e_1+\hat{e}_3+e_1=2e_1\\&=-2 e_1\wedge e_2(\hat{e}_1+e_2)=-2 e_1\wedge e_2(\Pi (E_2\wedge E_3)),\\ \Pi (e_{12}(E_3\wedge E_4))&=\Pi (E_1\wedge E_4)+\Pi (E_3\wedge -E_2)=-\hat{e}_1+e_2+\hat{e}_1+e_2=2e_2\\&=-2 e_1\wedge e_2(\hat{e}_3-e_1)=-2 e_1\wedge e_2(\Pi (E_3\wedge E_4)). \end{aligned}$$

$\square$

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 $\mathfrak {so}(3,2)$:

$$\begin{aligned} 2\hat{e}_1\wedge {\hat{e}}_2=\left[ \begin{array}{llllllllll} 0&{}2&{}&{}&{}&{}0&{}&{}&{}&{}\\ -2&{}0&{}&{}&{}&{}0&{}&{}&{}&{}\\ &{}&{}0&{}&{}&{}0&{}&{}&{}&{}\\ &{}&{}&{}0&{}&{}0&{}&{}&{}&{}\\ &{}&{}&{}&{}0&{}0&{}&{}&{}&{}\\ 0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0\\ &{}&{}&{}&{}&{}0&{}&{}&{}0&{}1\\ &{}&{}&{}&{}&{}0&{}&{}&{}-1&{}0\\ &{}&{}&{}&{}&{}0&{}0&{}1&{}&{}\\ &{}&{}&{}&{}&{}0&{}-1&{}0&{}&{}\\ \end{array}\right] ,\quad 2\hat{e}_3\wedge {\hat{e}}_1=\left[ \begin{array}{llllllllll} 0&{}&{}-2&{}&{}&{}0&{}&{}&{}&{}\\ &{}0&{}&{}&{}&{}0&{}&{}&{}&{}\\ 2&{}&{}0&{}&{}&{}0&{}&{}&{}&{}\\ &{}&{}&{}0&{}&{}0&{}&{}&{}&{}\\ &{}&{}&{}&{}0&{}0&{}&{}&{}&{}\\ 0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0\\ &{}&{}&{}&{}&{}0&{}&{}&{}1&{}0\\ &{}&{}&{}&{}&{}0&{}&{}&{}0&{}1\\ &{}&{}&{}&{}&{}0&{}-1&{}0&{}&{}\\ &{}&{}&{}&{}&{}0&{}0&{}-1&{}&{}\\ \end{array}\right] ,\\ 2\hat{e}_2\wedge \hat{e}_3=\left[ \begin{array}{llllllllll} 0&{}&{}&{}&{}&{}0&{}&{}&{}&{}\\ &{}0&{}2&{}&{}&{}0&{}&{}&{}&{}\\ &{}-2&{}0&{}&{}&{}0&{}&{}&{}&{}\\ &{}&{}&{}0&{}&{}0&{}&{}&{}&{}\\ &{}&{}&{}&{}0&{}0&{}&{}&{}&{}\\ 0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0\\ &{}&{}&{}&{}&{}0&{}&{}&{}0&{}1\\ &{}&{}&{}&{}&{}0&{}&{}&{}-1&{}0\\ &{}&{}&{}&{}&{}0&{}0&{}1&{}&{}\\ &{}&{}&{}&{}&{}0&{}-1&{}0&{}&{}\\ \end{array}\right] , \end{aligned}$$

$$\begin{aligned} 2\hat{e}_1\wedge e_1=\left[ \begin{array}{llllllllll} 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{array}\right] ,\qquad 2\hat{e}_1\wedge e_2=\left[ \begin{array}{llllllllll} 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{array}\right] ,\\ 2\hat{e}_2\wedge e_1=\left[ \begin{array}{llllllllll} 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{array}\right] ,\qquad 2\hat{e}_2\wedge e_2=\left[ \begin{array}{llllllllll} 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{array}\right] ,\\ 2\hat{e}_3\wedge e_1=\left[ \begin{array}{llllllllll} 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{array}\right] ,\qquad 2\hat{e}_3\wedge e_2=\left[ \begin{array}{llllllllll} 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{array}\right] , \end{aligned}$$

$$\begin{aligned} 2e_1\wedge e_2=\left[ \begin{array}{llllllllll} 0&{}&{}&{}&{}&{}0&{}&{}&{}&{}\\ &{}0&{}&{}&{}&{}0&{}&{}&{}&{}\\ &{}&{}0&{}&{}&{}0&{}&{}&{}&{}\\ &{}&{}&{}0&{}-2&{}0&{}&{}&{}&{}\\ &{}&{}&{}2&{}0&{}0&{}&{}&{}&{}\\ 0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0\\ &{}&{}&{}&{}&{}0&{}&{}&{}-1&{}0\\ &{}&{}&{}&{}&{}0&{}&{}&{}0&{}1\\ &{}&{}&{}&{}&{}0&{}1&{}0&{}&{}\\ &{}&{}&{}&{}&{}0&{}0&{}-1&{}&{}\\ \end{array}\right] . \end{aligned}$$

Recall that $2e_1\wedge {e}_2$ generates the isotropy algebra $\mathfrak {h}$.

Next we describe the element 2D:

$$\begin{aligned} 2D=\left[ \begin{array}{llllllllll} 2&{}&{}&{}&{}&{}0&{}&{}&{}&{}\\ &{}2&{}&{}&{}&{}0&{}&{}&{}&{}\\ &{}&{}2&{}&{}&{}0&{}&{}&{}&{}\\ &{}&{}&{}2&{}&{}0&{}&{}&{}&{}\\ &{}&{}&{}&{}2&{}0&{}&{}&{}&{}\\ 0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0\\ &{}&{}&{}&{}&{}0&{}1&{}&{}&{}\\ &{}&{}&{}&{}&{}0&{}&{}1&{}&{}\\ &{}&{}&{}&{}&{}0&{}&{}&{}1&{}\\ &{}&{}&{}&{}&{}0&{}&{}&{}&{}1\\ \end{array}\right] . \end{aligned}$$

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

$$\begin{aligned} \hat{e}_1=\left[ \begin{array}{llllllllll} &{}&{}&{}&{}&{}-1&{}&{}&{}&{}\\ &{}&{}&{}&{}&{}0&{}&{}&{}&{}\\ &{}&{}&{}&{}&{}0&{}&{}&{}&{}\\ &{}&{}&{}&{}&{}0&{}&{}&{}&{}\\ &{}&{}&{}&{}&{}0&{}&{}&{}&{}\\ 0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0\\ &{}&{}&{}&{}&{}0&{}&{}&{}&{}\\ &{}&{}&{}&{}&{}0&{}&{}&{}&{}\\ &{}&{}&{}&{}&{}0&{}&{}&{}&{}\\ &{}&{}&{}&{}&{}0&{}&{}&{}&{}\\ \end{array}\right] ,\qquad \hat{e}_2=\left[ \begin{array}{llllllllll} &{}&{}&{}&{}&{}0&{}&{}&{}&{}\\ &{}&{}&{}&{}&{}-1&{}&{}&{}&{}\\ &{}&{}&{}&{}&{}0&{}&{}&{}&{}\\ &{}&{}&{}&{}&{}0&{}&{}&{}&{}\\ &{}&{}&{}&{}&{}0&{}&{}&{}&{}\\ 0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0\\ &{}&{}&{}&{}&{}0&{}&{}&{}&{}\\ &{}&{}&{}&{}&{}0&{}&{}&{}&{}\\ &{}&{}&{}&{}&{}0&{}&{}&{}&{}\\ &{}&{}&{}&{}&{}0&{}&{}&{}&{}\\ \end{array}\right] ,\qquad \hat{e}_3=\left[ \begin{array}{llllllllll} &{}&{}&{}&{}&{}0&{}&{}&{}&{}\\ &{}&{}&{}&{}&{}0&{}&{}&{}&{}\\ &{}&{}&{}&{}&{}-1&{}&{}&{}&{}\\ &{}&{}&{}&{}&{}0&{}&{}&{}&{}\\ &{}&{}&{}&{}&{}0&{}&{}&{}&{}\\ 0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0\\ &{}&{}&{}&{}&{}0&{}&{}&{}&{}\\ &{}&{}&{}&{}&{}0&{}&{}&{}&{}\\ &{}&{}&{}&{}&{}0&{}&{}&{}&{}\\ &{}&{}&{}&{}&{}0&{}&{}&{}&{}\\ \end{array}\right] ,\\ e_1=\left[ \begin{array}{llllllllll} &{}&{}&{}&{}&{}0&{}&{}&{}&{}\\ &{}&{}&{}&{}&{}0&{}&{}&{}&{}\\ &{}&{}&{}&{}&{}0&{}&{}&{}&{}\\ &{}&{}&{}&{}&{}-1&{}&{}&{}&{}\\ &{}&{}&{}&{}&{}0&{}&{}&{}&{}\\ 0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0\\ &{}&{}&{}&{}&{}0&{}&{}&{}&{}\\ &{}&{}&{}&{}&{}0&{}&{}&{}&{}\\ &{}&{}&{}&{}&{}0&{}&{}&{}&{}\\ &{}&{}&{}&{}&{}0&{}&{}&{}&{}\\ \end{array}\right] ,\qquad e_2=\left[ \begin{array}{llllllllll} &{}&{}&{}&{}&{}0&{}&{}&{}&{}\\ &{}&{}&{}&{}&{}0&{}&{}&{}&{}\\ &{}&{}&{}&{}&{}0&{}&{}&{}&{}\\ &{}&{}&{}&{}&{}0&{}&{}&{}&{}\\ &{}&{}&{}&{}&{}-1&{}&{}&{}&{}\\ 0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0\\ &{}&{}&{}&{}&{}0&{}&{}&{}&{}\\ &{}&{}&{}&{}&{}0&{}&{}&{}&{}\\ &{}&{}&{}&{}&{}0&{}&{}&{}&{}\\ &{}&{}&{}&{}&{}0&{}&{}&{}&{}\\ \end{array}\right] . \end{aligned}$$

Finally we have the 4 basis elements of ${W}_2$:

$$\begin{aligned} 2E_1=\left[ \begin{array}{llllllllll} &{}&{}&{}&{}&{}0&{}&{}&{}&{}-2\\ &{}&{}&{}&{}&{}0&{}&{}&{}-2&{}\\ &{}&{}&{}&{}&{}0&{}&{}-2&{}&{}\\ &{}&{}&{}&{}&{}0&{}&{}2&{}&{}\\ &{}&{}&{}&{}&{}0&{}&{}&{}&{}2\\ 0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0\\ &{}&{}&{}&{}&{}-1&{}&{}&{}&{}\\ &{}&{}&{}&{}&{}0&{}&{}&{}&{}\\ &{}&{}&{}&{}&{}0&{}&{}&{}&{}\\ &{}&{}&{}&{}&{}0&{}&{}&{}&{}\\ \end{array}\right] ,\quad&2E_2=\left[ \begin{array}{llllllllll} &{}&{}&{}&{}&{}0&{}&{}&{}2&{}\\ &{}&{}&{}&{}&{}0&{}&{}&{}&{}-2\\ &{}&{}&{}&{}&{}0&{}2&{}&{}&{}\\ &{}&{}&{}&{}&{}0&{}2&{}&{}&{}\\ &{}&{}&{}&{}&{}0&{}&{}&{}2&{}\\ 0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0\\ &{}&{}&{}&{}&{}0&{}&{}&{}&{}\\ &{}&{}&{}&{}&{}-1&{}&{}&{}&{}\\ &{}&{}&{}&{}&{}0&{}&{}&{}&{}\\ &{}&{}&{}&{}&{}&{}&{}&{}&{}\\ \end{array}\right] \\ 2E_3=\left[ \begin{array}{llllllllll} &{}&{}&{}&{}&{}0&{}&{}-2&{}&{}\\ &{}&{}&{}&{}&{}0&{}2&{}&{}&{}\\ &{}&{}&{}&{}&{}0&{}&{}&{}&{}2\\ &{}&{}&{}&{}&{}0&{}&{}&{}&{}-2\\ &{}&{}&{}&{}&{}0&{}&{}-2&{}&{}\\ 0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0\\ &{}&{}&{}&{}&{}0&{}&{}&{}&{}\\ &{}&{}&{}&{}&{}0&{}&{}&{}&{}\\ &{}&{}&{}&{}&{}-1&{}&{}&{}&{}\\ &{}&{}&{}&{}&{}0&{}&{}&{}&{}\\ \end{array}\right] ,\quad&2E_4=\left[ \begin{array}{llllllllll} &{}&{}&{}&{}&{}0&{}2&{}&{}&{}\\ &{}&{}&{}&{}&{}0&{}&{}2&{}&{}\\ &{}&{}&{}&{}&{}0&{}&{}&{}-2&{}\\ &{}&{}&{}&{}&{}0&{}&{}&{}2&{}\\ &{}&{}&{}&{}&{}0&{}-2&{}&{}&{}\\ 0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0\\ &{}&{}&{}&{}&{}0&{}&{}&{}&{}\\ &{}&{}&{}&{}&{}0&{}&{}&{}&{}\\ &{}&{}&{}&{}&{}0&{}&{}&{}&{}\\ &{}&{}&{}&{}&{}-1&{}&{}&{}&{}\\ \end{array}\right] \end{aligned}$$

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 $\Lambda :\mathfrak {m}\times \mathfrak {m}\rightarrow \mathfrak {m}$, 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 $\Lambda ^\nabla _XY=\nabla _{X_0}Y-[X,Y]_0$, compare [15, Corollary 2.2, p.191]. By [15, Proposition X.2.3, p. 191] the torsion $T^\nabla$ of the connection corresponding to $\Lambda ^\nabla$ is given by

$$\begin{aligned} T^\nabla (X,Y)= \Lambda ^\nabla _X Y-\Lambda ^\nabla _Y X-[X,Y]_{\mathfrak {m}}. \end{aligned}$$

(15)

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

$$\begin{aligned} \widehat{[X,Y]}_{\mathcal {V}}=- \sum _{i=1}^3(2\alpha\Phi _i(\hat{X},\hat{Y})+2(\alpha-\delta)\eta_j\wedge\eta_k(\hat{X},\hat{Y}))\xi _i. \end{aligned}$$

Proof

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

$$\begin{aligned}d\eta _i(\hat{X},\hat{Y})&=\hat{X}(\eta _i(\hat{Y}))-\hat{Y}(\eta _i(\hat{X}))-\eta _i([\hat{X},\hat{Y}])\\&=\eta _i([\hat{X},\hat{Y}])-\eta _i([\hat{Y},\hat{X}])-\eta _i([\hat{X},\hat{Y}])=\eta _i([\hat{X},\hat{Y}])=-\eta _i(\widehat{[X,Y]}). \end{aligned}$$

Using ${\widehat{[X,Y]}_{\mathcal {V}}}= {\sum _{i=1}^3} {\eta _i} (\widehat{[X,Y]}) {\xi _i}$ and (1) yields the result. $\square$

Theorem 4.1.1

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

$$\begin{aligned} \Lambda ^\nabla _X Y=\left\{ \begin{array}{ll} \Lambda ^{g_N}_X Y &{} X,Y\in {{\mathcal {H}}}\\ \frac{\beta }{2\delta }[X,Y] &{} X,Y\in {\mathcal {V}}\\ {[}X,Y]-2\alpha \sum _{i=1}^3\eta _i(X)\varphi _i Y &{} X\in {\mathcal {V}},Y\in {{\mathcal {H}}}\\ 0 &{} X\in {{\mathcal {H}}},Y\in {\mathcal {V}}, \end{array}\right. \end{aligned}$$

where $\Lambda ^{g_N}:{\mathcal {H}}\times {\mathcal {H}}\rightarrow {\mathcal {H}}$ 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 (15) agrees with the canonical torsion

$$\begin{aligned} {T^\nabla} =2\alpha {\sum _{i=1}^3} {\eta _i} \wedge {\Phi _i^{\mathcal {H}}} +2(\delta -4\alpha )\eta _{123}. \end{aligned}$$

We begin with the case $X,Y\in {\mathcal {H}}$. Evaluating Lemma 4.1.1 in the origin $T_0M\cong \mathfrak {m}$ we obtain ${[X,Y]_{\mathcal {V}}}=-2\alpha {\sum _{i=1}^3} {\Phi _i}(X,Y){\xi _i}$. Thus,

$$\begin{aligned} {\Lambda^\nabla _X} Y- {\Lambda^\nabla _Y} X-[X,Y]&= {\Lambda^{g_N}_X} Y- {\Lambda^{g_N}_Y} X-{[X,Y]_{{\mathcal {H}}}}-{[X,Y]_{\mathcal {V}}}\\&={T^{g_N}} (X,Y)+2\alpha {\sum _{i=1}^3} {\Phi _i} (X,Y) {\xi _i} =0+2\alpha {\sum _{i=1}^3} {\Phi _i} (X,Y) {\xi _i}\\&={T^\nabla} (X,Y) \end{aligned}$$

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

$$\begin{aligned} {\Lambda^\nabla _X} Y- {\Lambda^\nabla _Y} X-[X,Y]=[X,Y]-2\alpha \varphi _i Y-[X,Y]=-2\alpha \varphi _i Y= {T^\nabla} (X,Y). \end{aligned}$$

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

$$\begin{aligned} \Lambda ^\nabla _X Y-\Lambda ^\nabla _YX-[X,Y]&=\frac{\beta }{2\delta }[X,Y]-\frac{\beta }{2\delta }[Y,X]-[X,Y]=\left( 2-\frac{4\alpha }{\delta }-1\right) [\xi _i,\xi _j]\\&=\left( 1-\frac{4\alpha }{\delta }\right) 2\delta \xi _k=T^\nabla (X,Y). \end{aligned}$$

We further need to verify that $\Lambda ^\nabla _X\in \mathfrak {so}(\mathfrak {m})$ for all $X\in \mathfrak {m}$, that is $g(\Lambda ^\nabla _XY,Z)+g(Y,\Lambda _X^\nabla Z)=0$ 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 $\Lambda ^\nabla _XY=0$ 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 [2, 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

$$\begin{aligned} g(\Lambda ^\nabla _XY,Z)=g(L_X Y-2\alpha \sum \eta _i(X)\varphi _iY,Z)=-g(Y,L_X Z-2\alpha \sum \eta _i(X)\varphi _i Z). \end{aligned}$$

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

$$\begin{aligned} L_X Z-2\alpha \sum \eta _i(X)\varphi _i Z\in {\mathcal {V}} \end{aligned}$$

is perpendicular to Y. Hence, $g(\Lambda ^\nabla _X Y,Z)=0=-g(Y,\Lambda ^\nabla _X Z)$. $\square$

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 [10, 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

$$\begin{aligned} \Lambda ^\nabla _X={\left\{ \begin{array}{ll} 0, &{} X\in {\mathcal {H}}\\ \frac{\beta }{2\delta }{\mathrm {ad}}\,X, &{} X\in {\mathcal {V}}. \end{array}\right. } \end{aligned}$$

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

$$\begin{aligned} 2\alpha \sum _{i=1}^3 \eta _i(X) \varphi _i|_{{\mathcal {H}}}=\frac{2\alpha }{\delta }\sum _{i=1}^3 \eta _i(X) {\mathrm {ad}}\,{\xi _i}=\frac{2\alpha }{\delta }{\mathrm {ad}}\,X. \end{aligned}$$

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

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

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

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 (3), or equivalently (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

$$\begin{aligned} \frac{1}{2} T(X,Y)&=\alpha \sum _{i=1}^3 \Phi _i(X,Y)\xi _i=\alpha \sum _{i=1}^3g(X,\varphi _i Y)\xi _i\\&=\frac{\alpha }{\delta }\sum _{i=1}^3 g(X,[\xi _i,Y])\xi _i=\frac{\alpha }{8\alpha \delta ^2(n+2)}\sum _{i=1}^3\kappa (X,[Y,\xi _i])\xi _i\\&= \frac{1}{8\delta ^2(n+2)}\sum _{i=1}^3\kappa ([X,Y],\xi _i)\xi _i\\&=-\frac{1}{2} \sum _{i=1}^3 g([X,Y]_\mathfrak {m},\xi _i)\xi _i=-\frac{1}{2} [X,Y]_\mathfrak {m}, \end{aligned}$$

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

$$\begin{aligned} \Lambda ^g_X Y=\Lambda ^\nabla _X Y -\frac{1}{2} T(X,Y)=\frac{1}{2} [X,Y]_\mathfrak {m} \end{aligned}$$

For vertical vectors $X=\xi _i,Y=\xi _j$, (ijk) an even permutation of (123), we find

$$\begin{aligned} \Lambda ^g_{\xi _i} \xi _j&=\Lambda ^\nabla _{\xi _i} \xi _j-(\delta -4\alpha )\eta _{123}(\xi _i,\xi _j,\cdot )=\frac{\beta }{2\delta } [\xi _i,\xi _j]- (\delta -4\alpha )\xi _k\\&=\beta \xi _k+(\delta -4\alpha )\xi _k=\delta \xi _k=[\xi _i,\xi _j] \end{aligned}$$

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

$$\begin{aligned} T(X,Y)&=2\alpha \sum _{i=1}^3\eta _i\wedge \Phi _i^{{\mathcal {H}}}(X,Y,\cdot \,)=2\alpha \sum _{i=1}^3\eta _i(X)\Phi _i^{{\mathcal {H}}}(Y,\cdot \,)\\&=-2\alpha \sum _{i=1}^3\eta _i(X)\varphi _iY=-\frac{2\alpha }{\delta }\left[ \sum _{i=1}^3\eta _i(X)\xi _i,Y\right] =-\frac{2\alpha }{\delta }[X,Y] \end{aligned}$$

and thus

$$\begin{aligned} \Lambda ^g_X Y&=\Lambda ^\nabla _X Y-\frac{1}{2} T(X,Y)=\left( 1-\frac{2\alpha }{\delta }\right) [X,Y]+\frac{\alpha }{\delta }[X,Y]=\left( 1-\frac{\alpha }{\delta }\right) [X,Y]. \end{aligned}$$

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

$$\begin{aligned} \Lambda ^g_X Y=\Lambda ^\nabla _X Y-\frac{1}{2} T(X,Y)&=\frac{1}{2} T(Y,X)=\frac{\alpha }{\delta }[X,Y]. \end{aligned}$$

$\square$

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 [3].