Curvature properties of 3-(α,δ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\alpha ,\delta )$$\end{document}-Sasaki manifolds

We investigate curvature properties of 3-(α,δ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\alpha ,\delta )$$\end{document}-Sasaki manifolds, a special class of almost 3-contact metric manifolds generalizing 3-Sasaki manifolds (corresponding to α=δ=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha =\delta =1$$\end{document}) that admit a canonical metric connection with skew torsion and define a Riemannian submersion over a quaternionic Kähler manifold with vanishing, positive or negative scalar curvature, according to δ=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta =0$$\end{document}, αδ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \delta >0$$\end{document} or αδ<0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \delta <0$$\end{document}. We shall investigate both the Riemannian curvature and the curvature of the canonical connection, with particular focus on their curvature operators, regarded as symmetric endomorphisms of the space of 2-forms. We describe their spectrum, find distinguished eigenforms, and study the conditions of strongly definite curvature in the sense of Thorpe.


Introduction
The present paper is devoted to the curvature properties of 3-(α, δ)-Sasaki manifolds, both of the Riemannian connection and the canonical connection and, most importantly, their interaction. We will be particularly concerned with the curvature operators, regarded as symmetric endomorphisms of the space of 2-forms, in order to investigate their spectrum, find distinguished eigenforms, and study the conditions of strongly definite curvature in the sense of Thorpe. 3-(α, δ)-Sasaki manifolds are a special class of almost 3-contact metric manifolds. They were introduced in [2] as a generalization of 3-Sasaki manifolds (corresponding to α = δ = 1), and as a subclass of canonical almost 3-contact metric manifolds, characterized by admitting a canonical metric connection with totally skew-symmetric torsion (skew torsion for brief). The vanishing of the coefficient β := 2(δ − 2α) defines parallel 3-(α, δ)-Sasaki manifolds, for which the canonical connection parallelizes all the structure tensor fields. The geometry of 3-(α, δ)-Sasaki manifolds has been further investigated in [3], where it was shown that they admit a locally defined Riemannian submersion over a quaternionic Kähler manifold with vanishing, positive or negative scalar curvature, according to δ = 0, αδ > 0 or αδ < 0. These coincide, respectively, with the defining conditions of degenerate, positive and negative 3-(α, δ)-Sasaki structures, which are all preserved by a special type of deformations, namely H-homothetic deformations. The vertical distribution of the canonical submersion, which turns out to have totally geodesic leaves, coincides with the 3-dimensional distribution spanned by the three Reeb vector fields ξ i , i = 1, 2, 3, of the structure. The canonical connection plays a central role in this picture, as it preserves both the vertical and the horizontal distribution, and in fact, when applied to basic vector fields, it projects onto the Levi-Civita connection of the quaternionic Kähler base space. Beyond this introduction, the remaining part of Section 1 will be devoted to a short review of the notions and results needed in this work.
In Sect. 2, we will see how the canonical curvature operator R is related to the Riemannian curvature operator R g N of the qK base space of the canonical submersion π : M → N . Introducing a suitable decomposition of R, we show that if R g N is non-negative, resp. nonpositive, then so is the operator R, provided that αβ ≥ 0 for non-negative definiteness (Theorem 2.3). The decomposition of the operator R also allows to determine a set of six orthogonal eigenforms of R, distinguished into two triples: i − ξ jk , and i + (n + 1)ξ jk , where (i jk) denotes an even permutation of (123), i are the fundamental 2-forms of the structure, and ξ jk := ξ j ∧ ξ k .
The goal of Sect. 3 is to interpret both triples i − ξ jk and i + (n + 1)ξ jk as eigenforms, not only of R, but also of the Riemannian curvature operator R g of M. We show that them being eigenforms of R g provides necessary and sufficient conditions for M to be Einstein, which precisely happens when δ = α or δ = (2n + 3)α, with dim M = 4n + 3 (Theorem 3.1). The result is obtained by taking into account the relation between the operators R and R g , involving two further symmetric operators G T and S T defined by means of the torsion of the canonical connection.
Section 4 is devoted to the investigation of conditions of strong definiteness for the Riemannian curvature of a 3-(α, δ)-Sasaki manifold. Recall that a Riemannian manifold (M, g) is said to have strongly positive curvature if for some 4-form ω the modified symmetric operator R g + ω is positive definite. On the one hand, this weakens the condition of positive definiteness of the Riemannian curvature operator (R g > 0), which forces the Riemannian manifold to be diffeomorphic to a space form [9]. On the other hand, this provides a stronger condition than positive sectional curvature as, for any 2-plane σ , sec(σ ) = (R g +ω)(σ ), σ . The method of modifying the curvature operator by a 4-form was originally introduced by Thorpe [16,17], and then developed by various authors [7,15,20]. In the same way, one can introduce a notion of strongly non-negative curvature. Considering a 3-(α, δ)-Sasaki manifold M with canonical submersion π : M → N , we determine sufficient conditions for strongly non-negative and strongly positive curvature (Theorem 4.1). We require a sufficiently large quotient δ/α 0, together with strongly non-negative or strongly positive curvature for the quaternionic Kähler base space N . Suitable 4-forms modifying the Riemannian curvature operator R g of M are constructed using the pullback π * ω of a 4-form ω which modifies the operator R g N , and the 4-form σ T = 1 2 dT , T being the torsion of the canonical connection; this 4-form is known to be a measure of the non-degeneracy of the torsion T , which explains its appearance in this context. We discuss the case of homogeneous 3-(α, δ)-Sasaki manifolds fibering over symmetric quaternionic Kähler spaces of compact type (Wolf spaces) and their non-compact duals. A construction of these spaces was given in [3], providing a classification in the compact case (αδ > 0). In this case, we show that if αβ ≥ 0, then the manifold is strongly non-negative. Strong positivity is much more restrictive, as the only spaces admitting a homogeneous structure with strict positive sectional curvature are the 7-dimensional Aloff-Wallach space W 1,1 , the spheres S 4n+3 , and real projective spaces RP 4n+3 . For these spaces, assuming αβ > 0, we provide explicit 4-forms modifying the Riemannian curvature operator to obtain strongly positive curvature (Theorem 4.5). In Sect. 4.3, we show strong positive curvature for a class of inhomogeneous 3-(α, δ)-Sasaski manifold obtained by 3-Sasaki reduction, compare [11,13].

Curvature endomorphisms and strongly positive curvature
We review notations and established properties of connections with skew torsion and their curvature. We refer to [1] for further details.
Let (M, g) be a Riemannian manifold, dim M = n. A metric connection ∇ is said to have skew torsion if the (0, 3)-tensor field T defined by is a 3-form. Then ∇ and the Levi-Civita connection ∇ g are related by ∇ X Y = ∇ g X Y + 1 2 T (X , Y ), and ∇ has the same geodesics as ∇ g . Assume further that T is parallel, i.e., ∇T = 0. Typical examples of manifolds admitting metric connections with parallel skew torsion include Sasaki, nearly parallel G 2 , nearly Kähler and several others (see also the recent paper [12]).
The fact that ∇T = 0 implies dT = 2σ T , where σ T is the 4-form defined by which implies the pair symmetry These identities trivially apply to the Levi-Civita connection ∇ g of (M, g) and its curvature R g . The Riemannian curvature R g is related to R by Recall that, given a Riemannian manifold (M, g), at each point x ∈ M the space p T x M of p-vectors of T x M can be endowed with the inner product defined by In particular, if {e r , r = 1, . . . , n} is an orthonormal basis of T x M, then {e i 1 ∧. . .∧e i p , 1 ≤ i 1 < . . . < i p ≤ n} is an orthonormal basis for p T x M. Furthermore, by means of the inner product, we identify p T x M with the space p T * x M of p-forms on T x M. The curvature tensor R induces by (1.2) a symmetric linear operator The sign − is due to our curvature convention, so that positive curvature operator R implies positive sectional curvature Any 4-form ω can be regarded as a symmetric operator In fact, the space of all symmetric linear operators splits as Then, ker b is the space of algebraic curvature operators, 1 i.e., operators satisfying the first Bianchi identity (1.1) for vanishing torsion. Definition 1. 1 We will denote by S T : 2 M → 2 M the symmetric operator associated to the 4-form σ T , i.e., We will also consider the (0, 4)-tensor field G T and the symmetric operator Owing to (1.3), we have (1.5) Definition 1.2 A Riemannian manifold (M, g) is said to have strongly positive curvature (resp. strongly non-negative curvature) if there exists a 4-form ω such that R g + ω is positive definite (resp. non-negative) at every point x ∈ M [7,16,17].
Such a notion is justified by the fact that for every 2-plane σ , being ω(σ ), σ = 0, one has sec(σ ) = (R g + ω)(σ ), σ , so that strongly positive curvature implies positive sectional curvature. In fact this is an intermediate notion between positive definiteness of the Riemannian curvature (R g > 0) and positive sectional curvature.

Review of 3-(˛, ı)-Sasaki manifolds and their basic properties
We now want to focus on the situation at hand. That is a 3-(α, δ)-Sasaki manifold and its canonical connection ∇. Let us recall the central definitions and key properties for later reference.
An almost 3-contact metric manifold is a differentiable manifold M of dimension 4n + 3 endowed with three almost contact metric structures (ϕ i , ξ i , η i , g), i = 1, 2, 3, sharing the same Riemannian metric g, and satisfying the following compatibility relations for any even permutation (i jk) of (123) [8]. The tangent bundle of M splits into the orthogonal sum T M = H ⊕ V, where H and V are, respectively, the horizontal and the vertical distribution, defined by In particular, H has rank 4n and the three Reeb vector fields ξ 1 , ξ 2 , ξ 3 are orthonormal. The manifold is said to be hypernormal if each almost contact metric structure (ϕ i , ξ i , η i , g) is normal. If the three structures are α-Sasaki, M is called a 3-α-Sasaki manifold, 3-Sasaki if α = 1. As a comprehensive introduction to Sasaki and 3-Sasaki geometry, we refer to [10]. We denote an almost 3-contact metric manifold by (M, ϕ i , ξ i , η i , g), understanding that the index is running from 1 to 3.
The distinction into degenerate, positive, and negative 3-(α, δ)-Sasaki manifolds stems from their behavior under a special type of deformations of the structure, called H-homothetic deformations, which turn out to preserve the three classes [2, Section 2.3].
We recall some basic properties of 3-(α, δ)-Sasaki manifolds. Any 3-(α, δ)-Sasaki manifold is hypernormal. Hence, for α = δ one has a 3-α-Sasaki manifold. Each Reeb vector field ξ i is Killing, and it is an infinitesimal automorphism of the horizontal distribution H, i.e., dη i (X , ξ j ) = 0 for every X ∈ H and i, j = 1, 2, 3. The vertical distribution V is integrable with totally geodesic leaves. In particular, the commutators of the Reeb vector fields are purely vertical and for every even permutation (i jk) of (123) they are given by Meanwhile, for any two horizontal vector fields X , Y , the vertical part of commutators is given by (1.7) Any 3-(α, δ)-Sasaki manifold is a canonical almost 3-contact metric manifold, in the sense of the definition given in [2], which is equivalent to the existence of a canonical connection. The canonical connection of a 3-(α, δ)-Sasaki manifold (M, ϕ i , ξ i , η i , g) is the unique metric connection ∇ with skew torsion such that for every even permutation (i jk) of (123), where β = 2(δ − 2α). The covariant derivatives of the other structure tensor fields are given by If δ = 2α, then β = 0 and the canonical connection parallelizes all the structure tensor fields. Any 3-(α, δ)-Sasaki manifold with δ = 2α, which is a positive 3-(α, δ)-Sasaki manifold, is called parallel.
The canonical connection plays a central role in the description of the transverse geometry defined by the vertical foliation: The base space N is equipped with a quaternion Kähler structure locally defined byφ i = π * • ϕ i • s, i = 1, 2, 3, where s : N → M is an arbitrary section of π. The scalar curvature of N is 16n(n + 2)αδ.
Here and in the following X ∈ T M denotes the horizontal lift of a vector field X ∈ T N under the Riemannian submersion π : M → N . We further denote the Levi-Civita connection on (N , g N ) by ∇ g N and analogously for its associated tensors, e.g., the curvature tensor R g N .
From the above theorem, it follows that any 3-(α, δ)-Sasaki manifold locally fibers over a quaternionic Kähler manifold of positive or negative scalar curvature if either αδ > 0 or αδ < 0, respectively, or over a hyper-Kähler manifold in the degenerate case.
Finally, we recall some properties for the torsion of the canonical connection. The torsion T of the canonical connection of a 3-(α, δ)-Sasaki manifold is given by where H i = i + η jk ∈ 2 (H) is the horizontal part of the fundamental 2-form i . Here we put η jk :=η j ∧ η k and η 123 :=η 1 ∧ η 2 ∧ η 3 . In particular, for every X , Y ∈ X(M), The symbol i, j,k S means the sum over all even permutations of (123). The torsion of the canonical connection satisfies ∇T = 0 and (1.12)

The canonical curvature and the canonical submersion
The canonical curvature is particularly well behaved on the defining tensors of a 3-(α, δ)-Sasaki manifold. We will make use of this to compute directly related curvature identities in the following two propositions. These, in turn, allowed us to prove the existence of the canonical submersion in [3].
Let ∇ be the canonical connection and R the curvature tensor of ∇. Then, the following equations hold: where X , Y , Z ∈ X(M) and (i jk) is an even permutation of (123).

Proposition 2.2
The curvature tensor R of the canonical connection of a 3-(α, δ)-Sasaki manifold satisfies for any X , Y , Z ∈ H and i, j, k, l = 1, 2, 3 the identities

4)
and for an even permutation (i jk) of (123) Proof Considering the symmetries of R, we immediately obtain the first three expressions from equation (2.2). Using (2.1) for ϕ j we obtain
Considering now the canonical submersion π : M → N defined in Theorem 1.1, in the next theorem we will relate the missing purely horizontal part of the canonical curvature tensor to the curvature of the quaternionic Kähler base space N . We recall a computational lemma from [3].

Lemma 2.1 ([3, Lemma 2.2.1]) For any vertical vector field X ∈ V and for any basic vector
where we have used (1.7) and Lemma 2.1. Plugging these identities into the curvature, we find

Decomposition of the canonical curvature operator
We now want to look at the canonical curvature as a curvature operator and consider its eigenvalues and definiteness. Recall that the canonical curvature operator R : 2 M → 2 M defines a symmetric operator. Rewriting (2.3) as operator identities we obtain showing that the canonical curvature operator vanishes on V ∧ H. Thus, it can be considered as a symmetric operator R : 2 V ⊕ 2 H → 2 V ⊕ 2 H. It does not restrict to the individual summands, but we can accomplish a more nuanced decomposition.

Remark 2.2
From here on out, we will freely identify T M and T * M as well as their exterior products. In particular, we write ξ jk :

Proposition 2.3
The curvature operator R can be decomposed as and R par is trivial outside of the horizontal part, i.e., R par | ( 2 H) ⊥ = 0.
Proof Equations (2.4) and (2.5) in terms of the curvature operator read We observe that identities (2.8) and (2.9) are of the form shows that R par :=R − αβR ⊥ is trivial outside 2 H → 2 H.
The notation R par is justified by the fact that in the parallel case (β = 0) we have R = R par . Taking into account the canonical submersion π : M → N , we may consider the Riemannian curvature operator R g N on the base N as a curvature operator 2 H → 2 H via the horizontal lift. From Theorem 2.1, we have Note the sign change due to our convention of sign in R compared to R. Comparing with the definition of R par in Proposition 2.3 and expanding β = 2(δ − 2α) yields (2.10)

Remark 2.3
Recall that the curvature operator of qK spaces is given by R g N = νR 0 + R 1 , where ν = 4αδ is the reduced scalar curvature, R 1 is a curvature operator of hyper-Kähler type, and is the curvature operator of HP n [4, Tables 1 and 2]. Here ∧ denotes the Kulkarni-Nomizu product viewed as an operator. A curvature operator is said to be of hyper-Kähler type if it is Ricci-flat and commutes with the quaternionic structure. Combining this with (2.10), we find Note that in the degenerate case, the picture simplifies, since then we just have R par = We will now show some crucial properties of the spectrum of the introduced operators. Before proving the next lemmas, we remark a few facts on the fundamental 2-forms i of a 3-(α, δ)-Sasaki structure. Each i can be expressed as where {e r , r = 1, . . . , 4n + 3} is a local orthonormal frame. A straightforward computation shows that for every i, j, k = 1, 2, 3, where i jk is the totally skew-symmetric symbol. We will also use adapted bases in the following sense. Therefore, in an adapted basis, the horizontal part H i = i + ξ jk is expressed as e r ∧ ϕ i e r + ϕ j e r ∧ ϕ k e r . (2.14) Lemma 2.2 R ⊥ has the only nonzero eigenvalue 2(n + 2) with eigenspace generated by i − ξ jk for i = 1, 2, 3.

Lemma 2.3
The kernel of R par contains the space generated by for any even permutation (i jk) of (123). Thus, Z ∧ϕ i Z +ϕ j Z ∧ϕ k Z ∈ ker R par . The second part of the statement follows immediately from (2.14) and R par | ( 2 H) ⊥ = 0.
As a first consequence, we can obtain a distinguished set of eigenforms of the canonical curvature operator R, which will have a special role in the characterization of the Einstein condition for a 3-(α, δ)-Sasaki manifold (see Theorem 3.1).

Theorem 2.2
The curvature operator R of the canonical connection of any 3-(α, δ)-Sasaki manifold (M, ϕ i , ξ i , η i , g) admits the following six orthogonal eigenforms: Proof From Lemma 2.3, all the forms are in the kernel of R par . Therefore, we only have to check that i −ξ jk and i +(n+1)ξ jk are eigenvectors of R ⊥ with the respective eigenvalues. Lemma 2.2 provides just that under the observation i + (n + 1)ξ jk , i − ξ jk = 0.
For later use, we observe that one can immediately obtain:

Remark 2.4
The discussion on the curvature operator R g N actually showed that the i are eigenvectors of R g N with eigenvalue 4αδ. Thus, only now we proved that the canonical submersion of a 7-dimensional 3-(α, δ)-Sasaki manifold has a quaternionic Kähler base under the stricter definition usually assumed. Compare the discussion ahead of [10, Definition 12.2.12].
The following theorem links the Riemannian curvature of the qK base to the canonical curvature of the total space, thus underlining the intricate relationship of these two connections. Proof By (2.18), if R g N is either non-negative or non-positive, then so is R par and the sign of αδ. Using (2.17), we obtain part (a) directly. For part (b), note that if αδ ≤ 0, then αβ = 2αδ − 4α 2 < 0.

Remark 3.1 Together with Theorem 2.2, we observe that exclusively in the Einstein case
i − ξ jk and i + (n + 1)ξ jk are joint eigenforms of R, R g and G T + S T . Since i + (n + 1)ξ jk ∈ ker R we have that the corresponding eigenvalue of G T + S T is 4λ, or 4λ 2 , respectively.
In order to prove Theorem 3.1, we will determine throughout the next propositions how R g acts on the forms i and ξ jk . Recall that by (1.5) the curvature operators R g and R are related by the operators S T and G T defined in Definition 1.1. They act on the forms i and ξ jk as follows. (M, ϕ i , ξ i , η i , g) be a 3-(α, δ)-Sasaki manifold. The torsion T of the canonical connection satisfies the following:

2)
Proof First we show that for every vector fields X , Y and for every even permutation (i jk) of (123) which is equivalent to (3.1), taking into account (1.4). Indeed, we compute We can also compute l,m,n Therefore, using (1.12) we get (3.3).
We have now gathered all necessary results to give a proof of the main theorem.

Proof of Theorem 3.1 The equivalence of (c) and (d) is known, see [2, Proposition 2.3.3].
From (3.9) and (3.10), we have that R g ( i − ξ jk ) = a i + bξ jk with Then i − ξ jk is an eigenform of R g if and only if a + b = 0, that is δ = α or δ = (2n + 3)α.

Strongly positive curvature
We now investigate strongly non-negative and even strongly positive curvature on (M, g). Recall that, by (1.5), the curvature operators R and R g are related by In particular, (M, g) is strongly non-negative with 4-form − 1 4 σ T if and only if Observe that G T is non-negative by definition, so we have directly strong non-negativity if R is non-negative. Theorem 2.3 thus yields (recall that β := 2(δ − 2α)) Corollary 4.1 Let M be a 3-(α, δ)-Sasaki manifold with αβ ≥ 0 and R g N ≥ 0. Then (M, g) is strongly non-negative with 4-form − 1 4 σ T .
As we later see, this will be sufficient for homogeneous spaces, but in general the condition R g N ≥ 0 is too strong. However, we can relax the condition on the base to strong nonnegativity, but we need an additional assumption on the 4-form. To do all this, we need some notation.
For i = 1, 2, 3, denote the 2-dimensional spaces N i := span{ H i , ξ jk }. Then decompose the space of 2-forms into orthogonal subbundles For a linear map A : 2 M → 2 M we denote A 1 :=A| 2 1 and correspondingly for the other spaces.
Let us motivate this decomposition. The obvious 2 = 0. However, R does not restrict to 2 V and 2 H, but to 2 1 and 2 2 . In fact, the characterization R = αβR ⊥ + R par is with respect to 2 1 and 2 2 as noted in the proof of Theorem 2.2. The space 2 1 can be seen as controlled by the 3-(α, δ)-Sasaki structure, while 2 2 resonates the geometry of the base N . This is emphasized by the fact that the common eigenforms discussed in Sect. 3 all lie in 2 1 .

Definition 4.1
We call a 4-form ω ∈ 4 N on a quaternionic Kähler space N adapted with minimal eigenvalue ν ∈ R if for every point p ∈ N the quaternionic bundle Q lies in the ν p -eigenspace of ω p , considered as an operator 2 T p N → 2 T p N , where the eigenvalues ν p are bounded below by ν.
are satisfied. Then M is strongly positive with 4-form π * ω − ( 1 4 + ε)σ T for some ε > 0 sufficiently small. The corollary will be proved as a byproduct of Theorem 4.1.

Remark 4.1
Observe that the conditions (4.1) will be fulfilled for δ/α 0 sufficiently big. This can be achieved by H-homothetic deformation, compare [2, Section 2.3]. In fact, the horizontal structure is only changed by global scaling via a parameter a inversely proportional to αδ. However, fixing a we can scale the Reeb orbits by a parameter c implying a quadratic change in δ α . Therefore, such a H-homothetic deformation does not change the horizontal structure and thereby fixes ν, but it increases the leading term of both polynomial conditions.
To prove these results, we need a more deliberate investigation of how G T acts on the spaces 2 i . From Eq. (1.11), it follows that the torsion T of the canonical connection satisfies Thus, G T preserves 2 3 = V ∧ H and by Proposition 3.2 2 1 as well. Therefore G T splits into a direct sum of operators G 1 ⊕ G 2 ⊕ G 3 on 2 1 ⊕ 2 2 ⊕ 2 3 . Consider some adapted basis e r , r = 1, . . . , 4n + 3 of M. We may define the quaternionic spaces H l = span{e 4l , e 4l+1 , e 4l+2 , e 4l+3 }, l = 1, . . . , n, and accordingly we have Note that these descriptions depend on the choice of adapted basis unlike the spaces 2 i themselves.

Proof
Again using (1.11), we find In particular, for the adapted basis e 4l , . . . , e 4l+3 of H l . Hence, the vectors e r ∧ξ 1 +ϕ 3 e r ∧ξ 2 −ϕ 2 e r ∧ξ 3 are 4 linearly independent eigenvectors with eigenvalue 12α 2 . In fact, these are all the eigenvectors with nonzero eigenvalues, since (4.3) shows that This implies that analogous to R ⊥ the sum αβR ⊥ + 1 4 G T is orthogonal to R par , i.e., it is trivial on the space 2 2 where R par is non-trivial. It is now time to include the 4-form ω.

Proof From Proposition 3.2, we obtain
By adaptedness of ω at every point, the two-forms H i ∈ π * Q lie inside some eigenspace with eigenvalue ν p ≥ ν ∈ R. Thus on N i with respect to the orthonormal basis 1 √ 2n H i and ξ jk the sum takes the matrix form Now the restriction to the 2-dimensional space N i is positive (non-negative) if and only if both the determinant and the trace are. We have Since the ν p are bounded below by ν, the trace is positive if the quadratic polynomial in δ satisfies δ 2 + 4nαδ − 6nα 2 + ν > 0. The determinant is given by and, thus, the operator αβR ⊥ + 1 4 G 1 + (π * ω) 1 is positive if the cubic polynomial 4nα(δ − 2α) 3 + δ 2 ν is positive as well.
In the unaltered case, or equivalently ν = 0, we can quantify the condition more nicely in terms of αβ.
Proof of Theorem 4. 1 We have seen that under R ⊥ , R par , G T and π * ω the spaces 2 1 , 2 2 and 2 3 are invariant. Thus, we may decompose By assumption is positive (non-negative), where we have used the identification 2 2 = π * Q ⊥ . The results so far are summarized in Table 1. In fact, in the non-negative case we are done.
In order to prove strong positivity, we need to prove that −εσ T provides strict positivity on the kernel of G 3 . For sufficiently small ε, it will do so without destroying positivity where already established. Indeed, Lemma 4.3 shows that σ T is negative definite on the kernel of G 3 .

Lemma 4.3 The operator S T corresponding to σ T is negative definite on the kernel of G 3 if and only if αβ > 0.
Proof As in the proof of Lemma 4.1, we may split G 3 into n copies ofĜ 3 on each quaternionic subspace. Let e 4l , . . . , e 4l+3 ∈ H l be an adapted basis of one such subspace. Then from the same proof we find that T (e r ∧ ξ i ) = 2αϕ i e l = −T (ϕ j e r ∧ ξ k ). Thus, By definition of S T , we have , ξ a )). We thus obtain the full expression Finally we compute S T on kerĜ 3 to obtain the result As a word of caution we should state where this theorem might and might not be applicable. By assumption, the quaternionic Kähler orbifold is strongly positive and thereby has positive sectional curvature. M. Berger investigated such manifolds in [5]. As observed in [13], Berger's argument is purely local. It therefore extends to quaternionic Kähler orbifolds.  (M 4n , g, Q) is locally isometric to HP n with its standard quaternionic Kähler structure.
Thus, the strong positivity result of Theorem 4.1 can only be applicable on 3-(α, δ)-Sasaki manifolds of dimension 7 or on finite quotients of S 4n+3 . We will see in the next section that indeed both cases appear for homogeneous manifolds.

The homogeneous case
We would like to apply the positivity discussion to homogeneous 3-(α, δ)-Sasaki manifolds, more precisely to those that fiber over Wolf spaces and their non-compact duals. We recall their construction from our previous publication [3], extending the similar discussion for homogeneous 3-Sasaki manifolds by [14].
Note that the homogeneous 3-(α, δ)-Sasaki structure on RP 4n+3 is not directly obtained by this construction but as the quotient of S 4n+3 = Sp(n + 1)/Sp(n) by Z 2 . Here the local structure is the same as for S 4n+3 given in the theorem. With this exception, we have that all positive homogeneous 3-(α, δ)-Sasaki manifolds are obtained from the theorem. In the negative case more exist so we will restrict ourselves in the following discussion to those over symmetric base spaces. Proof In the positive case, M has to fiber over a symmetric base, compare [3]. In this case, the base is a compact symmetric space; hence, the curvature operator R g N is non-negative. In part (b), the base is a non-compact symmetric space by assumption, hence R g N ≤ 0. Therefore in both cases it fulfills the requirement of Theorem 2.3. In the positive case also, Corollary 4.1 applies.
We will next focus on strong positivity. This is much more restrictive than strong nonnegativity. In particular, strong positivity implies strict positive sectional curvature and homogeneous manifolds with strictly positive sectional curvature have been classified [6,18,19]. Out of these only the 7-dimensional Aloff-Wallach space W 1,1 , the spheres S 4n+3 and real projective spaces RP 4n+3 admit homogeneous 3-(α, δ)-Sasaki structures. We will thus prove

Remark 4.2
The strong positivity of these spaces W 1,1 and S 4n+3 , RP 4n+3 , can actually be proven by the Strong Wallach Theorem in [7]. We compare to our case: (i) Observe that all positive homogeneous 3-(α, δ)-Sasaki manifolds are given by a homogeneous fibration In the case of S 4n+3 , the fiber is Sp(1) instead. (ii) In their strong Wallach theorem [7], the authors consider the metrics g t = t Q| V + Q| H for 0 < t < 1, where Q is a negative multiple of the Killing form. If we set Q = −κ 8αδ(n+2) as in the 3-(α, δ)-Sasaki setting then t = 2α δ and, thus, the condition 0 < t < 1 is equivalent to β > 0. (iii) We have dim G 0 /H = 3 and G 0 /H = SO(3) = RP 3 , S 3 in the case of S 4n+3 , with a scaled standard metric. In particular, the fiber is of positive sectional curvature. (iv) They require a strong fatness property for the homogeneous fibration. Adapted to our notation the bundle is strongly fat if there is a 4-form τ such that F + τ : Thus by the previous lemma τ = −εσ T accomplishes strong fatness for sufficiently small ε. (v) The final condition is for the base to be one of S 4n , RP 4n , CP 2n , HP n . The only homogeneous 3-(α, δ)-Sasaki manifolds such that this holds are S 4n+3 , RP 4n+3 which fiber over HP n , and W 1,1 which fibers over CP 2 .
Note that (i)-(iv) are valid for all positive homogeneous examples not only for the spheres, real projective spaces and W 1,1 .

Proof of Theorem 4.5
Since our discussion is pointwise we will identify tensors on N with those on H.
In [13], the author shows that under the assumption √ 2 min p i > max p i a certain deformation of the 3-Sasaki metric, corresponding to a H-homothetic deformation in our notation, admits positive sectional curvature. We make use of a key step of his showing that their underlying quaternionic Kähler orbifolds have positive sectional curvature, [13,Theorem 2]. In order to make the jump from positive sectional curvature to strongly positive curvature we make use of the fact that O( p 1 , p 2 , p 3 ) is 4-dimensional. In this dimension, Thorpe proves the following [16,Corollary 4.2]. Theorem 4.8 ([16, 17]) Let V be a 4-dimensional vector space and R any algebraic curvature operator on V . If λ is the minimal sectional curvature of R, then there is a unique ω ∈ 4 V such that λ is the minimal eigenvalue of R + ω.
We are finally ready to state our main theorem. Theorem 4.9 Let p 1 , p 2 , p 3 be coprime integers with √ 2 min p i > max p i . Then there is a H-homothetic deformation of S( p 1 , p 2 , p 3 ) that has strongly positive curvature. Proof Thorpe's theorem proves that the orbifold O( p 1 , p 2 , p 3 ) has not only positive sectional curvature but strongly positive curvature. Since we are in dimension 4 the form ω is necessarily a multiple of the volume form ω = ν p dVol. As before the volume form has eigenspaces 2 ± where 2 + = Q and 2 − = Q ⊥ . In particular, ω is an adapted 4-form with minimal eigenvalue ν = min ν p . The minimum exists since the orbifolds are quotients of compact spaces and, thus, compact themselves. All in all we may apply Theorem 4.1. Note that by Remark 4.1 we obtain a H-homothetic deformation of S( p 1 , p 2 , p 3 ) with δ/α 0 sufficiently big while not changing the metric g O on O( p 1 , p 2 , p 3 ).