J.reine angew. Math. 455 (1994), 183-220  Journal für die reine und

angewandte Mathematik

The Geometry and Topology of 3-Sasakian Manifolds

Charles P. Boyer Krzysztof Galicki Benjamin M. Mann

In recent years quaternionic Kähler and hyperkähler manifolds have received a great deal of attention. They appear in many different areas of mathematics and mathematical physics. It has been argued that these recent advances in quaternionic geometry vindicate Hamilton’s conviction that the algebra of quaternions should play an important role in mathematical physics [At, Hi1]. The purpose of this paper is to describe the geometry and topology of a class of Riemannian Einstein manifolds that is closely related to both hyperkähler and quaternionic Kähler manifolds. These manifolds, known as manifolds with a Sasakian 3-structure, first appeared in a paper by Kuo in 1970 [Ku] which was published a few years before Ishihara and Calabi introduced the now commonly accepted terms “quaternionic Kähler” and “hyperkähler”, respectively. We shall refer to manifolds with a Sasakian 3-structure as 3-Sasakian manifolds.

Historically Sasakian structures grew out of research in contact manifolds and were studied extensively in the 1960’s especially by the Japanese school (See [YK] and references therein). In 1970 three more papers, [KuTach], [TachYu], and [Tan1], were published in the Japanese literature discussing Sasakian 3-structures. These structures were then vigorously studied by Japanese mathematicians from 1970-1975, culminating with an important paper of Konishi in 1975 [Kon] which shows the existence of a Sasakian 3-structure on a certain principal SO(3) bundle over any quaternionic Kähler manifold of positive scalar curvature. Earlier on, in 1973 Ishihara [I2] had shown that if the distribution formed by the three Killing vector fields which define the Sasakian 3-structure is regular then the space of leaves is a quaternionic Kähler manifold. This then led Ishihara to his foundational work on quaternionic Kähler manifolds [I1]. It is notable that in this early period the only examples of 3-Sasakian manifolds that were given were those of constant curvature.

Unlike the current interest in both quaternionic Kähler and hyperkähler structures, Sasakian 3-structures appear to have been largely neglected in recent years. For example, in Besse’s comprehensive book on Einstein manifolds [Bes], there is a chapter devoted to quaternionic Kähler and hyperkähler manifolds. By contrast, there is no explicit mention of 3-Sasakian manifolds which appear only implicitly and very briefly as examples of homogeneous Einstein spaces. For instance, Besse proves the existence of two Einstein metrics on a certain principal SO(3) bundle Y over any quaternionic Kähler manifold with positive scalar curvature [Bes: 14.85]. The manifold Y with one of these two metrics is an example of a 3-Sasakian manifold, and the fibration is the one given in the lower right hand corner of diagram 0.1 below. Although this result was discovered by Konishi [Kon] almost 20 years ago, Besse did not seem to be aware of this fact.

During the preparation of this work all three authors were supported by NSF grants.

We were led to study this geometry because it appears as a natural object in a new quotient construction for certain hyperkähler manifolds [BGM1]. We found that 3-Sasakian manifolds provided a natural piece of a puzzle that links together three other different geometric structures. In particular, for any quaternionic Kähler manifold M of positive scalar curvature there exists a commutative diagram

0.1  \begin{aligned}& & U \\ & ↙^{ℂ^{∗}/ℤ_{2}} & & ↖^{ι} \\ Z & & ↓ℍ^{∗}/ℤ_{2} & & S, \\ & ↘^{ℂℙ^{1}} & & ↙^{ℝℙ^{3}} \\ & & M \end{aligned}

where U is hyperkähler (the Swann bundle associated to M [Sw]), Z is Kähler-Einstein (the twistor space associated to M [Sal1]), and S is 3-Sasakian (the Konishi bundle associated to M [Kon]). The map ι : S ↪ U is the inclusion of a level set of a natural real valued function while all the other maps in diagram 0.1 are fibrations where we have denoted each map by its associated fiber. Furthermore, both Z and S are compact, of positive scalar curvature, and S is a principal circle bundle over Z.

All four geometries in diagram 0.1 are Einstein. The important observation, due to Kashiwada [Ka], that 3-Sasakian manifolds are Einstein and the relationship of S to the other quaternionic geometries appearing in diagram 0.1 motivated our study of these spaces. Our first main result in this paper is

Theorem A: Let (S, g, ξ^{a}) be a 3-Sasakian manifold of dimension 4n + 3 such that the Killing vector fields ξ^{a} are complete for a = 1, 2, 3. Then

(i) (S, g, ξ^{a}) is an Einstein manifold of positive scalar curvature equal to 2(2n+1)(4n+3).

(ii) The metric g is bundle-like with respect to the foliation F, defined by {ξ^{a}}_{a=1,2,3}.

(iii) Each leaf L of the foliation F is a 3-dimensional homogeneous spherical space form.

(iv) The space of leaves S/F is a quaternionic Kähler orbifold of dimension 4n with positive scalar curvature equal to 16n(n + 2).

Hence, every complete 3-Sasakian manifold is compact with finite fundamental group and diameter less than or equal to π.

Thus, in order to recapture the close relationship between 3-Sasakian and quaternionic Kähler geometries, one must generalize diagram 0.1 to allow the base space M to be a quaternionic Kähler orbifold. 3-Sasakian manifolds are much more plentiful than quaternionic Kähler manifolds of positive scalar curvature [BGM1]. LeBrun and Salamon [LS] have recently shown that there are only a finite number of quaternionic Kähler manifolds of positive scalar curvature (up to homothety) in each quaternionic dimension. Moreover, all known quaternionic Kähler manifolds are symmetric spaces and appear in Wolf’s classication [Wo]. By contrast, a 3-Sasakian manifold must be of real dimension 4n + 3 and in each such allowable dimension we have found infinitely many compact examples which range through infinitely many distinct homotopy types in every dimension. Moreover, in dimension 7, we found countable families of strongly inhomogeneous 3-Sasakian manifolds, that is manifolds which are not homotopy equivalent to any compact Riemannian homogeneous space. These examples, which are discussed in detail in the later part of this paper, are, to the best of our knowledge, the only known examples of complete strongly inhomogeneous Einstein manifolds of positive scalar curvature.

In section three we prove the converse of the result presented in [BGM1], where we showed that 3-Sasakian manifolds occur naturally as level sets of the hyperkähler potential function ν arising from an Sp(1) action on hyperkähler manifolds with a certain tensor field ϕ being constant. More precisely we prove

Theorem B: Let (S, g_{S}, ξ^{a}) be a complete 3-Sasakian manifold. Then the product manifold M = S × ℝ^{+} with the cone metric g_{M} = dr^{2} + r^{2}g_{S} is hyperkähler in such a way that the obstruction section ϕ associated to the natural Sp(1) action is constant.

Theorem B and our constructions in the later sections can be used to give many new examples of compact hypercomplex manifolds. For details see Corollary 3.6. In section four we classify all 3-Sasakian homogeneous spaces, that is 3-Sasakian manifolds with transitive action of the group of automorphisms of the Sasakian 3-structure.

Theorem C: Let (S, g, ξ^{a}) be a 3-Sasakian homogeneous space. Then S is precisely one of the following homogeneous spaces:

\frac{Sp(n)}{Sp(n − 1)} ≃ S^{4n−1}, \frac{Sp(n)}{Sp(n − 1) × ℤ_{2}} ≃ ℝℙ^{4n−1}, \frac{SU(m)}{S(U(m − 2) × U(1))},

\frac{SO(k)}{SO(k − 4) × Sp(1)}, \frac{G_{2}}{Sp(1)}, \frac{F_{4}}{Sp(3)}, \frac{E_{6}}{SU(6)}, \frac{E_{7}}{Spin(12)}, \frac{E_{8}}{E_{7}} .

Here n ≥ 1, Sp(0) denotes the identity group, m ≥ 3, and k ≥ 7. Furthermore, the fiber F over the Wolf space is Sp(1) in only one case which occurs precisely when (S, g, ξ^{a}) is simply connected with constant curvature; that is, when S = S^{4n−1}. In all other cases F = SO(3).

Our technique for constructing new examples of 3-Sasakian manifolds is the reduction procedure described in section five. Explicitly, we prove

Theorem D: Let (S, g_{S}, ξ^{a}) be a 3-Sasakian manifold with a connected compact Lie group G acting on S by 3-Sasakian isometries. Let μ_{S} be the corresponding 3-Sasakian moment map and assume both that 0 is a regular value of μ_{S} and that G acts freely on the submanifold μ_{S}^{−1}(0). Furthermore, let ι : μ_{S}^{−1}(0)⟶S and π : μ_{S}^{−1}(0)⟶μ_{S}^{−1}(0)/G denote the corresponding embedding and submersion. Then (Š = μ_{S}^{−1}(0)/G, ǧ_{S}, ξ̌^{a}) is a smooth 3-Sasakian manifold of dimension 4(n − dim 𝔤) − 1 with metric ǧ_{S} and Sasakian vector fields ξ̌^{a} determined uniquely by the two conditions ι^{∗}g_{S} = π^{∗}ǧ_{S} and π_{∗}(ξ^{a} |μ_{S}^{−1}(0)) = ξ̌^{a}.

Theorem D is then used in the next two sections to obtain explicit new families of 3-Sasakian manifolds. First, in section six, we give an explicit description of the Riemannian metric for the Sasakian 3-structure on the coset spaces \frac{U(n)}{U(n−2)×U(1)} and \frac{SO(n)}{SO(n−4)×Sp(1)}. These spaces are obtained from the reduction Theorem D as 3-Sasakian quotients of the unit sphere S^{4n−1} with its canonical metric by standard diagonal actions of U(1) ⊂ U(n) ⊂ Sp(n) and Sp(1) ⊂ Sp(n), respectively. Here the group Sp(n) is the automorphism group of the Sasakian 3-structure on S^{4n−1}. Second, in section seven, we reduce by a general circle subgroup of the maximal torus in Sp(n). In particular, in sections seven and eight, we prove

Theorem E: Let p = (p_{1}, …, p_{n}) ∈ ℤ_{+}^{n} be an n-tuple of pairwise relatively prime positive integers. Let S(p) be the quotient of the complex Stiefel manifold V_{n,2}^{ℂ} by a free circle action depending on p (see equation 7.1). Then S(p) is a compact, simply connected (4n − 5)-dimensional 3-Sasakian manifold and, as rings,

H^{∗}(S(p), ℤ) ≅ (\frac{ℤ[b_{2}]}{[b_{2}^{n} = 0]} ⊗ E[f_{2n−1}])/R(S(p))

where the subscripts on b_{2} and f_{2n−1} denote the cohomological dimension of each generator. The relations R(S(p)) are given by σ_{n−1}(p)b_{2}^{n−1} = 0 and f_{2n−1}b_{2}^{n−1} = 0. Here σ_{n−1}(p) is the (n − 1)^{st} elementary symmetric polynomial in p.

The computation of the integral cohomology ring of S(p) is based on techniques developed by Eschenburg [Esch]. Notice that Theorem E immediately implies that in every dimension of the form (4n−5) for n ≥ 3 there are infinitely many distinct homotopy types of complete 3-Sasakian manifolds. All of them, except in the case when p = (1, …, 1), are inhomogeneous Einstein manifolds. Moreover, some of our examples are not even homotopy equivalent to any compact homogeneous Riemannian space. In section nine, combining a result of Eschenburg and Theorem E we prove

Theorem F: If σ_{2}(p) = p_{1}p_{2} + p_{2}p_{3} + p_{3}p_{1} ≡ 2 (mod 3) then S(p_{1}, p_{2}, p_{3}) is strongly inhomogeneous; that is, it is not homotopy equivalent to any compact Riemannian homogeneous space. In particular, for any odd positive integer c, S(c, c + 1, c + 2) is strongly inhomogeneous. Thus, there exists a countable family of compact, simply connected, strongly inhomogeneous Einstein spaces of positive scalar curvature.

We would like to thank Gerardo Hernandez for introducing us to Sasakian 3-structures. We also thank Jim Milgram and Stephan Stolz for helpful discussions, and the Max-Planck Institute and its director F. Hirzebruch for their hospitality during a visit of the second author.

§1. Orbifolds and Riemannian Foliations

In this section we review some important properties of both orbifolds and related Riemannian foliations. Roughly speaking orbifolds are like differentiable manifolds except that instead of being modelled on ℝ^{n} they locally look like ℝ^{n}/Γ, where Γ is a discrete group of diffeomorphisms of ℝ^{n}. This idea was first introduced by Satake [Sat] and he called these spaces V -manifolds. They also became known as Satake manifolds or orbifolds. We will use the term orbifold which has gained recent acceptance in the literature. Orbifolds appear naturally as the space of leaves of certain nicely behaved Riemannian foliations. In this paper we will not be concerned with the most general type of orbifold nor the most general type of foliation, but rather only with those orbifolds O that arise as the quotient space of a locally free action of a compact Lie group G on a smooth manifold M (in our case G will be either SU(2) or SO(3)). In this case the fundamental vector fields of the action of G on M define a foliation F on M and the space of leaves M/F has the structure of an orbifold. Thus, the smooth manifold M can be viewed as a desingularization of the orbifold M/F. More generally, the leaf space M/F of any Riemannian foliation (M, F, g) with compact leaves is an orbifold. Satake’s original article [Sat] is a good reference for the theory of orbifolds and the books of Molino [Mo] and Reinhart [Rei1] are good references for the theory of Riemannian foliations.

Following Satake [Sat] and Molino [Mo] we define orbifolds and smooth maps between them. Let O be a second countable Hausdorff space, and U ⊂ O an open set. A local uniformizing system (l.u.s.) for U is a triple {Ũ, Γ, π} where Ũ ⊂ ℝ^{n} is an open subset of ℝ^{n}, Γ is a finite group of diffeomorphisms on Ũ, and π : Ũ⟶U is a continuous map satisfying

(i) π ∘ σ = π for all σ ∈ Γ,

(ii) π induces a homeomorphism ϕ : U⟶Ũ/Γ.

The pair (U, ϕ) is called a local chart of O.

In particular, let O = Ũ/Γ and V ⊂ Ũ/Γ any open set. Then {Ṽ , Γ, π} is a local uniformizing system for V with Ṽ = {p ∈ Ũ | π̃(p) ∈ V } and π : Ṽ ⟶V equal to the restriction to Ṽ of the natural projection π̃ : Ũ⟶Ũ/Γ. In this case the local chart ϕ is the restriction of the identity map. Now consider open sets Ũ ⊂ ℝ^{n} and Ũ′ ⊂ ℝ^{m} together with finite groups of diffeomorphisms Γ and Γ′ acting on Ũ and Ũ′, respectively. We say that a continuous map f : Ũ/Γ⟶Ũ′/Γ′ is smooth if f lifts locally to a smooth map, that is, for every point p ∈ U/Γ there are neighborhoods V ⊂ Ũ/Γ of p and V′ ⊂ Ũ′/Γ′ of f(p), local uniformizing systems {Ũ, Γ, π} and {Ũ′, Γ′, π′} for V and V′, respectively, and a smooth map f̃: Ṽ ⟶Ṽ′ such that the diagram

1.1  \begin{matrix}Ṽ & ⟶^{f̃} & Ṽ′ \\ ↓π & & ↓π′ \\ V & ⟶^{f} & V′ \end{matrix}

commutes. The rank of the smooth map f is defined to be the rank of f̃. The notions of orbifold immersions, submersions, diffeomorphisms, etc., are defined in a similar manner.

Notice that a smooth map f : Ũ/Γ⟶Ũ′/Γ′ defines a group homomorphism Γ⟶Γ′ as follows: Let σ ∈ Γ then by the commutativity of diagram 1.1 f̃(σ(p)) lies in the fiber π^{′−1}(f(π(p)) ≃ Γ′. Hence, there is a unique σ′ ∈ Γ′ such that f̃∘ σ = σ′ ∘ f̃. One easily checks that the map σ ↦ σ′ is a homomorphism. If f is a diffeomorphism onto its image f(Ũ/Γ) ⊂ Ũ′/Γ, then the map σ ↦ σ′ is a group monomorphism. In particular, if f̃ covers the identity map f = id then Satake [Sat] calls f̃ an injection.

Definition 1.2: Let O be a second countable Hausdorff space. A smooth orbifold atlas for O is a cover {U_{i}} of O by open sets U_{i} with a local uniformizing system {Ũ_{i}, Γ_{i}, π_{i}} for each U_{i} such that the homeomorphisms ϕ_{i} satisfy the condition that

ϕ_{i} ∘ ϕ_{j}^{−1} : ϕ_{j}(U_{i} ∩ U_{j}) ⟶ ϕ_{i}(U_{i} ∩ U_{j})

is a diffeomorphism for each i, j with U_{i} ∩ U_{j} ≠ ∅. Then O together with a maximal orbifold atlas is called an orbifold.

A point of p ∈ O is regular if p has a neighborhood U that has a local uniformizing system with Γ = id. Otherwise p is called singular. The regular points of O form a dense open set.

Remark 1.3: Satake’s original definition included the requirement that the fixed point set of any of the finite groups Γ_{i} be of codimension at least 2. This restrictive definition is not theoretically convenient at this point. For example, Molino’s theorem stated below (see 1.8) does not hold for Satake’s less general definition of orbifold. However, the orbifolds that we construct in this paper do have singular sets of codimension greater than or equal two.

Satake then generalizes such standard notions of differential geometry as bundles, differential forms, Riemannian metrics, etc., to the orbifold category (see [Sat] for further details). As mentioned above our interest in orbifolds stems from the fact that they occur naturally as the leaf space of certain Riemannian foliations.

Recall that a foliation F on a manifold M is given by an integrable subbundle V of the tangent bundle T M, that is, a subbundle whose smooth sections form a Lie subalgebra of the Lie algebra of smooth vector fields on M. There is an exact sequence of vector bundles

1.4  0 ⟶ V ⟶ T M ⟶ T M/V ⟶ 0.

A Riemannian metric g on M splits this exact sequence to yield T M = V ⊕ H defining the horizontal subbundle H. The integrable subbundle V is called the vertical subbundle. In general, and in our case in particular, the horizontal subbundle H is not integrable. A Riemannian manifold (M, g) together with a foliation F on M is called a foliated Riemannian manifold and is denoted by (M, g, F). The following definition is due to Reinhardt.

Definition 1.5: [Rei2] Let (M, F, g) be a foliated Riemannian manifold. The metric g is said to be bundle-like if for any horizontal vector fields X, Y in the normalizer of V under the Lie bracket, the equation V g(X, Y) = 0 holds for any vertical vector field V.

Remarks 1.6:

1. Vector fields belonging to the normalizer of V are often called foliate [Mo].

2. Functions that are annihilated by all vertical vector fields, as in Definition 1.5 above, are known as basic [Mo].

3. Definition 1.5 is equivalent to the condition that the horizontal distribution H be totally geodesic (see [Mo] or [Rei1] for details).

We shall make use of the following lemma:

Lemma 1.7: Let (M, g, F) be a foliated Riemannian manifold, and suppose that the vertical distribution V is spanned by Killing vector fields. Then g is bundle-like.

Proof: It is enough to show that the condition in definition 1.5 holds when V is any Killing vector field. Let X, Y be horizontal vector fields on M belonging to the normalizer of V, then we have V g(X, Y) = (L_{V} g)(X, Y) + g([V, X], Y) + g(X, [V, Y]) = 0, where L_{V} denotes the Lie derivative with respect to V. The first term vanishes since V is a Killing vector field. The two remaining terms vanish since X and Y are horizontal and the terms in brackets are vertical.

The following result, which is given in Molino, is fundamental to our work:

Theorem 1.8: [Mo: Proposition 3.7] Let (M, F, g) be a Riemannian foliation of codimension q with compact leaves and bundle-like metric g. Then the space of leaves M/F admits the structure of a q dimensional orbifold such that the natural projection π : M⟶M/F is an orbifold submersion.

Another important concept in the theory of foliations is that of the leaf holonomy group. This group is a certain image of the fundamental group of a leaf in the local group of germs of diffeomorphisms of a transverse submanifold to the leaf. It measures how transversals change as one moves along a loop in the leaf. In our situation we have the following:

Proposition 1.9: Let (M, F, g) be a Riemannian foliation with compact leaves and bundle like metric as in theorem 1.8. The dense open set (M/F)^{o} of regular points of M/F is precisely the set of leaves with trivial holonomy and there is a unique Riemannian metric ǧ on (M/F)^{o} such that the natural projection π : M⟶M/F restricts to a locally trivial Riemannian submersion on π^{−1}((M/F)^{o}). At a singular point of M/F the finite group Γ is precisely the holonomy of the leaf. In particular, if there are no singular points, the projection π : M⟶M/F is a locally trivial Riemannian fibration.

Proof: Except for the local triviality statement, this follows from Molino [Mo: §3.6]. The local triviality is a consequence of the Ehresmann fibration theorem.

Theorem 1.8 and Proposition 1.9 allow one to talk about the quotient M/F as a Riemmanian orbifold. The metric ǧ is defined only on the dense open set (M/F)^{o}; however, the transverse part g_{T} of the metric g is a metric on the horizontal distribution H which satisfies π^{∗}ǧ = g_{T} on the points of π^{−1}((M/F)^{o}). Thus, g_{T} can be interpreted as describing the metric on the whole orbifold including its singular locus. Accordingly all of O’Neill’s standard formulae for Riemannian submersions hold for these more general orbifold Riemannian submersions (see [Rei1: page 160]). Hence, we shall freely apply these well-known formulae to this more general setting.

§2. Some Old and New Results on 3-Sasakian Manifolds

In this section we review some known results about Riemanian manifolds which admit 3-Sasakian structures and then give an important generalization. Following Ishihara and Konishi [IKon], we begin by recalling the definitions of Sasakian and 3-Sasakian structures on a Riemannian manifold.

Definition 2.1: Let (S, g) be a Riemannian manifold and let ∇ denote the Levi-Civita connection of g. Then (S, g) has a Sasakian structure if there exists a Killing vector field ξ of unit length on S so that the tensor field Φ of type (1, 1), defined by Φ = ∇ξ satisfies the condition (∇_{X}Φ)(Y) = η(Y)X − g(X, Y)ξ for any pair of vector fields X and Y on S. Here η denotes the 1-form dual to ξ with respect to g, i.e. g(Y, ξ) = η(Y) for any vector field Y, and satisfies the dual equation (∇_{X}η)(Y) = g(Y, ΦX). We write (Φ, ξ, η) to denote the specific Sasakian structure on (S, g) and refer to S with such a structure as a Sasakian manifold.

It is straightforward to verify that the following equations hold.

Proposition 2.2: Let (S, g, ξ) be a Sasakian manifold and X and Y any pair of vector fields on S. Then Φ∘Φ(Y) = −Y + η(Y)ξ, Φξ = 0, and η(ΦY) = 0. Moreover, g(X, ΦY) + g(ΦX, Y) = 0, g(ΦY, ΦZ) = g(Y, Z) − η(Y)η(Z), and dη(Y, Z) = 2g(ΦY, Z). Furthermore, the Nijenhuis torsion tensor

N_{Φ}(Y, Z) = [ΦY, ΦZ] + Φ^{2}[Y, Z] − Φ[Y, ΦZ] − Φ[ΦY, Z]

of Φ satisfies N_{Φ}(Y, Z) = −dη(Y, Z) ⊗ ξ.

We now define our main objects of interest.

Definition 2.3: Let (S, g) be a Riemannian manifold that admits three distinct Sasakian structures {Φ^{a}, ξ^{a}, η^{a}}_{a=1,2,3} such that g(ξ^{a}, ξ^{b}) = δ^{ab} and [ξ^{a}, ξ^{b}] = 2ϵ^{abc}ξ^{c} for a, b, c = 1, 2, 3. Then (S, g) is a 3-Sasakian manifold with Sasakian 3-structure (S, g, ξ^{a}).

It follows directly from the definition that every 3-Sasakian manifold admits a local action of either Sp(1) or SO(3) as local isometries, and if the vector fields ξ^{a} are complete these are global isometries. We refer to this action as the standard Sp(1) action on S. In the remainder of this paper we shall assume that the vector fields ξ^{a} are complete. This structure has several important implications. First, it is not difficult to verify that the following relations between the Sasakian structures hold:

2.4  η^{a}(ξ^{b}) = δ^{ab}, Φ^{a}ξ^{b} = −ϵ^{abc}ξ^{c}, Φ^{a} ∘ Φ^{b} − ξ^{a} ⊗ η^{b} = −ϵ^{abc}Φ^{c} − δ^{ab}id.

The following result is well-known:

Theorem 2.5: Every 3-Sasakian manifold (S, g, ξ^{a}) has dimension 4n + 3 and defines a Riemannian foliation (S, F) of codimension 4n with totally geodesic leaves of constant curvature 1. Furthermore (S, g, ξ^{a}) is an Einstein manifold.

The first result is due to Kuo and Tachibana [KuTach] and the second is due to Kashiwada [Ka]. For more general almost contact 3-structures Kuo [Ku] has proven:

Theorem 2.6: [Ku] The structure group of any manifold with an almost contact 3structure is reducible to Sp(n) × 𝕀_{3} where 𝕀_{3} denotes the three by three identity matrix.

Corollary 2.7: Every 3-Sasakian manifold is a spin manifold.

In addition, using harmonic theory on compact Sasakian manifolds, Kuo [Ku] has also shown that on a compact (4n + 3)-dimensional 3-Sasakian manifold S the i^{th} Betti number, b_{i}(S), must be of the form 4q whenever i is odd and i < 2n + 2.

Much of the previous work on 3-Sasakian manifolds has concentrated on the regular case when (S, g, ξ^{a}) is the total space of a Riemannian submersion [IKon,I2]. The following is a theorem of Ishihara.

Theorem 2.8: [I2] Let (S, g, ξ^{a}) be a 3-Sasakian manifold such that the space of leaves S/F is a Riemannian manifold and the natural projection π : (S, g)⟶(S/F, ǧ) is a Riemannian submersion. Then (S/F, ǧ) is a quaternionic Kähler manifold.

A converse of this theorem was obtained by Konishi.

Theorem 2.9: [Kon] Let (M, ǧ) be a quaternionic Kähler manifold with positive scalar curvature. Then there is a principal SO(3) bundle S over M whose total space admits a metric g with an associated 3-Sasakian structure.

Konishi also considers the case when the quaternionic Kähler manifold has negative scalar curvature. This gives a Sasakian 3-structure on S with indefinite signature (3, 4n). We shall not consider this case. In the positive scalar curvature case, there is an obstruction to lifting the SO(3) bundle to an SU(2) bundle. This obstruction is the Marchiafava-Romani class ϵ [MaR] of the quaternionic Kähler manifold M, and a result of Salamon [Sal1] says that if the quaternionic Kähler manifold is complete with positive scalar curvature then ϵ vanishes if and only if M = ℍℙ^{n}. This result does not hold in the case of quaternionic Kähler orbifolds, nor if the completeness assumption is dropped. In Proposition 7.15 below we give a class of 3-Sasakian manifolds each of which fibers, in the orbifold sense, over a quaternionic Kähler orbifold with the generic fibre equal to Sp(1). These classical results are generalized by Theorem A stated in the introduction.

Proof of Theorem A: The last statement is a direct consequence of the first statement and Myers’ theorem. Next we prove the first statement. Since the vector fields ξ^{a} are Killing vector fields, the metric g is bundle-like by lemma 1.7. Furthermore, since these vector fields are complete the foliation F has compact leaves. So, by Molino’s theorem 1.8, the space of leaves S/F is an orbifold O of dimension 4n, and by proposition 1.9 the natural projection π : S⟶S/F is a Riemannian submersion on the dense open set of regular leaves. The fact that the space of leaves S/F has a quaternionic Kähler structure follows from Ishihara’s theorem 2.8 applied to the dense open set of leaves. Now the dimension of each leaf is three, and all leaves are totally geodesic of constant curvature 1 by 2.5. In particular, O’Neill’s tensor field T vanishes, and so for each a = 1, 2, 3 the tensor field Φ^{a} restricted to any leaf L defines a Sasakian structure there. Hence, (g, Φ^{a}, ξ^{a})_{a=1,2,3} restricted to L makes L a 3-Sasakian manifold of dimension 3. These were classified by Sasaki [Sas], and it follows that each leaf is a 3-dimensional homogeneous spherical space form with scalar curvature 6. This proves everything except for the last statement about the scalar curvature.

To compute the scalar curvature we first determine the Einstein constant λ of g. Using proposition 1.9 we can apply [Bes: 9.62] to the dense open set of regular points

to give λ = \frac{1}{3}(6+ |A|^{2}), where A is O’Neill’s tensor. In particular, λ is positive and it remains to compute the O’Neill tensor A to determine the scalar curvature explicitly. Again proposition 1.9 shows that π : S⟶O is a locally trivial orbifold bundle. The generic fibres have the form SU(2)/Γ where Γ is a discrete subgroup of SU(2). In the case that Γ = id or ℤ_{2}, S is a principal orbifold bundle with group SU(2) or SO(3), respectively.

Otherwise S is an associated bundle. In either case we show that the three 1-forms η^{a} for a = 1, 2, 3 define the components of a connection in the orbifold bundle S, and that we can compute the tensor field A from the curvature of this connection. We have

Lemma 2.10: The three 1-forms η^{a} with a = 1, 2, 3 are the components of a connection 1-form in the orbifold bundle S.

Proof: Choosing a basis e_{a} where a = 1, 2, 3 for the Lie algebra 𝔰𝔭(1) and defining η = η^{a}e_{a} we obtain a Lie algebra valued 1-form which annihilates the distribution H orthogonal to V with respect to the metric g. To see that the horizontal distribution H is equivariant and thus defines a connection on S with connection form η, we use Proposition 2.2 to compute

0 = 2g(Φ^{a}ξ^{b}, X) = dη^{a}(ξ^{b}, X) = ξ^{b}η^{a}(X) − Xη^{a}(ξ^{b}) − η^{a}([ξ^{b}, X]) = η^{a}([ξ^{b}, X]).

Here X is a horizontal vector field and 1 ≤ a, b ≤ 3. But this implies that [ξ^{a}, X] is horizontal for all a = 1, 2, 3.

Lemma 2.11: For any pair of horizontal vector fields X and Y on S

(i) A_{X}Y = ∑_{a=1}^{3} g(Φ^{a}X, Y)ξ^{a}.

(ii) A_{X}ξ^{a} = Φ^{a}X.

Proof: (i) follows from Proposition 2.2 and the fact that if Ω is the curvature two form

of the principal connection η, then [Bes: 9.54] shows that A_{X}Y = −\frac{1}{2}θ^{−1}Ω(X, Y), where θ : V⟶𝔰𝔭(1) denotes the isomorphism between the vertical vector space V at a point of S and the Lie algebra 𝔰𝔭(1). (ii) now follows from equation (i) and [Bes: 9.21d].

Returning to computation of the scalar curvature in the proof of Theorem A, we let X_{i} for 1 ≤ i ≤ 4n denote a local orthonormal basis of the horizontal distribution H, and compute using lemma 2.11, Proposition 2.2, and [Bes: 9.33a], viz.

|A|^{2} = ∑_{i=1}^{4n} g(A_{X_{i}}, A_{X_{i}}) = ∑_{i=1}^{4n} ∑_{a=1}^{3} g(A_{X_{i}}ξ^{a}, A_{X_{i}}ξ^{a}) = ∑_{i=1}^{4n} ∑_{a=1}^{3} g(Φ^{a}X_{i}, Φ^{a}X_{i}) = 12n.

Substituting this equation into the expression for the Einstein constant determines the scalar curvature and establishes Theorem A part (iv).

Remark 2.12: The homogeneous spherical space forms in dimension 3 are well known. They are Sp(1)/Γ where Γ is either the trivial group, the cyclic group of order m, a binary dihedral group with m is an integer greater than 2, the binary tetrahedral group, the binary octahedral group, or the binary icosahedral group. Thus, Theorem A part (iii) shows that every leaf of any 3-Sasakian manifold is of this form.

An interesting corollary of Theorem A and a theorem of Bérard-Bergery [BéBer], which rescales the metric along the fibres, is

Corollary 2.13: Every 3-Sasakian manifold has two distinct Einstein metrics of positive scalar curvature. The first is given in Theorem A part (i) and the second Einstein metric

has scalar curvature 2(2n + 1)(4n + 9) − \frac{12n}{2n + 3}. By distinct here we mean nonhomothetic.

Proof: The first statement follows from theorem A and [Bes: 9.73]. It only needs to be checked that the connection 1-form η defined by Lemma 2.10 is a Yang-Mills connection. In fact, one can show directly that this connection 1-form η is anti-self-dual in the sense of [GPo] and [M-CS]. To compute the scalar curvature for the second Einstein metric we use [Bes: 9.70d] and [Bes: 9.74].

Remark 2.14: Given any Einstein metric we can easily obtain a one parameter family of Einstein metrics by scaling the metric. The scale factor, however, for the 3-Sasakian metric g is fixed by the 3-Sasakian structure. This is not the case for the second Einstein metric. Perhaps a more meaningful invariant for the second Einstein metric is not its scalar curvature, but the ratio of the scalar curvature of the second (non 3-Sasakian) Einstein metric to the scalar curvature of the first (3-Sasakian) Einstein metric. This ratio is given

by 1 + \frac{6(n + 1)}{(2n + 3)(2n + 1)}.

Now let O be any quaternionic Kähler orbifold of positive scalar curvature. In general Konishi’s principal SO(3) bundle over the dense open set of regular points of O extends to an orbifold bundle over O whose total space is an orbifold, but not necessarily a smooth manifold. We will say that the quaternionic Kähler orbifold is a good orbifold if the total space S of the principal SO(3) bundle over O is a smooth manifold. We have the following corollary of our main theorem.

Corollary 2.15: There is a one-to-one correspondence (up to covering) between simply connected 3-Sasakian manifolds of dimension 4n+3 and good quaternionic Kähler orbifolds of dimension 4n with positive scalar curvature equal to 16n(n + 2).

Remark 2.16: A complete quaternionic Kähler manifold M of positive scalar curvature is necessarily simply connected [Sal1]. This follows from a theorem of Kobayashi [Kob] which says that any complete Kähler manifold with positive definite Ricci curvature is simply connected, and a theorem of Salamon [Sal1] saying that the twistor space of a quaternionic Kähler manifold with positive scalar curvature is Kähler-Einstein with positive scalar curvature. It would be interesting to see whether this result generalizes to the case of quaternionic Kähler orbifolds.

Finally, we give some general results concerning the curvature of any 3-Sasakian manifold. Since the curvature of any Riemannian manifold is completely determined by its sectional curvature and the sectional curvature of any Sasakian manifold [YK] is completely determined by the Φ-sectional curvature, we essentially give the latter.

Proposition 2.17: Let (S, g, ξ^{a}) be a 3-Sasakian manifold and let K and Ǩ denote the sectional curvatures of g and its transverse component ǧ, respectively. Then if X is any horizontal vector field of unit length on S, we have

(i) K(ξ^{a}, ξ^{b}) = 1 where a + 1 ≡ b (mod 3).

(ii)) K(X, ξ^{a}) = 1.

(iii) K(X, Φ^{a}X) = Ǩ(X, Φ^{a}X) − 3.

Proof: (i) follows from theorem 2.5, but is also easy to compute directly. Next, notice that proposition 1.9 implies that we can use [Bes: 9.29] applied to the dense open set of regular points of S/F and that the equations in proposition 2.2 show that if X is horizontal of unit length, then the set {X, Φ^{1}X, Φ^{2}X, Φ^{3}X} is an orthonormal 4-frame. Thus, (ii) follows from [Bes: 9.29b] and part (ii) of lemma 2.11. Finally part (i) of lemma 2.11 implies A_{X}Φ^{a}X = ξ^{a} and thus (iii) follows from this fact and [Bes: 9.29c].

§3. An Embedding Theorem for 3-Sasakian Manifolds

In [BGM1] we showed how certain 3-Sasakian manifolds naturally arise as the level sets of hyperkähler manifolds with certain additional properties. In this section we prove a converse to this result by embedding every 3-Sasakian manifold (S, g, ξ^{a}) in a hyperkähler manifold. To begin we recall

Theorem 3.1: [BGM1] Let G be either Sp(1) or SO(3) and let M be a hyperkähler manifold admitting a locally free isometric action of G permuting the complex structures on M. Then there is an Sp(1) invariant function ν and an obstruction section ϕ of the fourth order symmetric product of the spin bundle S^{4}H on M. If this obstruction section ϕ is constant on M then each level set of ν admits a 3-Sasakian structure.

To prove Theorem B of the introduction first notice that the Cartesian product manifold S × ℝ^{+} has a natural Sp(1) action defined to be the standard Sp(1) action on S and the trivial action on ℝ^{+}.

Proof of Theorem B: We construct a hyperkähler structure on M = S × ℝ^{+} in such a way that the obstruction section ϕ associated to the natural Sp(1) action is constant. To begin let Ψ = r \frac{∂}{∂r} denote the Euler field on M and for each a = 1, 2, 3 define smooth sections I^{a} of End T M by the formulae

3.2  I^{a}Y = −Φ^{a}Y + η^{a}(Y)Ψ and I^{a}Ψ = −ξ^{a}.

Here Y is any vector field on M that is tangent to S. The action of Sp(1) on M = S × ℝ^{+} extends the ξ^{a} to vector fields on M and, by abuse of notation, we let η^{a} denote the 1-forms η^{a} on S pulled back to M. Now Kuo [Ku] shows that the I^{a}’s form an almost quaternionic structure on M, and a standard computation shows that for any vector fields X, Y on M we have g_{M}(I^{a}X, I^{a}Y) = g_{M}(X, Y). Hence, g_{M} is almost hyperhermitian.

To prove that (M, g_{M}) is hyperkähler we show that the complex structures I^{a} are parallel, i.e., that ∇I^{a} = 0. This then implies that the almost complex structures I^{a} are integrable and that Hermitian structure (M, g_{M}, I^{a}) is Kähler for each a = 1, 2, 3. We begin by computing the second fundamental form of the embedding S ↪ M obtained as the level set r = 1. Actually, it is just as easy to compute the second fundamental form for the family of embeddings determined by arbitrary r.

Lemma 3.3: Let S be a Riemannian manifold of dimension n, and M = S × ℝ^{+} the cone on S with cone metric given above. Then the second fundamental form s of the embedding S ↪ M as the level set for any fixed nonzero r is given by s(X, Y) = −g_{S}(X, Y)Ψ. Hence, the embedding is totally umbilical.

Proof: Let {θ^{i}} denote a local orthonormal coframe for the metric g_{S} on S, then we obtain a local orthonormal coframe {ϕ^{μ}} for the cone metric g_{M} on M by settingϕ^{i} = rθ^{i} and ϕ^{0} = dr where 1 ≤ i ≤ n + 1. The first Cartan structure equations for M are dϕ^{μ}+ω_{ν}^{μ}∧ϕ^{ν} = 0. Here ω_{ν}^{μ} denotes the connection 1-forms with respect to the Levi-Civita connection ∇^{M} on M, where 0 ≤ μ, ν, ≤ n + 1. Separating out the structure equations on S one sees that ω_{0}^{i} = θ^{i}. This fact and Gauss’ formula prove the lemma.

Returning to the proof of theorem B, we next show that ∇^{M}I^{a} = 0. First let X and Y be vector fields on M that are both tangent to S. Then, using equation 3.2, Gauss’ formula, and lemma 3.3, we obtain

3.4 ∇_{X}^{M}I^{a}(Y) = −∇_{X}^{S}(Φ^{a}Y) + g_{S}(X, Φ^{a}Y)Ψ + Xη^{a}(Y)Ψ + η^{a}(Y)∇_{X}^{M}Ψ + Φ^{a}(∇_{X}^{S}Y) − η^{a}(∇_{X}^{S}Y)Ψ + g_{S}(X, Y)I^{a}Ψ.

Weingarten’s equation and lemma 3.3 imply that ∇_{X}^{M}Ψ = X. Thus, using 2.1 and 2.2, one checks that equation 3.4 becomes

[X(η^{a}(Y)) − (∇_{X}^{S}η^{a})(Y) − η^{a}(∇_{X}^{S}Y)]Ψ + [−(∇_{X}^{S}Φ^{a})(Y) + η^{a}(Y)X − g_{S}(X, Y)ξ^{a}].

Clearly, the first term in brackets vanishes, and 2.1 implies that the second term in brackets also vanishes. This shows that I^{a} is parallel when X and Y are both tangent to S. Similar computations show that (∇_{X}^{M}I^{a})(Ψ) = 0. Finally, we note from the proof of lemma 3.3 that the connection 1-forms ω_{ν}^{μ} have no dr component. This implies that ∇_{Ψ}^{M}Y = 0 for any vector field Y on M, and hence, that ∇_{Ψ}^{M}I^{a} = 0. This completes the proof that (M, g_{M}) is hyperkähler.

To compute the obstruction section ϕ defined in [BGM1] notice that equations 2.4 and 3.2 imply that I^{a}ξ^{b} = ϵ^{abc}ξ^{c} + δ^{ab}Ψ. Comparing this with equation 2.16 of [BGM1] does indeed show that the obstruction section ϕ is constant. This completes the proof of Theorem B.

Remarks 3.5:

1. S × ℝ^{+} is not complete with respect to metric g_{M} and cannot be completed by filling in the cone point unless S = S^{4n+3} with its standard Sasakian 3-structure. This follows from a result of Yano as pointed out in [BGM1].

2. The proof given here that I^{a} is parallel with respect to the Levi-Civita connection ∇^{M} shows that any Sasakian manifold embeds into a Kähler manifold with a cone metric. This is a previously known result of Tashiro [Tas].

3. Hernandez [Her] has shown that the embedding of S in M given in Theorem B provides S with the structure of an extrinsic sphere in the sense of Nomizu. Hernandez also shows that this embedding is the only way a 3-Sasakian manifold can embed as a hypersurface in a hyperkähler manifold.

4. Using the second Cartan structure equations it is easy to show that any cone metric g_{M} is Einstein if and only if g_{S} is Einstein. In particular, the second Einstein metric on our 3-Sasakian manifold S gives an Einstein metric on M with positive scalar curvature. Of course, the 3-Sasakian Einstein metric on S induces a Ricci flat metric on M, as it must since M is hyperkähler.

5. Both [FK] and [Bär] use Killing spinors to study 3-Sasakian manifolds with the later paper considering the relation to hyperkähler geometry.

Finally, we can use Theorem B to give a generalization of the standard Hopf surface construction which we can then use to construct many new compact hypercomplex manifolds. Consider the manifold S × S^{1} obtained from S × ℝ^{+} as the quotient by the multiplicative action of ℤ on ℝ^{+} generated by r ↦ ar where a ≠ 1 is a fixed positive real number.

Corollary 3.6: Let S be a complete 3-Sasakian manifold, then the manifold S × S^{1} constructed above has a naturally induced hypercomplex structure. In fact, the product metric is locally conformally hyperkähler.

Proof: The Euler vector field Ψ passes to the quotient manifold and generates the standard circle action on S^{1}. Thus, it follows from equation 3.2 that tensor fields I^{a} pass to the quotient and define an almost hypercomplex structure on S × S^{1}. Moreover, the proof of Theorem B implies that the hypercomplex structure is integrable. Setting r = e^{u} we see that the product metric du^{2} + g_{S} is conformally equivalent to the cone metric restricted to an open set in the fundamental domain of the multiplicative action given above.

Combining Corollary 3.6 with the results of sections 4 and 6 give explicit examples of homogeneous hypercomplex manifolds while combining Corollary 3.6 with the results of section 7 give, for each n ≥ 2, infinitely many homotopically distinct, non-homogeneous, 4n-dimensional hypercomplex manifolds (4-dimensional compact manifolds that admit a hypercomplex structure were classified in [Boy]). Of course, these manifolds described by Corollary 3.6 are not simply connected. However, Joyce [Joy] noticed that by twisting an associated space with a certain circle bundle one can obtain simply connected hypercomplex manifolds. Thus, twisting the manifolds constructed in sections 4 and 6 give explicit examples of simply connected, homogeneous hypercomplex manifolds. Moreover, Joyce’s twisting construction generalizes to the orbifold category to give, together with the results of section 7, non-homogeneous, simply connected, hypercomplex manifolds in dimension 4n for n ≥ 2. These manifolds are analyzed in [BGM4].

§4. The Classification of 3-Sasakian Homogeneous Manifolds.

In this section we classify 3-Sasakian homogeneous spaces. To begin notice that the Killing vector fields ξ^{1}, ξ^{2}, and ξ^{3} which give a Riemannian manifold (S, g) a Sasakian 3-structure generate non-trivial isometries. Thus, every 3-Sasakian manifold (S, g, ξ^{a}) has a non-trivial isometry group and we denote the connected component of the identity by I(S, g). Let I_{0}(S, g) denote the subgroup of I(S, g) consisting of those isometries that leave the tensor fields Φ^{a} invariant for all a = 1, 2, 3. We refer to elements of I_{0}(S, g) as 3-Sasakian isometries. The following theorem was proven by Tanno.

Theorem 4.1: [Tan1] Let (S, g, ξ^{a}) be a complete 3-Sasakian manifold which is not of constant curvature. Then dim I(S, g) = dim I_{0}(S, g) + 3.

Furthermore, the Killing vector fields ξ^{a} generate the three dimensional subspace of isometries that are not 3-Sasakian isometries. Let 𝔦 and 𝔦_{0} denote the Lie algebras of I(S, g) and I_{0}(S, g), respectively. Hence, Tanno’s theorem says that if (S, g) is not of constant curvature, then 𝔦 = 𝔦_{0} + 𝔰𝔭(1), where + indicates vector space direct sum. However, more is true, namely

Lemma 4.2: This direct sum is a direct sum of Lie algebras, i.e., 𝔦 = 𝔦_{0} ⊕ 𝔰𝔭(1).

This lemma is implicit in Tanno’s work, although he never stated it explicitly there. It follows immediately from

Lemma 4.3: Let (S, g, ξ^{a}) be a 3-Sasakian manifold and X ∈ 𝔦 be a Killing vector field on S. Let L_{X} denote the Lie derivative with respect to X. Then, for a = 1, 2, 3, the following conditions are equivalent

(i) L_{X}Φ^{a} = 0,

(ii) L_{X}η^{a} = 0,

(iii) L_{X}ξ^{a} = 0.

Furthermore, if any (hence, all) of the conditions above is satisfied, then for any vector field Y on S we have

(iv) Xη^{a}(Y) = η^{a}([X, Y]).

Proof: Let X ∈ 𝔦, then it follows from 2.2 and the definition of the Nijenhuis tensor that the following three conditions are equivalent L_{X}Φ^{a} = 0, L_{X}N_{Φ^{a}} = 0, and L_{X}dη^{a} = 0, and so X ∈ 𝔦_{0} if any one of the these conditions is satisfied. Thus, 2.2 implies that

0 = L_{X}N_{Φ^{a}} = L_{X}(dη^{a} ⊗ ξ^{a}) = (L_{X}dη^{a}) ⊗ ξ^{a} + dη^{a} ⊗ (L_{X}ξ^{a}) = dη^{a} ⊗ (L_{X}ξ^{a}).

This shows that (i) implies (iii). But since η^{a} is dual to ξ^{a} through the metric g and X is a Killing vector field, (iii) holds if and only if (ii) holds. Next we show that (iii) implies (i): Since any infinitesimal isometry is an infinitesimal affine transformation with respect to the Levi-Civita connection, we have for any vector field Y

L_{X}(Φ^{a}Y) = L_{X}∇_{Y} ξ^{a} = ∇_{Y} L_{X}ξ^{a} + ∇_{[X,Y]}ξ^{a} = Φ^{a}[X, Y].

But the left hand side is (L_{X}Φ^{a})(Y) + Φ^{a}[X, Y] which proves (i). Finally (iv) follows easily from (ii).

Notice that any of the first three conditions in Lemma 4.3 can be used to describe the Lie subalgebra 𝔦_{0} ∈ 𝔦. Moreover, the equivalence of conditions (iii) and (i) says that the Lie algebra 𝔠(𝔰𝔭(1)) of the centralizer of Sp(1) in I(S, g) is precisely 𝔦_{0}. Globally, on the group level we have:

Proposition 4.4: Let (S, g, ξ^{a}) be a complete 3-Sasakian manifold. Then both the isometry groups I(S, g) and I_{0}(S, g) are compact. Furthermore, if (S, g, ξ^{a}) is not of constant curvature then either I(S, g) = I_{0}(S, g) × Sp(1) or I(S, g) = I_{0}(S, g) × SO(3). Finally, if (S, g, ξ^{a}) does have constant curvature then I(S, g) strictly contains either I_{0}(S, g)×Sp(1) or I_{0}(S, g) × SO(3) as a proper subgroup and I_{0}(S, g) is the centralizer of Sp(1) or SO(3).

Proof: The first assertion follows from Theorem A and a standard result of Myers and Steenrod (cf. [Bes]). Next, since I_{0}(S, g), Sp(1), and SO(3) are all compact, the direct sum on the Lie algebra level given in lemma 4.2 also gives a direct product of Lie groups. The last assertion follows immediately from lemma 4.3

We are particularly interested in the case of a transitive isometry group.

Definition 4.5: A 3-Sasakian homogeneous space is a 3-Sasakian manifold (S, g, ξ^{a}) on which I_{0}(S, g) acts transitively.

Proposition 4.6: Let (S, g, ξ^{a}) be a 3-Sasakian homogeneous space. Then all leaves are diffeomorphic and S/F is a quaternionic Kähler manifold where the natural projection π : S⟶S/F is a locally trivial Riemannian fibration. Furthermore, I_{0}(S, g) acts transitively on the space of leaves S/F.

Proof: Let ψ : I_{0}(S, g) × S⟶S denote the action map so that, for each a ∈ I_{0}(S, g), ψ_{a} = ψ(a, ·) is a diffeomorphism of S to itself. Proposition 4.3 implies that the isometry group I(S, g) contains I_{0}(S, g)×Sp(1) where either Sp(1) acts effectively or its ℤ_{2} quotient SO(3) ≃ Sp(1)/ℤ_{2} acts effectively. Since the Killing vector fields ξ^{a} for a = 1, 2, 3 are both the infinitesimal generators of the group Sp(1) and a basis for the vertical distribution V, it follows that Sp(1) acts transitively on each leaf with isotropy subgroup of a point some finite subgroup Γ ⊂ Sp(1). Now let p_{1} and p_{2} be any two points of S and let L_{1} and L_{2} denote the corresponding leaves through p_{1} and p_{2}, respectively. Since I_{0}(S, g) acts transitively on S, there exists an a ∈ I_{0}(S, g) such that ψ_{a}(p_{1}) = p_{2}. Now ψ_{a} restricted to L_{1} maps L_{1} diffeomorphically onto its image, and, since the Sp(1) factor acts transitively on each leaf and commutes with I_{0}(S, g), the image of ψ_{a} lies in L_{2}. But the same holds for the inverse map ψ_{a^{−1}} with L_{1} and L_{2} interchanged, so the leaves must be diffeomorphic. Thus, the leaf holonomy is trivial and π : S⟶S/F is a locally trivial Riemannian fibration by proposition 1.9. The fact that the space of leaves S/F is a quaternionic Kähler manifold now follows from Ishihara’s theorem 2.8. Finally, the constructions above shows directly that I_{0}(S, g) acts transitively on S/F.

The following proposition is now immediate from Proposition 4.5 and Theorem A.

Proposition 4.7: Let (S, g, ξ^{a}) be a 3-Sasakian homogeneous space. Then S is the total space of a locally trivial Riemannian fibration over a quaternionic Kähler homogeneous space M of positive scalar curvature (i.e., a Wolf space) with fibre F = Sp(1)/Γ where Γ is one of the finite subgroups of Sp(1) (cf. Remark 2.12).

While this proposition enumerates a complete list of possibilities for all the 3-Sasakian homogeneous spaces Theorem C of the introduction shows that not all of them actually arise.

Proof of Theorem C: If S is a 3-Sasakian homogeneous manifold then each fibre must be a 3-Sasakian homogeneous 3-manifold. But the fibres are all of the form Sp(1)/Γ where Γ is a finite subgroup of Sp(1). These space forms are both homogeneous and 3-Sasakian [Sas]; however, they are not 3-Sasakian homogeneous unless Γ = id or ℤ_{2}. To see this notice that there are two equivalent Sasakian 3-structures on Sp(1) ≃ S^{3} both with the constant curvature bi-invariant metric. One Sasakian 3-structure is obtained from the right invariant vector fields on Sp(1) while the other structure comes from the left invariant vector fields. Consider the right invariant structure. Then to obtain a compatible 3-Sasakian homogeneous structure Γ must act on Sp(1) from the left. But if Γ is neither the identity subgroup nor ℤ_{2} then Γ is not in the center of Sp(1). Hence, the centralizer of Γ in Sp(1) is a proper subgroup of Sp(1). So its dimension is less than two, and thus cannot act transitively on Sp(1). This proves that the fibre is either Sp(1) or Sp(1)/ℤ_{2} ≃ SO(3).

It follow from this fact and proposition 4.6 that S is a principal Sp(1) or SO(3) bundle over a Wolf space W. The Wolf spaces are well known [Wo] to have the form G/L_{1} · Sp(1) where G is a simple compact Lie group and L_{1} (Wolf’s notation) is a certain subgroup of G. Wolf showed that each homogeneous quaternionic Kähler manifold W = G/L_{1} · Sp(1) is the base space of an S^{2} bundle whose total space is one of the homogeneous complex contact manifolds Z ≃ G/L_{1} · S^{1} (since identified as the twistor space of W) which were classified by Boothby [Boo]. Let 𝔤^{ℂ} denote the complexification of the Lie algebra 𝔤 of G. Swann [Sw] has identified the total space of the dual of the contact line bundle on Z with the highest root nilpotent adjoint orbit N in 𝔤^{ℂ}. The nilpotent orbits N are well known [Kro] to have a hyperkähler structure and Swann has further identified N with his ℍ^{∗}/ℤ_{2} bundle U(W). It follows from [BGM1: Proposition 4.21] (see also theorem 3.1) that the level set ν^{−1}(1/2) of the hyperkähler potential has a Sasakian 3-structure. This level set is easily identified with S ≃ G/L_{1}. It is a principal SO(3) bundle over W and a principal S^{1} bundle over Z. Furthermore, as explained in the remark about the Marchiafava-Romani class made after Theorem 2.9 the only time that this SO(3) bundle lifts to a Sp(1) bundle is when the base space is ℍℙ^{n−1}. The theorem now follows from the classification of Wolf spaces [Wo] (cf. [Bes: pg. 409]) or the classification of homogeneous complex contact manifolds [Boo].

Using a similar result for Wolf spaces or homogeneous complex contact manifolds we have the following immediate corollary.

Corollary 4.8: There is a one-to-one correspondence between the simple Lie algebras and the simply connected 3-Sasakian homogeneous manifolds.

As mentioned above Sasaki [Sas], classified 3 dimensional 3-Sasakian manifolds. They are precisely the homogeneous spherical space forms Sp(1)/Γ where the finite subgroups Γ are listed in remark 2.12. However, as we have just seen they are not 3-Sasakian homogeneous manifolds unless Γ = id or ℤ_{2}. Sasaki asked the natural question: Which spherical space forms in dimension 4n − 1 admit a Sasakian 3-structure? We do not solve this problem here, but only mention that Sasaki also noticed that taking the quotient of the diagonal embedding Γ⟶Sp(n + 1) gives a 3-Sasakian manifold Γ\S^{4n−1}. In this case I_{0}(S, g) = Sp(n + 1) acts on the left where the sphere S^{4n−1} is represented by a quaternionic valued column vector of unit length. The infinitesimal isometries which generate multiplication by a unit quaternion on each component from the right then give S^{4n−1} and hence Γ\S^{4n−1} a Sasakian 3-structure. However, as the homogeneous structure and Sasakian 3-structure are not compatible, Γ\S^{4n−1} is not a 3-Sasakian homogeneous manifold. This construction of 3-Sasakian manifolds appears to be special to the spheres. If one attempts a similar procedure for the other homogeneous spaces, one obtains a double coset space Γ\G/L_{1} which, in general, is an orbifold.

§5. 3-Sasakian Reduction

In this section we give a general 3-Sasakian reduction procedure which constructs new 3-Sasakian manifolds from a given 3-Sasakian manifold with a non-trivial 3-Sasakian isometry group. The key to this construction is the quaternionic reduction of 3-Sasakian manifolds constructed in [BGM1]. Actually, this is a reduction that is associated with a quadruple of spaces, namely, the quaternionic Kähler space, the corresponding twistor space, Swann bundle, and the 3-Sasakian Konishi bundle. It incorporates the quaternionic Kähler, twistor space, and hyperkähler reductions as well as the 3-Sasakian reduction presented in this section. For example, diagram 6.1 given in the next section pictorially represents how all these various reductions follow from the flat hyperkähler metric on ℍ^{n} \ {0} in the homogeneous case.

To begin let (S, g_{S}, ξ^{a}) be a 3-Sasakian manifold with a nontrivial group I_{0}(S, g_{S}) of 3-Sasakian isometries. By the embedding Theorem B, M = S × ℝ^{+} is a hyperkähler manifold with respect to the cone metric g_{M}. The isometry group I_{0}(S, g_{S}) extends to a group I_{0}(M, g_{M}) ≅ I_{0}(S, g_{S}) of isometries on M by defining each element to act trivially on ℝ^{+}. Furthermore, it follows easily from the definition of the complex structures I^{a} given in equation 3.2 that these isometries I_{0}(M, g_{M}) are hyperkähler; that is, they preserve the hyperkähler structure on M. Recall [HKLR] shows that any subgroup G ⊂ I_{0}(M, g_{M}) gives rise to a hyperkähler moment map μ : M⟶𝔤^{∗} ⊗ ℝ^{3}, where 𝔤 denotes the Lie algebra of G and 𝔤^{∗} is its dual. Thus, we can define a 3-Sasakian moment map

5.1  μ_{S} : S ⟶ 𝔤^{∗} ⊗ ℝ^{3}

by restriction μ_{S} = μ | S. We denote the components of μ_{S} with respect to the standard basis of ℝ^{3}, which we have identified with the imaginary quaternions, by μ_{S}^{a}. Recall that ordinarily moment maps determined by Abelian group actions (in particular, those associated to 1-parameter groups) are only specified up to an arbitrary constant. This is not the case for 3-Sasakian moment maps since we require that the group Sp(1) generated by the Sasakian vector fields ξ^{a} acts on the level sets of μ_{S}. However, we shall see that 3-Sasakian moment maps are given by a particularly simple expression.

Proposition 5.2: Let (S, g_{S}, ξ^{a}) be a 3-Sasakian manifold with a connected compact Lie group G acting on S by 3-Sasakian isometries. Let τ be an element of the Lie algebra 𝔤 of G and let X^{τ} denote the corresponding infinitesimal isometry. Then there is a unique 3-Sasakian moment map μ_{S} such that the zero set μ_{S}^{−1}(0) is invariant under the group Sp(1) generated by the vector fields ξ^{a}. This moment map is given by

5.3  < μ_{S}^{a}, τ > = \frac{1}{2}η^{a}(X^{τ}).

Furthermore, the zero set μ_{S}^{−1}(0) is G invariant.

Proof: Using the embedding Theorem B we can define the 2-forms ω_{S}^{a} on S as the restriction of the hyperkähler 2-forms ω^{a}. Then any 3-Sasakian moment map μ_{S}^{a}(τ) determined by τ ∈ 𝔤 satisfies 2dμ_{S}^{a}(τ) = 2X^{τ} ⌋ω_{S}^{a} = −X^{τ} ⌋dη^{a}. As X^{τ} is a 3-Sasakian infinitesimal isometry, lemma 4.2 implies that d(2 < μ_{S}^{a}, τ > −η^{a}(X^{τ})) = 0. Hence, locally we have

5.4  2 < μ_{S}^{a}, τ > = η^{a}(X^{τ}) + C_{τ}^{a}

for some locally defined constants C_{τ}^{a}. Now the Lie bracket relations appearing in definition 2.3 and lemma 2.10 imply that L_{ξ^{b}}η^{a} = −2ϵ^{abc}η^{c}. Using this equation and 5.4 we can compute the Lie derivative of the moment map to obtain

L_{ξ^{b}} < μ_{S}^{a}, τ > = −2ϵ^{abc} < μ_{S}^{c}, τ > +ϵ^{abc}C_{τ}^{c}.

It follows that μ_{S}^{−1}(0) is invariant under the group generated by ξ^{a} if and only if the constants C_{τ}^{a} vanish. So locally 5.4 becomes 5.3, and locally the moment map μ_{S}^{a}(τ) is clearly unique. But the functions η^{a}(X^{τ}) are globally defined on S so equation 5.3 must hold globally.

To prove the last statement we can work infinitesimally since G is compact. Let ζ, τ ∈ 𝔤, then by 4.3.iv we have X^{τ} η^{a}(X^{ζ}) = η^{a}([X^{τ} , X^{ζ}]). Since the bracket in the last term is in 𝔤 this term vanishes on the zero set μ_{S}^{−1}(0) which proves the G invariance.

Henceforth by the 3-Sasakian moment map, we shall mean the moment map μ_{S} determined in Proposition 5.2. Hence, the Embedding Theorem B, Proposition 5.2, and the results of [BGM1] now imply Theorem D of the introduction.

We conclude this section with the following fact concerning 3-Sasakian isometries:

Proposition 5.5: Assume that the hypothesis of Theorem D holds. In addition assume that (S, g_{S}) is complete and hence compact. Let C(G) ⊂ I_{0}(S, g_{S}) denote the centralizer of G in I_{0}(S, g_{S}) and let C_{0}(G) denote the subgroup of C(G) given by the connected component of the identity. Then C_{0}(G) acts on the submanifold μ_{S}^{−1}(0) as isometries with respect to the restricted metric ι^{∗}g_{S} and the 3-Sasakian isometry group I_{0}(Š, ǧ_{S}) of the quotient (Š, ǧ_{S}) determined in Theorem D contains an isomorphic copy of C_{0}(G). Furthermore, if C_{0}(G) acts transitively on Š, then Š is a 3-Sasakian homogeneous space.

Proof: By Proposition 4.4 I_{0}(S, g_{S}) is compact and connected, so it suffices to prove the corresponding result on the Lie algebra level. Let 𝔦_{0}(S, g_{S}), 𝔤, and 𝔠(𝔤) denote the Lie algebras of I_{0}(S, g_{S}), G, and C_{0}(G) respectively. For any x ∈ 𝔦_{0}(S, g_{S}) we let X^{x} denote the corresponding vector field on S. Then lemma 4.3 implies that for any y ∈ 𝔠(𝔤) and for all τ ∈ 𝔤 we have X^{y}η^{a}(X^{τ}) = η^{a}([X^{y}, X^{ξ}]) = 0. Hence, C_{0}(G) acts on the zero set μ_{S}^{−1}(0). Furthermore, this action is an isometry on μ_{S}^{−1}(0) since the metric is the restricted metric and C_{0}(G) ⊂ I_{0}(S, g_{S}). This proves the first statement.

Next, by Proposition 5.2, G acts by isometries on μ_{S}^{−1}(0) so the action of C_{0}(G) on μ_{S}^{−1}(0) passes to an action of C_{0}(G) on the quotient Š = μ_{S}^{−1}(0)/G. It is easy to check that C_{0}(G) acts as 3-Sasakian isometries on (Š, ǧ_{S}, ξ̌^{a}). Notice here that if G is commutative then G ⊂ C_{0}(G), and so we do not require that I_{0}(Š, ǧ_{S}) acts effectively.

§6. The Classical 3-Sasakian Homogeneous Metrics

We now apply the reduction procedure given in Theorem D to the round unit sphere S^{4n−1} to explicitly construct the Riemannian metrics for the 3-Sasakian homogeneous manifolds arising from the simple classical Lie algebras. These metrics are precisely the ones associated to the three infinite families appearing in Theorem C. We used this reduction technique in [BGM1] to show that these homogeneous spaces admit a Sasakian 3-structure; however, the general 3-Sasakian reduction construction given in Theorem D was not formulated there and, except for the case of the S^{4n−1} sphere, the Riemannian metrics were not explicitly given.

Recall that the unit sphere S^{4n−1} with its canonical metric g_{can} is the simplest example of a 3-Sasakian manifold and that the quaternionic Hopf fibration exhibits this sphere as the total space projecting to the quaternionic projective space ℍℙ^{n−1} with fibre Sp(1). This is a locally trivial Riemannian fibration where the base space ℍℙ^{n−1} has its standard quaternionic Kähler metric. Notice that the canonical metric on S^{4n−1} is not the standard homogeneous metric on the homogeneous space Sp(n)/Sp(n − 1) with respect to the reductive decomposition 𝔰𝔭(n) ≃ 𝔰𝔭(n − 1) + 𝔪. It is, of course, the standard homogeneous metric with respect to the naturally reductive decomposition of the orthogonal Lie algebra, 𝔬(4n) ≃ 𝔬(4n − 1) + 𝔪. This is quite special to the sphere and orthogonal group. As we shall see, in general the 3-Sasakian metrics given in Theorem C are not naturally reductive with respect to any reductive decomposition.

The following diagram schematically represents the reduction in this homogeneous case from the flat hyperkähler metric on ℍ^{n} \{0} of the hyperhähler, quaternionic Kähler, twistor, and 3-Sasakian reductions which result in the corresponding Swann bundle, quaternionic Kähler base space, twistor space, and 3-Sasakian Konishi bundle. We will consider examples of the more general orbifold reduction in the next section.

6.1 \begin{aligned}& & ℍ^{n} \ \{0\} & & & ⟶^{Q(𝔽)} & & & N \\ & {ℝ^{+} \atop ↙} & & {ℂ^{∗} \atop ↘} & & & & {t(ℂ) \atop ↙} & & {t(ℝ) \atop ↘} \\ S^{4n−1} & & ↓ℍ^{∗} & & ℂℙ^{2n−1} & ⟶^{Q_{2}(𝔽)} & Z & & ↓t(ℍ) & & S \\ & ↘ & & ↙ & & & & ↘ & & ↙ \\ & & ℍℙ^{n−1} & & & ⟶^{Q_{2}(𝔽)} & & & W \end{aligned}

Here 𝔽 denotes any of the three (skew) fields ℝ, ℂ, and ℍ, 𝔽^{∗} denotes the group of nonzero elements of 𝔽, and ℝ^{+}, the positive reals, is the component of ℝ^{∗} connected to the identity. Moreover Q(𝔽) ⊂ 𝔽^{∗} denotes the subgroup of 𝔽^{∗} consisting of elements of norm one so that Q(ℝ) = ℤ_{2}, Q(ℂ) = U(1), and Q(ℍ) = Sp(1), respectively. Finally, Q_{2}(𝔽) = Q(𝔽)/ℤ_{2}, t(𝔽) = 𝔽^{∗}/ℤ_{2}, and n ≥ 1 + [𝔽 : ℝ] where [𝔽 : ℝ] is the dimension of 𝔽 over ℝ.

To carry out this reduction we must set some conventions. We describe the unit sphere S^{4n−1} by its embedding in flat space and we represent an element u = (u_{1}, ⋯, u_{n}) ∈ ℍ^{n} as a column vector. The quaternionic components of this vector are denoted by u^{0} for the real component and by u^{a} for the three imaginary components. Then the quotients by the groups ℝ^{+}, ℂ^{∗}, and ℍ^{∗} in the left most diagram in 6.1 are given by right scalar multiplication, i.e., u ↦ uq where q ∈ ℝ^{+}, q ∈ ℂ^{∗}, and q ∈ ℍ^{∗}, respectively. In particular, the infinitesimal generators of the subgroup Sp(1) ⊂ ℍ^{∗} acting from the right are the defining vector fields ξ^{a} for the Sasakian 3-structure. These vector fields are given by

6.2  ξ_{r}^{a} = u^{0} · \frac{∂}{∂u^{a}} − u^{a} · \frac{∂}{∂u^{0}} − ϵ^{abc}u^{b} · \frac{∂}{∂u^{c}} ,

where the dot indicates sum over the vector components u_{i} and the subscript r means that these vector fields are the generators of the right action.

The group Q(𝔽) ⊂ Sp(1) used for the reduction procedure is then given by left scalar multiplication; i.e., u ↦ σu for σ ∈ Q(𝔽). The non-commutativity of the quaternions distinguishes these two actions. Notice, however, that any r ∈ ℝ commutes with any u ∈ ℍ^{n} and this gives rise to the ℤ_{2} factor that appears in the reduction. Our choice of hyperkähler structure on ℍ^{n}\{0}, and hence the 3-Sasakian structure on S^{4n−1}, is such that the left action preserves the hyperkähler structure and hence the 3-Sasakian structure. Notice here that the corresponding induced left actions on the quotients ℂℙ^{2n−1} and ℍℙ^{n−1} preserve the corresponding complex contact and quaternionic Kähler structures, respectively. The infinitesimal generators of the group Q(ℍ) = Sp(1) acting from the left are

6.3  ξ_{l}^{a} = u^{0} · \frac{∂}{∂u^{a}} − u^{a} · \frac{∂}{∂u^{0}} + ϵ^{abc}u^{b} · \frac{∂}{∂u^{c}} .

Our first task in the reduction procedure is to find the zero set of the moment map. We identify the imaginary quaternions ℝ^{3} with the Lie algebra 𝔰𝔭(1) in equation 5.1 and let 𝔮(𝔽) denote the Lie algebra of Q(𝔽), so the moment map is μ_{S} : S^{4n−1} ⟶ 𝔮(𝔽)^{∗} ⊗ 𝔰𝔭(1). Now 𝔮(𝔽) can be identified with the pure imaginary elements in the field 𝔽. Notice that 𝔮(ℝ) = 0, so that μ_{S} is the zero map and μ_{S}^{−1}(0) is the entire sphere S^{4n−1} in the real case. Thus, it is convenient to make the following definition.

Definition 6.4: Let 𝔽 = ℝ, ℂ, ℍ. Then N(𝔽) is the zero set μ_{S}^{−1}(0).

Next, we need to recall some facts about Stiefel manifolds. Let 𝔽^{n} denote the ndimensional vector space over 𝔽 with its natural inner product which we denote by ū · v, where u ↦ ū denotes conjugation in 𝔽 on each component. Let U(n, 𝔽) denote the subgroup of GL(n, 𝔽) that preserves this inner product so U(n, ℝ) = O(n), U(n, ℂ) = U(n), and U(n, ℍ) = Sp(n). Now let V_{n,k}^{𝔽} denote the Stiefel manifold 𝔽-orthonormal k dimensional frames in 𝔽^{n}. It is convenient to introduce the notion of an “opposite field” 𝔽^{op} as follows ℝ^{op} = ℍ, ℂ^{op} = ℂ, and ℍ^{op} = ℝ. The Stiefel manifolds that appear in our 3-Sasakian reduction are V_{n,[𝔽:ℝ]}^{𝔽^{op}}. As usual these can be represented in matrix terminology as follows. Let M_{n,k}(𝔽) denote the n by k matrices over 𝔽, then

6.5  V_{n,[𝔽:ℝ]}^{𝔽^{op}} = {𝔸 ∈ M_{n,[𝔽:ℝ]}(𝔽^{op}) | 𝔸^{∗}𝔸 = 𝕀_{[𝔽:ℝ]}},

where ∗ denotes transpose together with conjugation in 𝔽 and 𝕀_{k} denotes the k by k identity matrix. There is a natural Riemannian metric on V_{n,[𝔽:ℝ]}^{𝔽^{op}} given by restricting the flat metric h = tr(d𝔸^{∗} · d𝔸) on M_{n,[𝔽:ℝ]}(𝔽^{op}) to V_{n,[𝔽:ℝ]}^{𝔽^{op}}. We denote this restricted metric by h_{1}.

Proposition 6.6: Under a rescaling N(𝔽) is precisely the Stiefel manifold V_{n,[𝔽:ℝ]}^{𝔽^{op}}. Thus, ι : N(𝔽) ↪ S^{4n−1} is a smooth compact submanifold of dimension 4n + 2 − 3[𝔽 : ℝ] on which Q(𝔽) acts freely. Furthermore, the Riemannian metrics are related by the equation

ι^{∗}g_{can} = \frac{1}{[𝔽 : ℝ]}h_{1}.

Proof: As mentioned above when 𝔽 = ℝ we have N(𝔽) = S^{4n−1} which is just the Stiefel manifold V_{n,1}^{ℍ}. For the remaining two cases we compute the moment map 5.3. First, let 𝔽 = ℍ. Proposition 5.2 shows to compute μ_{S} we need the 1-forms η^{a} which we can easily obtain from the flat space metric g_{0} and the Sasakian vector fields ξ_{r}^{a} given in equation 6.2. Then a straightforward computation shows that the components of the moment map < μ_{S}^{a}, ξ_{l}^{b} > are quadratic functions of the quaternionic Cartesian coordinates u. These components lie in 𝔰𝔭(1)^{∗} ⊗ 𝔰𝔭(1) ≃ 𝔰𝔭(1) ⊗ 𝔰𝔭(1) and decomposing this into irreducible representations under the action of Sp(1), namely the identity representation, the adjoint representation 𝔰𝔭(1), and the 5-dimensional representation of three by three traceless symmetric matrices over ℝ we see that the zero set of the moment map is determined precisely the condition that the four vectors (u^{0}, u^{1}, u^{2}, u^{3}) are mutually orthogonal and of norm \frac{1}{4}. Thus, a simple scale transformation identifies N(ℍ) with the Stiefel manifold V_{n,4}^{ℝ} in 6.5 and Riemannian metrics are related by the factor of \frac{1}{4.}

Finally, when 𝔽 = ℂ, the group Q(ℂ) is a U(1) subgroup of Sp(1). This corresponds to stabilizing one component in the imaginary quaternions ℝ^{3}, say the b = 1 component. Then defining the complex vectors z^{1} = u^{0} + iu^{1} and z^{2} = u^{2} + iu^{3} permits us to rewrite u = z^{1} +jz̄^{2}, and the zero set of the moment map < μ_{S}, ξ_{l}^{1} > is represented by the equations z̄^{1} ·z^{1} = z̄^{2} ·z^{2} = \frac{1}{2} and z̄^{2} ·z^{1} = 0. Again a simple scale transformation identifies N(ℂ) with the complex Stiefel manifold V_{n,2}^{ℂ} with the Riemannian metrics correspondingly related. In each case the Stiefel manifolds are known to be smooth compact submanifolds of S^{4n−1} of dimension 4n + 2 − 3[𝔽 : ℝ]. Lastly, it is easy check that the left action of Q(𝔽) is free in each case.

The free action of Q(𝔽) on N(𝔽) makes N(𝔽) a principal Q(𝔽) bundle over the quotient N(𝔽)/Q(𝔽). Moreover, given the metric ι^{∗}g_{can} on N(𝔽) there is a unique Riemannian metric ǧ on N(𝔽)/Q(𝔽) such that π : N(𝔽) ⟶ N(𝔽)/Q(𝔽) is a Riemannian submersion. The group I(S^{4n−1}, g_{can}) of 3-Sasakian isometries acting on S^{4n−1} is precisely Sp(n) acting from the left and we have

Lemma 6.7: The centralizer C(Q(𝔽)) of Q(𝔽) in Sp(n) in precisely U(n, 𝔽^{op}).

Proof: Of course Sp(n) ⊂ M_{n,n}(ℍ) ≃ M_{n,n}(ℝ) ⊗_{ℝ} ℍ. By ℝ linearity it suffices to prove the result on a simple element A ⊗ q ∈ M_{n,n}(ℝ) ⊗_{ℝ} ℍ. But Q(𝔽) ⊂ ℍ is a subgroup for 𝔽 = ℝ, ℂ, ℍ and, in every case, is embedded in Sp(n) as the diagonal embedding σ ↦ σ𝕀_{n}. So finding C(Q(𝔽)) amounts to finding the centralizer C_{Q(𝔽)}(ℍ) of Q(𝔽) in ℍ. But it is direct to check that C_{Q(𝔽)}(ℍ) = 𝔽^{op}. So A ⊗ q commutes with σ𝕀_{n} for all σ ∈ Q(𝔽) if and only if q ∈ 𝔽^{op}.

Now the Stiefel manifold V_{n,[𝔽:ℝ]}^{𝔽^{op}} with the Riemannian metric h_{1} is a homogeneous

Riemannian manifold with homogeneous structure given by \frac{U(n, 𝔽^{op})}{U(n − [𝔽 : ℝ], 𝔽^{op})}. Thus,

Lemma 6.7, Theorem D, Proposition 5.5, and Proposition 6.6 imply the following fact.

Theorem 6.8: With the restrictions on n given in Theorem C, the Riemannian manifold (N(𝔽)/Q(𝔽), ǧ) is one of the classical 3-Sasakian homogeneous manifolds

\frac{U(n, 𝔽^{op})}{U(n − [𝔽 : ℝ], 𝔽^{op}) × Q(𝔽)}

listed in Theorem C. Furthermore, the metric ǧ is given explicitly by ǧ = \frac{1}{[𝔽 : ℝ]}π_{∗}h_{1}.

Remarks 6.9:

1. Except for the real case 𝔽 = ℝ, when the reduction is just given by taking a ℤ_{2} quotient, the metric ǧ is not naturally reductive with respect to the homogeneous space structure and hence ǧ is not the standard homogeneous metric on S. In fact, the standard homogeneous metric for 𝔽 = ℂ, ℍ is not Einstein [Bes].

2. There are some duplications in low dimensions due to the following isomorphisms of the classical Lie groups SO(5) ≃ Sp(2)/ℤ_{2} and SO(6) ≃ SU(4)/ℤ_{2}.

Thus far we have not been able to explicitly obtain the metrics in the cases of the exceptional groups appearing in Theorem C by a reduction procedure from the canonical unit sphere S^{4n−1}. Nevertheless, Theorem C guarantees the existence of corresponding 3-Sasakian homogeneous metrics. As in the classical case they can not be naturally reductive. This follows from the fact that naturally reductive homogeneous metrics of a compact Lie group have non-negative sectional curvature (cf. [Bes: 9.87]). But then Proposition 2.17 implies that the Wolf space W with its symmetric quaternionic Kähler metric would have positive sectional curvature greater than or equal to 3. For n ≥ 3 a theorem of Berger ([Bes: 14.43]) implies that W = ℍℙ^{n−1}. If n = 2 then W = ℂℙ^{2} with its Fubini-Study metric which has positive sectional curvature. However, with our normalization one can check that the sectional curvature takes on all values between 2 and 5. Summarizing we have

Proposition 6.10: Let (S, g) be a 3-Sasakian homogeneous manifold which is not of constant curvature. Then the metric g is not naturally reductive with respect to its homogeneous structure.

§7. Reductions by Weighted Circle Actions

In this section we apply the 3-Sasakian reduction technique described in the previous section to a general circle action giving rise to new infinite families of homotopy distinct 3-Sasakian manifolds. Instead of considering a circle S^{1} embedded in Sp(n) = I_{0}(S^{4n−1}, g_{can}) as a diagonal subgroup with equal weights, we now consider the most general circle subgroup of the maximal torus of Sp(n) embedded diagonally but with unequal weights. These manifolds can be thought of as “discrete deformations” of the 3-Sasakian homogeneous manifolds \frac{U(n)}{U(n−2)×U(1)} obtained by reduction from the diagonal circle action used in section 6.

We begin by considering a maximal torus T^{n} of Sp(n) = I_{0}(S^{4n−1}, g_{can}). Up to conjugacy T^{n} is unique and can be taken to act on S^{4n−1} as the subgroup of norm preserving diagonal matrices acting on the quaternionic coordinates u of ℍ^{n}. Here, as in the paragraph preceding 6.2, we view u as a column vector and the action is given by matrix multiplication from the left. Explicitly, we have that the action θ̃: T^{n} × S^{4n−1} ⟶ S^{4n−1} is given by θ̃(t, u) = (e^{2πit_{1}}u_{1}, ⋯, e^{2πit_{n}}u_{n}), where t_{j} ∈ ℝ and u_{j} ∈ ℍ denote the j^{th} component of t and u, respectively. Consider a sequence p = (p_{1}, ⋯, p_{n}) of nonzero integers. For each p ∈ (ℤ^{∗})^{n}, we can define a weighted circle subgroup U(1)_{p} ⊂ T^{n} by setting t_{i} = p_{i}t for each 1 ≤ i ≤ n where t ∈ ℝ. Then the action θ̃ restricts to the circle action

7.1  θ_{p} : S^{1} × S^{4n−1} ⟶ S^{4n−1}

given by θ_{p}(t, u) = (e^{2πip_{1}t}u_{1}, ⋯, e^{2πip_{n}t}u_{n}). Notice that the case p = 1 = (1, ⋯, 1) is precisely the circle action of the previous section.

Next we compute the moment map μ_{S}(p) : S^{4n−1}⟶iℝ^{3} associated to the circle action 7.1. Here we identify the Lie algebra 𝔲(1) with the pure imaginary numbers iℝ, and we shall write μ_{S}(p)(u) for < μ_{S}(p)(u), i > . The fundamental vector field corresponding to the action of the circle subgroup U(1)_{p} is given by

7.2  ξ_{l}(p) = ∑_{j=1}^{n} p_{j}(u_{j}^{0} \frac{∂}{∂u_{j}^{1}} − u_{j}^{1} \frac{∂}{∂u_{j}^{0}} + u_{j}^{2} \frac{∂}{∂u_{j}^{3}} − u_{j}^{3} \frac{∂}{∂u_{j}^{2}} ) = ∑_{j=1}^{n} p_{j}H_{j},

where the H_{j} are the infinitesimal generators for the action of the maximal torus T^{n}. The moment map can now be obtained from 5.3 and 7.2. This computation yields the following lemma.

Lemma 7.3: In terms of the complex coordinates z^{1} = u^{0} + iu^{1} and z^{2} = u^{2} + iu^{3} introduced in section 6 the components of the moment map μ_{S}(p) of the circle action given by 7.1 are

2μ_{S}^{1}(p)(z^{1}, z^{2}) = −i ∑_{j=1}^{n} p_{j}(|z_{j}^{1} |^{2} − |z_{j}^{2} |^{2}),

μ_{S}^{−}(p)(z^{1}, z^{2}) = −i ∑_{j=1}^{n} p_{j}z̄_{j}^{2}z_{j}^{1}, where μ_{S}^{−} = μ_{S}^{2} − iμ_{S}^{3}.

Notice that these equations specialize in the case that p = 1 to the equations for the moment map given in section 6. Just as in this homogeneous case the zero set of the moment map is a fundamental object of interest.

Definition 7.4: N(p) = μ_{S}(p)^{−1}(0).

Proposition 7.5: For each p ∈ (ℤ^{∗})^{n} the zero set N(p) of the moment map μ_{S}(p) is diffeomorphic to the complex Stiefel manifold V_{n,2}^{ℂ}. Thus, N(p) is a smooth compact submanifold of S^{4n−1} of dimension 4n−4. Furthermore, the circle action θ_{p} on S^{4n−1} restricts to a circle action on N(p) which is free if the absolute values |p_{j} | of the components of p are pairwise relatively prime.

Proof: For each p ∈ (ℤ^{∗})^{n} we define a linear map T_{p} : ℍ^{n}⟶ℍ^{n} by the equation

7.6  T_{p}u = (ϵ(p_{1})\sqrt{|p_{1} |}u_{1}, ⋯, ϵ(p_{n})\sqrt{|p_{n} |}u_{n})

where ϵ(p) = 1 if p is positive and ϵ(p) = j, the imaginary quaternion, if p is negative. Clearly T_{p} is an isomorphism of quaternionic vector spaces for each p ∈ (ℤ^{∗})^{n}; however, it is norm preserving only if |p_{j} |= 1 for all j = 1, ⋯, n. Thus, for each p ∈ (ℤ^{∗})^{n} we define a diffeomorphism ϕ_{p} : S^{4n−1}⟶S^{4n−1} by ϕ_{p}(u) =∥ T_{p}u ∥^{−1} T_{p}u, where ∥ · ∥ denotes the standard norm in ℍ^{n}. Let μ_{S}(1) and μ(1) denote the 3-Sasakian and hyperkähler moment maps, respectively, of the homogeneous case given in the previous section. Then using lemma 7.3 and the fact that the components of the moment map are homogeneous functions of degree 2 in the variables u we have the equalities

7.7  μ_{S}(p)(u) = μ(1)(T_{p}u) = μ(1)(∥ T_{p}u ∥ ϕ_{p}(u)) =∥ T_{p}u ∥^{2} μ_{S}(1)(ϕ_{p}(u)).

Hence, u ∈ N(p) if and only if ϕ_{p}(u) ∈ N(ℂ). Thus, the diffeomorphism ϕ_{p} restricts to a diffeomorphism ϕ_{p} : N(p)⟶N(ℂ). But recall that the identification of N(ℂ) with V_{n,2}^{ℂ} in Proposition 6.7 requires a dilation by a factor of \sqrt{2}. Composing ϕ_{p} with this dilation gives the required diffeomorphism.

Since the circle action θ_{p} is a linear map on ℍ^{n}, it clearly restricts to the zero set N(p). Now assume that the integers |p_{j} | are pairwise relatively prime. It follows that the only fixed points on the S^{4n−1} can occur along a quaternionic coordinate axis, say the k^{th} axis given by u_{j} = 0 for all j ≠ k. In this case the isotropy subgroup of any such point is the cyclic group ℤ_{p_{k}}. Hence, by Lemma 7.3 the vanishing of the moment map restricted to the k^{th} quaternionic coordinate axis gives p_{k} | z_{k}^{1} |= p_{k} | z_{k}^{2} | and p_{k}z̄_{k}^{2}z_{k}^{1} = 0. These two equations imply that z_{k}^{1} = z_{k}^{2} = 0 which cannot happen on N(p) ⊂ S^{4n−1}. Thus, the circle action θ_{p} is free on N(p).

Remark 7.8: Notice that there is nothing in the proof that the zero sets N(p) are compact submanifolds of S^{4n−1} diffeomorphic to the Stiefel manifolds V_{n,2}^{ℂ} that prohibits p from being any real vector in (ℝ^{∗})^{n}. In this sense the N(p) can be thought of as smooth deformations of V_{n,2}^{ℂ}. Of course, for general p ∈ (ℝ^{∗})^{n} the quotients defined below in Definition 7.14 will not be manifolds. Nevertheless, we can think of 3-Sasakian manifolds S(p) defined below as “discrete” deformations of the 3-Sasakian homogeneous manifold S(1).

Let ι_{p} : N(p) ↪ S^{4n−1} denote the embedding given by the zero set of the moment map μ_{S}(p). We define a Riemannian metric g(p) on N(p) by restricting the canonical metric on S^{4n−1}, that is g(p) = ι_{p}^{∗}g_{can}. We shall make use of the following simple observation: Any infinitesimal isometry (Killing vector field) on (S^{4n−1}, g_{can}) that has the property that when it is restricted to N(p) it is tangent to N(p) is also an infinitesimal isometry of (N(p), g(p)). In particular, the 3-Sasakian vector fields ξ_{r}^{a} and the infinitesimal generator ξ_{l}(p) of the circle group U(1)_{p} satisfy this property by Proposition 5.2. More generally, it follows from (iv) of Lemma 4.4 that any Killing vector field in 𝔠(𝔲(1)_{p}) also satisfies this property. In particular, the maximal torus T^{n} acts as isometries on (N(p), g(p)). Let W_{n} denote the Weyl group of Sp(n). W_{n} is isomorphic to the semidirect product (ℤ_{2})^{n}^{>}◁ Σ_{n} where Σ_{n} is the symmetric group on n letters, and W_{n} acts on the Lie algebra 𝔱_{n} of the maximal torus by permutations and sign changes. Notice that (ℤ_{2})^{n}^{>}◁ Σ_{n} also acts naturally on (ℤ^{∗})^{n} by permutations and sign changes. We shall identify this group with the Weyl group and denote this action by p ↦ wp for w ∈ W_{n}. Now the Weyl group W_{n} can be realized as a subgroup of Sp(n) = I_{0}(S^{4n−1}, g_{can}) by the following action on the quaternionic coordinates u ∈ ℍ^{n}: The symmetric group acts by permutations of the vector components (u_{1}, ⋯, u_{n}) ∈ ℍ^{n}. The l^{th} reflection of W_{n} acts by sending the l^{th} component u_{l} of u to ju_{l} and leaving all other components fixed. Taking the direct product of these actions gives an action of W_{n} on S^{4n−1} × (ℤ_{2})^{n}. It is now easy to check that

Proposition 7.9: The action of W_{n} on S^{4n−1}×(ℤ^{∗})^{n} described above induces an isometry between (N(wp), g(wp)) and (N(p), g(p)) which preserves the 3-Sasakian vector fields ξ_{r}^{a}.

It follows from this proposition that without loss of generality we can take the integers p_{j} to be positive integers and order p = (p_{1}, ⋯, p_{n}) such that p_{1} ≤ p_{2} ≤ ⋯ ≤ p_{n}. Henceforth, we shall assume this to be the case unless otherwise specified.

Proposition 7.10: Let p = (p_{1}, ⋯, p_{n}) be an n-tuple of pairwise relatively prime positive integers, and let k be the number of 1’s in p. Then the centralizer C(U(1)_{p}) of U(1)_{p} in Sp(n) is U(k) × T^{n−k}.

Proof: An argument similar to that given in the proof of Lemma 6.7 shows that C(U(1)_{p}) must lie in U(n). It is direct to check that the centralizer is as stated.

Let F : V_{n,2}^{ℂ}⟶N(p) denote the diffeomorphism of Proposition 7.6, and consider the metric F^{∗}g(p) on V_{n,2}^{ℂ}. Proposition 7.10 now implies.

Corollary 7.11: Let p ≠ 1. Then the metric F^{∗}g(p) is not U(n) invariant; hence, it is not homothetic to the homogeneous metric \frac{1}{2}h_{1} of Proposition 6.8.

We now come to our major objects of study.

Definition 7.12: Let p = (p_{1}, p_{2}, ⋯, p_{n}) be an n-tuple of relatively prime ordered positive integers. Then (S(p), ǧ(p)) is the Riemannian manifold N(p)/U(1)_{p} with the unique Riemannian metric ǧ(p) that makes π : N(p)⟶N(p)/U(1)_{p} a Riemannian submersion.

The following theorem is a direct corollary of Theorems A, D, and Corollary 2.13.

Theorem 7.13: For each n-tuple p of ordered relatively prime positive integers, the Riemannian manifold (S(p), ǧ(p)) is a compact 3-Sasakian manifold; hence, it is a compact Einstein manifold of positive scalar curvature equal to 2(2n − 3)(4n − 5). The space of leaves S(p)/F is a compact 4(n − 2)-dimensional quaternionic Kähler orbifold O(p) of scalar curvature equal to 16n(n − 2). Furthermore, S(p) has a second homothety class of Einstein metrics nonhomothetic to ǧ(p) with positive scalar curvature.

Our next task is to understand the manifolds S(p) group theoretically. The Stiefel manifold V_{n,2}^{ℂ} is the homogeneous space U(n)/U(n−2) viewed as the space of right cosets. The subgroup U(1)_{p} ⊂ U(n) is the diagonal embedding and acts from the left giving

Proposition 7.14: The 3-Sasakian manifold S(p) can be identified with the double coset space U(1)_{p}\U(n)/U(n − 2). If p = 1 then the subgroup U(1)_{p} is central and the the 3-Sasakian manifold S(1) is the homogeneous space given in Theorem 6.8 when 𝔽 = ℂ.

Remark 7.15: Notice that for any p ≠ 1, it follows from Proposition 7.10 that S(p) is not homogeneous with respect to the U(n) action described above. In fact, since S(p) is not of constant curvature, Proposition 4.4 and the proof of Proposition 4.6 imply that S(p) is not Riemannian homogeneous. Actually, in section 9 we shall prove a much stronger result in dimension 7 for a certain infinite subset of the p; namely, that those S(p) are not homotopy equivalent to any homogeneous space.

Proposition 7.16: Let p = (p_{1}, ⋯, p_{n}) be an n-tuple of positive pairwise relatively prime integers, so that S(p) is a smooth compact 3-Sasakian manifold. A generic leaf of the foliation F on S(p) is isomorphic to:

(i) SO(3) if all p_{j} are odd,

(ii) Sp(1) otherwise.

The isotropy subgroup of any singular leaf is a cyclic subgroup of the circle group U(1) ⊂ Sp(1)_{r} corresponding to the complex direction i.

Proof: The proof of the first statement amounts to whether or not the central ℤ_{2} in Sp(1) lies in the circle group U(1)_{p}. The conditions for this are that for all i, j = 1, ⋯, n there

exist positive integers k_{i}, k_{j} such that \frac{2k_{i} + 1}{p_{i}} = \frac{2k_{j} + 1}{p_{j}}. Now take j > i, then p_{j} ≥ p_{i} and

equality holds if and only if p_{i} = p_{j} = 1, so the condition becomes p_{j} −p_{i} = 2(k_{j}p_{i} −k_{i}p_{j}) whose left hand side is even if p_{j} are odd for all j. On the other hand, the condition that the p_{j}’s be pairwise relatively prime implies that at most one p_{j} is even, so if the p_{j} are not all odd the equation above cannot hold over the integers.

To prove the second statement we compute first on S^{4n−1}. The condition for the existence of fixed points is that e^{2πip_{l}t}u_{l}σ = u_{l} for some σ ∈ Sp(1) and each 1 ≤ l ≤ n. Recall from the proof of Proposition 7.5 that on N(p) at least two such u_{l} must be nonvanishing. In terms of the complex coordinates of Lemma 7.3 these conditions are e^{2πip_{l}t}z_{l}^{1}σ = z_{l}^{1}, and e^{−2πip_{l}t}z_{l}^{2}σ = z_{l}^{2}, where not all z’s vanish. This implies that σ must be of the form e^{2πis} for some real number s; that is, σ ∈ U(1) corresponding to the complex direction i. But the isotropy subgroup of a leaf must be of the form given in 2.12 and the only such groups that are subgroups of a circle are the cyclic groups.

The singular locus Σ(p) ⊂ O(p) of the 4(n − 2)-dimensional quaternionic Kähler orbifold O(p) can be rather complicated depending on the choice of p. We shall describe Σ(p) in detail, when n = 3, in the Section 9. Here, we make the following two observations. First, notice that O(p) is a leaf space of a Seifert fibration [OWag]. Recall that the group Sp(1)_{r} generated by the 3-Sasakian vector fields ξ_{r}^{a} acts as isometries on (N(p), g(p)). Moreover, since Sp(1)_{r} acts freely on S^{4n−1}, it acts freely on the submanifold N(p). Thus, N(p) is a principal Sp(1) bundle over its quotient N(p)/Sp(1).

Definition 7.17: M(p) = N(p)/Sp(1)_{r}.

Proposition 7.18: The group U(1)_{p} acts locally freely on M(p), and thus defines a Seifert fibration π_{s} : M(p)⟶O(p) over the quaternionic Kähler orbifold O(p). Furthermore, M(p) is a submanifold of quaternionic projective space ℍℙ^{n−1} and is diffeomorphic to the homogeneous space \frac{U(n)}{U(n−2)×SU(2).}

Proof: First we can check as in the proof of Proposition 7.16 above that the circle group U(1)_{p} acts locally freely on M(p). The remainder of the proof then follows from Proposition 7.5 and the following commutative diagram

S(p) ^{\underleftarrow{π_{p}}} N(p) ↪ S^{4n−1}
↓π_{0}  ↓  ↓
O(p) ^{\underleftarrow{π_{s}}} M(p) ↪ ℍℙ^{n−1}.

The embedding M(p) ↪ ℍℙ^{n−1} is realized as the inclusion of the zero set of the quaternionic Kähler moment map of Galicki and Lawson [GL]. Also when n = 3 it is easy to see that M(p) ≃ S^{5}. We discuss this case in section 9.

§8. The Integral Cohomology Ring of S(p)

Notice that Theorem E of the introduction has the corollary.

Corollary 8.1: There are infinitely many non-homotopy equivalent simply-connected compact 3-Sasakian manifolds in dimension 4n − 5 for every n ≥ 3.

Proof: Let p(n, d) = (1, …, 1, d). Then S(p(n, d)) is a simply-connected, compact, 4n−5 dimensional 3-Sasakian manifold with H^{2n−2}(S(p), ℤ) ≅ ℤ_{(n−1)d+1}.

Theorem E follows directly from a spectral sequence argument of Eschenburg [Esch: §3]. However, we include sufficient detail in order to make the discussion here reasonably self-contained.

Proof of Theorem E: Using Proposition 7.14 Eschenburg’s analysis shows that there is a commutative diagram of fibrations

8.2  \begin{aligned}& & U(n) & ≅ & U(n) \\ & & ↓ & & ↓ \\ S(p) & ≃ & U(1)_{p}\U(n)/U(n − 2) & ⟶^{ρ̂} & B_{U(n)} \\ & & ↓p & & ↓Δ \\ & & B_{U(1)_{p}×U(n−2)} & ⟶^{ρ} & B_{U(n)^{2}} \end{aligned}

which is an equivalence on the fibres.

Recall that H^{∗}(U(n); ℤ) ≅ E[e_{1}, e_{3}, …, e_{2n−1}] is an exterior algebra on generators e_{2i−1} of dimension 2i − 1 for 2 ≤ i ≤ n whereas H^{∗}(B_{U(n)}; ℤ) ≅ ℤ[c_{1}, c_{2}, …, c_{n}], where c_{i} is the restriction of the i^{th} universal Chern class to U(n) and is thus of dimension 2i. Thus,

8.3  H^{∗}(B_{U(n)^{2}}; ℤ) ≅ ℤ[a_{1}, …, a_{n}, b_{1}, …, b_{n}],

where a_{i} = c_{i} ⊗ 1 and b_{i} = 1 ⊗ c_{i}.

The Serre spectral sequence associated to the right hand column of 8.2 has the form E_{2}^{∗,∗}(Δ) ≅ H^{∗}(B_{U(n)^{2}}; ℤ) ⊗ H^{∗}(U(n); ℤ) as the cohomology of the base and fibre are torsion free. Let k_{j}^{∗} : H^{∗}(B_{U(n)^{2}}; ℤ)⟶E_{j}^{∗,0}(Δ) denote the natural projection of the E_{2}^{∗,0}(Δ) term along the base.

Proposition 8.4: [Bor1] Δ^{∗} = k_{∞} : H^{∗}(B_{U(n)^{2}}; ℤ)⟶E_{∞}^{∗,0}(Δ) ⊂ H^{∗}(U(n); ℤ).

Using this proposition of Borel and the fact that Δ^{∗} induces the cup product in H^{∗}(B_{G}) so Δ^{∗}(u⊗1) = Δ^{∗}(1⊗u) = u, Eschenburg computed the differentials in E_{j}^{∗,∗}(Δ).

Lemma 8.5: [Esch] For all n ≥ 3 the differentials d_{j} : E_{j}^{s,t}(Δ)⟶E_{j}^{s+j,t−j+1}(Δ) in the cohomology spectral sequence E_{j}^{∗,∗}(Δ) converging to H^{∗}(B_{U(n)}; ℤ) are generated by 1. d_{j}(e_{2i−1}) = 0 for j ≤ i,

2. d_{2i}(e_{2i−1}) = ±k_{2i}(a_{i} − b_{i}) for 1 ≤ i ≤ n.

Thus, by naturality of the Serre spectral sequence, we have

Lemma 8.6: For all n ≥ 3 the differentials d_{j} : E_{j}^{s,t}(p)⟶E_{j}^{s+j,t−j+1}(p) in the cohomology spectral sequence E_{j}^{∗,∗}(p) converging to H^{∗}(S(p); ℤ) are generated by

1. d_{j}(e_{2i−1}) = 0 for j ≤ i,

2. d_{2i}(e_{2i−1}) = ±k_{2i}ρ^{∗}(a_{i} − b_{i}) for 1 ≤ i ≤ n.

Once again, k_{j}^{∗} : H^{∗}(B_{U(1)_{p}×U(n−2)}; ℤ)⟶E_{j}^{∗,0}(p) denotes the natural projection of the E_{2}^{∗,0}(p) term along the base in the spectral sequence.

Proof: That these differentials exist follows directly from naturality. Moreover, the

identity map gives an isomorphism E_{2}^{0,∗}(Δ) ^{id ; ≅} E_{2}^{0,∗}(π) along the fibres and the differentials in the first cohomology spectral sequence are all transgressively generated.

Once we compute ρ^{∗} : H^{∗}(B_{U(n)^{2}}; ℤ) ⟶ H^{∗}(B_{U(1)_{p}}; ℤ) ⊗ H^{∗}(B_{U(n−2)}; ℤ) we can apply lemma 8.4 to compute the differentials in the Serre spectral sequence converging to H^{∗}(S(p); ℤ) to establish Theorem E. The inclusion U(1)_{p} × U(n − 2) ⟶ U(n) × U(n) which is the product of the composition mapping ι(Δ_{p}) : U(1)⟶T^{n}⟶U(n) on the first factor and the natural inclusion j_{n} : U(n − 2)⟶U(n) on the second factor. Here the maximal torus T^{n} includes as diagonal matrices into U(n) in the standard way, and the maps Δ_{p} and j_{n} are given in section 7.

We now compute in cohomology. The inclusion map on the second factor implies that

8.7  ρ^{∗}(b_{i}) = ρ^{∗}(1 ⊗ c_{i}) = 1 ⊗ y_{i}

for 1 ≤ i ≤ n − 2. Here the classes c_{i} and y_{i} are the i^{th} Chern classes in H^{∗}(BU(n); ℤ) and H^{∗}(BU(n − 2); ℤ), respectively and b_{i} is defined in equation 8.3.

Recall that if T^{n} = U(1) × U(1) × ⋯ × U(1) is a maximal torus in U(n) then the homomorphism ι_{n}^{∗} : H^{∗}(B_{U(n)}; ℤ)⟶H^{∗}(T^{n}; ℤ) induced by the natural inclusion is an injection. In fact, ι_{n}^{∗} is an isomorphism onto the polynomial subalgebra of Σ_{n} invariant polynomials which are freely generated by the elementary symmetric functions in the x_{i}’s where x_{i} ∈ H^{2}(B_{U(1)} = ℂℙ^{∞}; ℤ) is the two dimensional generator of the i^{th} factor.

Next, recall that if x ∈ H^{2}(B_{U(1)_{p}}; ℤ) is the two dimensional generator then Δ_{p}^{∗}(x_{i}) = p_{i}x for 1 ≤ i ≤ n. This immediately implies that, for 1 ≤ i ≤ n,

8.8  ρ^{∗}(a_{i}) = ρ^{∗}(c_{i} ⊗ 1) = σ_{i}(p)x^{2i} ⊗ 1,

where σ_{i}(p) is the i^{th} elementary symmetric function of the coordinates of p and a_{i} is defined in equation 8.3.

The E_{2}^{∗,∗}(S(p)) term for the spectral sequence converging to H^{∗}(S(p); ℤ) is isomorphic to

8.9  ℤ[x ⊗ 1, 1 ⊗ y_{1}, …, 1 ⊗ y_{n−2}] ⊗ E[e_{1}, …, e_{2n−1}].

Equations 8.7, 8.8, and lemma 8.6 imply there are differentials

1. d_{j}(e_{2i−1}) = 0 for j ≤ i,

2. d_{2i}(e_{2i−1}) = ±k_{2i}(σ_{i}(p)x^{2i} ⊗ 1 − 1 ⊗ y_{i}) for 1 ≤ i ≤ n − 2,

3. d_{2i}(e_{2i−1}) = ±k_{2i}(σ_{i}(p)x^{2i} ⊗ 1) for n − 1 ≤ i ≤ n.

A direct calculation using the first two families of differentials shows that

8.10  E_{2n−3}^{∗,∗} ≅ ℤ[x ⊗ 1] ⊗ E[e_{2n−3}, e_{2n−1}],

where the classes on the right hand side are understood to be the E_{2n−3} level equivalence classes. Theorem E now follows by using the last two differentials above and the fact that σ_{n−1}(p) and σ_{n}(p) are relatively prime.

§9. Strongly Inhomogeneous Einstein Spaces

We now consider the 7-dimensional 3-Sasakian manifolds S(p_{1}, p_{2}, p_{3}) in more detail. In 1982 Eschenburg [Esch] introduced the concept of a strongly inhomogeneous space. Such a space is a compact topological space which is not homotopy equivalent to any compact Riemannian homogeneous space. He classified homogeneous spaces homotopy equivalent to a compact, closed, oriented connected 7-dimensional smooth manifold M such that π_{1}(M) = 0, π_{2}(M) = ℤ, π_{3}(M) = ℤ, π_{4}(M) = 0. In particular, Eschenburg deduced that if H^{4}(M; ℤ) is a finite cyclic group of order r then r ≢ 2 (mod 3). Thus, the analysis in [Esch: §4], Theorem E, and a trivial calculation imply Theorem F of the introduction.

In addition to exhibiting strongly inhomogeneous examples, the case when n = 3 is special as V_{3,2}^{ℂ} ≃ SU(3). This lets us rewrite S(p) as a quotient of SU(3) by a certain circle action and this alternative description permits us to analyze the quaternionic Kähler orbifold O(p).

Consider the diffeomorphism α : SU(3) × U(1)⟶U(3) defined as the composition

9.1  SU(3) × U(1) ⟶ U(3) × U(3) ⟶^{μ} U(3),

where the first map is the natural inclusion on the first factor and j_{3}(τ) = diag(1, 1, τ) on the second factor. The second map μ is group multiplication in U(3). Notice that α is not a group homomorphism; however, U(1) acts on SU(3) × U(1) by multiplication in the second factor, and on U(3) by the inclusion j_{3} followed by right multiplication. Furthermore, α intertwines these two actions and thus α induces a map of homogeneous spaces α̂ : SU(3) ⟶ \frac{U(3)}{U(1)} = V_{3,2}^{ℂ} given as the composition of the natural inclusion SU(3)⟶U(3) followed by the natural projection U(3)⟶U(3)/j_{3}(U(1)). To describe the inverse map we recall that any 𝔹 ∈ U(n) can be viewed as n column vectors in ℂ^{n} that are mutually orthogonal with respect to the standard Hermitian inner product in ℂ^{n}. Thus, writing any 𝔹 ∈ U(3) in terms of its column vectors, 𝔹 = (b_{1}, b_{2}, b_{3}), the map α̂^{−1} is given by α̂^{−1}(𝔹j_{3}(U(1))) = (n_{1}, n_{2}, n_{3}) where n_{i} =∥ b_{i} ∥^{−1} b_{i}. It is easy to check that this is independent of the coset representative.

Now consider the circle subgroup U(1)_{p} ⊂ T^{3} of the maximal torus of U(3) with the conventions adopted in section 7; in particular, p = (p_{1}, p_{2}, p_{3}) and the action on the zero set N(p) = N(p_{1}, p_{2}, p_{3}) is given by restricting the action θ_{p} given in equation 7.1. We have a diffeomorphism given by the composition

9.3  G : N(p_{1}, p_{2}, p_{3}) ⟶^{F^{−1}} V_{3,2}^{ℂ} ⟶^{α̂^{−1}} SU(3),

where F is the diffeomorphism given in Proposition 7.5. Thus, the free circle action θ_{p} on N(p) induces a free circle action ϑ_{p} on SU(3). A straightforward computation shows that

9.4  ϑ_{p}(τ, 𝔸) = diag(τ^{p_{1}}, τ^{p_{2}}, τ^{p_{3}}) 𝔸 diag(1, 1, τ^{−(p_{1}+p_{2}+p_{3})}),

where p = (p_{1}, p_{2}, p_{3}). Notice that the image of U(1)_{p} under ϑ_{p} is a subgroup of S(U(3)× U(3)) acting on SU(3) by left-right multiplication. We shall denote the quotient of SU(3) by this circle action by T (p_{1}, p_{2}, p_{3}). Thus, we get an diffeomorphism S(p_{1}, p_{2}, p_{3}) ≃ T (p_{1}, p_{2}, p_{3}), and the 3-Sasakian structure on S(p_{1}, p_{2}, p_{3}) is transported to T (p_{1}, p_{2}, p_{3}) by this diffeomorphism giving T (p_{1}, p_{2}, p_{3}) as an isomorphic model for S(p_{1}, p_{2}, p_{3}).

We shall now see why this alternative model is useful. To begin recall the manifold M(p) = N(p)/Sp(1)_{r} from Definition 7.16. Here M(p) is actually isomorphic to S^{5} ≃ SU(3)/SU(2) under the isomorphism above. The circle action ϑ_{p} on SU(3) described above commutes with the standard SU(2) action and thus passes to act on the quotient S^{5}. However, this action is, in general, not free but only locally free. There is a commutative diagram

9.5  \begin{matrix}SU(3) & ⟶ & SU(3)/U(1)_{p} \\ ↓ & & ↓π_{0} \\ S^{5} & ⟶^{π_{s}} & O(p_{1}, p_{2}, p_{3}), \end{matrix}

where the maps π_{0} and π_{s} are orbifold submersions. In fact, π_{s} is a Seifert fibration. Now according to Proposition 7.15 the generic fibre of π_{0} is either SO(3) or Sp(1) according to whether σ_{1}(p) is odd or even, so we need to distinguish the two cases.

Proposition 9.6: Let (p_{1}, p_{2}, p_{3}) be a triple of ordered pairwise relatively prime positive integers. Then the quaternionic Kähler orbifolds appearing in diagram 9.5 are:

(i) O(p_{1}, p_{2}, p_{3}) = ℂℙ^{2}(p_{1} + p_{2}, p_{1} + p_{3}, p_{2} + p_{3}) when p_{1} + p_{2} + p_{3} is even,

(ii) O(p_{1}, p_{2}, p_{3}) = ℂℙ^{2}(\frac{p_{1}+p_{2}}{2} , \frac{p_{1}+p_{3}}{2} , \frac{p_{2}+p_{3}}{2} ) when p_{1} + p_{2} + p_{3} is odd.

Here the terms on the right are weighted projective spaces.

Proof: The circle action 9.4 on SU(3) passes to the action on the quotient S^{5} given by

9.7  (z_{1}, z_{2}, z_{3}) ↦ (e^{−2πi(p_{2}+p_{3})t}z_{1}, e^{−2πi(p_{1}+p_{3})t}z_{2}, , e^{−2πi(p_{1}+p_{2})t}z_{3}).

For each triple (p_{1}, p_{2}, p_{3}) the quotient space of S^{5} by this action is known to be a weighted projective spaces (cf. [GL]). If p_{1} + p_{2} + p_{3} is even, then precisely one of the p_{j} is even, so two of the sums p_{i} + p_{j} are odd and one is even; whereas, if p_{1} + p_{2} + p_{3} is odd, all the p_{j}’s are odd and so all the sums p_{i} + p_{j} are even.

Next we analyze the singular locus Σ(p_{1}, p_{2}, p_{3}) of the orbifold S(p_{1}, p_{2}, p_{3}). This is a straightforward exercise using the action 9.7. Again we distinguish two case:

Case 1: Let p_{1} + p_{2} + p_{3} be even. There are two possibilities for Σ(p_{1}, p_{2}, p_{3}) :

1. Three isolated points. This occurs when the entries in the triple (p_{1}+p_{2}, p_{1}+p_{3}, p_{2}+p_{3}) are pairwise relatively prime.

2. A single copy of S^{2} and an isolated point. This occurs when two of the p_{i} + p_{j} have a common factor.

Case 2: Let p_{1} + p_{2} + p_{3} be odd. Then there are four possibilities for Σ(p_{1}, p_{2}, p_{3}):

1. The empty set. This occurs when p = (1, 1, 1) and corresponds to the regular case when S(1, 1, 1) is homogeneous and fibres over the standard ℂℙ^{2}.

2. Three isolated points. This occurs when the entries in the triple (p_{1}+p_{2}, p_{1}+p_{3}, p_{2}+p_{3}) are pairwise relatively prime modulo 2.

3. A single copy of S^{2}. This occurs when p = (1, 1, 2k + 1).

4. A single S^{2} with an additional isolated orbifold point. This occurs in all the other cases not covered in items 1,2, and 3.

Thus we see that the orbifold locus Σ(p_{1}, p_{2}, p_{3}) is either empty (only in the one regular case), or it consists of either three isolated orbifold points, a 2-sphere with an additional isolated orbifold point, or a single 2-sphere.

Summarizing the results of this section with the results of previous sections implies the following theorem.

Theorem 9.8: Let (p_{1}, p_{2}, p_{3}) be a triple of ordered pairwise relatively prime positive integers. Then the manifold T (p_{1}, p_{2}, p_{3}) is isometric to the 3-Sasakian manifold S(p_{1}, p_{2}, p_{3}). Therefore T (p_{1}, p_{2}, p_{3}) admits a 3-Sasakian structure with a non-homogeneous Einstein metric with scalar curvature equal to 42. The leaf space O(p_{1}, p_{2}, p_{3}) of the associated 3-Sasakian foliation is an orbifold which is smoothly equivalent to the weighted projective space

(i) ℂℙ^{2}(p_{1} + p_{2}, p_{1} + p_{3}, p_{2} + p_{3}) when p_{1} + p_{2} + p_{3} is even,

(ii) ℂℙ^{2}(\frac{p_{1}+p_{2}}{2} , \frac{p_{1}+p_{3}}{2} , \frac{p_{2}+p_{3}}{2} ) when p_{1} + p_{2} + p_{3} is odd.

Here the base orbifold has the quaternionic Kähler orbifold metric (with a fixed scale) constructed by Galicki and Lawson [GL], and the singular locus Σ(p_{1}, p_{2}, p_{3}) that is described above.

Remark 9.9: The homotopy types of the weighted projective spaces ℂℙ^{2}(a, b, c) are wellknown [Kaw].

It is worthwhile to notice that there is a close connection between our 7-manifolds T (p_{1}, p_{2}, p_{3}) and a construction of Eschenburg [Esch]. More precisely, Eschenburg considers quotients of SU(3) by free actions of circle subgroups of SU(3)_{L} × SU(3)_{R}. It is easy to see that all such circle actions are given by

9.10  𝔸 ⟶^{τ} diag(τ^{k}, τ^{l}, τ^{−(k+l)}) 𝔸 diag(τ^{m}, τ^{n}, τ^{−(m+n)}),

where 𝔸 ∈ SU(3), τ ∈ S^{1}, and the quadruple of integers (k, l, m, n) must satisfy some additional conditions in order that the circle action be free (see [Esch: 21] for the precise constraints). When the action is free, Eschenburg denotes the quotient space by M_{klmn} which is a simply connected, compact 7-manifold. Eschenburg’s M_{klmn} manifolds are related to several other manifolds of general interest. For example, when m = n = 0, Eschenburg’s construction recovers the homogeneous Aloff-Wallach spaces M_{kl} studied in [AlWal, KrSt]. Eschenburg computed H^{∗}(M_{klmn}; ℤ) as a graded ring. The result is almost identical to the graded ring appearing in Theorem E with the single modification that the finite cyclic group H^{4}(M_{klmn}; ℤ) has order r = |k^{2} + l^{2} + kl − (m^{2} + n^{2} + mn)|. In this way he was able to find strongly inhomogeneous 7-manifolds. A straightforward computation shows

Proposition 9.11: If p_{1}+p_{2}+p_{3} ≡ 0 (mod 3) then the 3-Sasakian manifold T (p_{1}, p_{2}, p_{3}) is diffeomorphic to the Eschenburg manifold M_{klmn} where (k, l, m, n) is determined by the equations k = \frac{1}{3}(2p_{1} − p_{2} − p_{3}), l = \frac{1}{3}(2p_{2} − p_{3} − p_{1}), m = n = \frac{1}{3}(p_{1} + p_{2} + p_{3}).

However, the M_{klmn} Riemannian manifolds with the metrics constructed by Eschenburg are not 3-Sasakian manifolds so the diffeomorphisms in Proposition 9.11 are not isometries.

We chose to highlight the S(c, c + 1, c + 2) manifolds in Theorem F because

Corollary 9.12: S(c, c + 1, c + 2) is diffeomorphic to M_{−1,0,c+1,c+1} for all odd c.

Finally, Corollary 9.12 has two corollaries which follow from [Esch]. Recall that Cheeger [Ch1,Ch2] has constructed a distance ρ^{∗}((M, g), (M′, g′)) between two compact n-dimensional Riemannian manifolds.

Corollary 9.13: Let c be odd. Then the S(c, c + 1, c + 2) manifolds converge in the Cheeger ρ^{∗}-topology to the homogeneous space S(1, 1, 1) with respect to the Aloff-Wallach metrics as c⟶∞.

Corollary 9.14: For all sufficiently large odd positive integers c, S(c, c + 1, c + 2) admits a metric of positive sectional curvature.

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Department of Mathematics and Statistics Revised Version, November 1993 University of New Mexico Albuquerque, NM 87131 email: cboyer@math.unm.edu, galicki@math.unm.edu, mann@math.unm.edu