Annals of Global Analysis and Geometry Volume 14 (1996), 81-105

Hypercomplex Structures on Stiefel Manifolds

Charles P. Boyer Krzysztof Galicki Benjamin M. Mann

Abstract. This paper describes a family of hypercomplex structures {I^{a}(p)}_{a=1,2,3} depending on n real non-zero parameters p=(p_{1},...,p_{n}) on the Stiefel manifold of complex 2-planes in ℂ^{n} for all n>2. Generally, these hypercomplex structures are inhomogeneous with the exception of the case when all the p_{i}’s are equal. We also determine the Lie algebra of infinitesimal hypercomplex automorphisms for each structure. Furthermore, we solve the equivalence problem for the hypercomplex structures in the case that the components of p are pairwise commensurable. Finally, some of these examples admit discrete hypercomplex quotients whose topology we also analyze.

Introduction

A hypercomplex structure on a smooth manifold M is a G-structure where G = GL(n, ℍ) that admits a necessarily unique torsion free connection, the Obata connection [Bon,Ob]. In particular, every such M has three complex structures I, J, and K which satisfy the relations of the algebra of imaginary quaternions and thus generate an entire two-sphere’s worth of complex structures on M. Until recently, there were few known examples of compact, irreducible, hypercomplex manifolds in dimension 8 and higher. The first class of such examples are the hyperkähler twisted products of K3 surfaces constructed by Beauville [Bea]. Examples of hypercomplex manifolds that are not hyperkähler were very scarce, the simplest ones being the Hopf manifolds S^{4n+3} × S^{1} which are locally conformally hyperkähler. Recently the authors [BGM2] gave a class of new compact locally conformally hyperkähler manifolds by replacing S^{4n+3} with any 3-Sasakian manifold. Similar examples involving the quaternionic Heisenberg group were found by Hernandez [Her]. None of these examples, however, are simply connected.

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

In contrast to the 4-dimensional case, where all hypercomplex structures on compact 4-manifolds are locally conformally hyperkähler [Boy], in higher dimensions this is no longer true. A class of hypercomplex manifolds that are not locally conformally hyperkähler was studied by physicists interested in supersymmetric σ-models. In this regard, Spindel et. al. [SSTP] classified compact Lie groups which admit hypercomplex structures. This is the generalization to the hypercomplex category of the classic work of Samelson [Sam] and Wang [W] on the classification of compact Lie groups [Sam] and homogeneous spaces [W] admitting complex structures, respectively. Using different methods, Joyce [Joy1] later recovered this [SSTP] classification and developed a theory of homogeneous hypercomplex manifolds which generalizes Wang’s [W] result.

In this paper, guided by our previous work on 3-Sasakian manifolds [BGM1,BGM2], we prove the existence of uncountably many distinct hypercomplex structures on Stiefel manifolds of complex 2-planes in complex n-space. Our main results are as follows.

Theorem A: Let n > 2 and p = (p_{1}, …, p_{n}) ∈ (ℝ^{∗})^{n} be an n−tuple of non-zero real numbers. For each such p there is a compact hypercomplex manifold (N (p), I^{a}(p)), where N (p) is diffeomorphic to 𝕍_{n,2}^{ℂ}, the Stiefel manifold of 2-frames in ℂ^{n}.

Since 𝕍_{n,2}^{ℂ} is (2n − 4)-connected, it cannot admit a locally conformally hyperkähler structure. A priori, all the hypercomplex structures {I^{a}(p)} on 𝕍_{n,2}^{ℂ} could be equivalent. In particular, permuting the coordinates of p or changing their signs does not change the hypercomplex structure. Thus, we can assume that p is an n-tuple of positive, nondecreasing, real numbers; i.e., p is an element in the positive cone C_{n} = {p ∈ ℝ^{n} | 0 < p_{1} ≤ p_{2} ≤ … ≤ p_{n}}. However, we prove

Theorem B: If p and q are both commensurable sequences in the positive cone C_{n} then the hypercomplex manifolds N (p) and N (q) are hypercomplex equivalent if and only if p = q. Here p is said to be commensurable if each of the ratios \frac{p_{i}}{p_{j}} is a rational number. Furthermore, the manifold N (p) is hypercomplex homogeneous if and only if p = λ(1, …, 1) for some λ ∈ ℝ^{∗}.

The one parameter family of distinct U(n)-homogeneous hypercomplex structures on 𝕍_{n,2}^{ℂ} given in Theorem B was found implicitly by Joyce [Joy1] (see also [Bat]); however the remaining inhomogeneous hypercomplex structures on 𝕍_{n,2}^{ℂ} are new. They are analogous to the inhomogeneous complex structures found by Griffiths [Gr] in the versal deformation space of homogeneous complex structures. The equivalence problem in the case of incommensurable sequences p appears to be beyond the scope of the techniques used in this paper and is relegated to a future work.

Our next main result determines the connected component of the group of hypercomplex automorphisms of N (p). In particular, it is shown that this group depends only on the multiplicities m_{i} of the components p_{i} of p. More precisely,

Theorem C: For all p ∈ (ℝ^{∗})^{n} the Lie algebra 𝔥(p) of infinitesimal hypercomplex automorphisms of N (p) is isomorphic to ⨁_{i=1}^{k} 𝔲(m_{i}). Hence, the connected component of the

group of hypercomplex automorphisms of N (p) is ∏_{i=1}^{k} U(m_{i}). In particular, there exists a

natural hyperhermitian metric h(p) on N (p) such that every infinitesimal automorphism is an infinitesimal isometry with respect to h(p).

Notice that Theorem C implies that the set of multiplicities in p is an invariant of the hypercomplex structure even when p is not a commensurable sequence.

Many of our examples admit discrete quotients which are also hypercomplex. To state this result we need some additional notation. A commensurable sequence p is called basic if all the coordinates are integers and the greatest common divisor of all the coordinates is one. A basic sequence is said to be coprime if the coordinates are pairwise relatively prime. If p is an integer multiple of a basic sequence and if the triples (p_{i}, p_{j}, k) have no common factor for all 1 ≤ i < j ≤ n then p is called k-coprime.

Theorem D: Let p be k-coprime. Then there is a compact hypercomplex manifold H(p, k) with universal cover ρ_{k} : N (p)⟶H(p, k) such that π_{1}(H(p, k)) ≅ ℤ_{k} and ρ_{k} is a hypercomplex map. Moreover, H(p, k) is never locally conformally hyperkähler and is hypercomplex homogeneous if and only if p = (p, p, …, p).

Theorems B and C immediately extend to these non-simply connected examples so H(p, k) is hypercomplex equivalent to H(q, l) if and only if p = q and k = l. Furthermore, the Lie algebra of infinitesimal hypercomplex automorphisms of H(p, k) is the Lie algebra 𝔥(p) given in Theorem C. Moreover, if p is commensurable but not k-coprime then H(p, k) is a hypercomplex orbifold.

This paper is organized as follows: In section one we construct by the method of symmetry reduction two equivalent models for the n-parameter family of hypercomplex structures on the Stiefel manifold 𝕍_{n,2}^{ℂ}. Both of these models are then used in section 2 together with a crucial scaling argument in the parameter space to completely describe the connected component to the identity of the group of hypercomplex automorphisms for each hypercomplex structure. The precise statement is Theorem C above. Next we turn our attention to the question of when the hypercomplex structures on 𝕍_{n,2}^{ℂ} defined by different p are inequivalent and prove Theorem B. This is done in the case of commensurable p by analyzing the holonomy groups associated to certain foliations canonically attached to the hypercomplex structures. We briefly address a few questions in the more general case when some of the \frac{p_{i}}{p_{j}} ratios are irrational. This is done in sections 3 and 4. This leads to the study of the moduli space of hypercomplex structures on the complex Stiefel manifold 𝕍_{n,2}^{ℂ}. Finally, in the last section we construct the hypercomplex structures on certain discrete quotients H(p, k) and analyze their topology. An announcement of the results presented here has appeared in [BGM3].

We would like to thank Dr. H. Hernandez and Dr. T. Nitta for interesting remarks and discussions. The second author would like to thank Professor F. Hirzebruch and the Max-Planck Institute for support and hospitality. This article was written in part during his visit there.

§1. The Hypercomplex Geometry of Stiefel Manifolds

Recall that the Stiefel manifold 𝕍_{n,2}^{ℂ} of complex 2-planes in ℂ^{n} is the homogeneous manifold U(n)/U(n−2). Alternatively, 𝕍_{n,2}^{ℂ} can be described as an embedded submanifold 𝕍_{n,2}^{ℂ} ⊂ S^{4n−1} ⊂ ℍ^{n}. It is the zero set of a 3-Sasakian moment map μ : S^{4n−1} → ℝ^{3} associated with the central U(1) subgroup of U(n) [BGM2].

We shall induce the hypercomplex structures on 𝕍_{n,2}^{ℂ} from the flat hypercomplex geometry of ℍ^{n}. This is the flat G-structure with G = GL(n, ℍ). Since the first prolongation 𝔤𝔩(n, ℍ)^{(1)} vanishes the automorphism group of this flat G-structure is the semidirect product GL(n, ℍ)^{>}◁ Q where Q is the quaternionic translation group generated by u ↦ u + q. Owing to the noncommutativity of the quaternions there are two (equivalent) flat hypercomplex structures on ℍ^{n}. With respect to the standard quaternionic coordinates (u_{1}, …, u_{n}) they are given by

1.1  I_{±}^{a} = ∑ (\frac{∂}{∂u_{j}^{a}} ⊗ du_{j}^{0} − \frac{∂}{∂u_{j}^{0}} ⊗ du_{j}^{a} ± ϵ^{abc} \frac{∂}{∂u_{j}^{b}} ⊗ du_{j}^{c}) .

The two hypercomplex structures are distinguished as follows: On ℍ^{n} there is a natural action of Sp(1) · Sp(1) ≃ Sp(1) × Sp(1)/ℤ_{2} acting diagonally on each coordinate u_{i} by u_{i} ↦ lu_{i}r, where l, r ∈ Sp(1). This action induces representations of Sp(1) · Sp(1) on the trivial vector bundles V_{±} spanned by I_{±}^{a}, respectively. These representations are id ⊗ Ad on V_{+} and Ad⊗id on V_{−}. We shall mainly confine ourselves to the hypercomplex structure defined by I_{+}^{a}. Let ξ_{+}^{a} denote the infinitesimal generators of the right Sp(1) action on ℍ^{n}, and let Ψ denote the Euler field that generates dilatations on ℍ^{n}. Together this gives an action of ℍ^{∗} on ℍ^{n} that acts freely on the open submanifold ℍ^{n} \ {0}. The hypercomplex structure I_{+}^{a} on ℍ^{n} restricts to give a hypercomplex structure, also denoted by I_{+}^{a}, on ℍ^{n} \ {0} with automorphism group GL(n, ℍ). The Lie algebra 𝔤𝔩(n, ℍ) of GL(n, ℍ) has the Levi decomposition

1.2  𝔤𝔩(n, ℍ) ≃ 𝔰𝔩(2n, ℍ) ⊕ 𝔯,

where the radical 𝔯 is the central one-dimensional Lie algebra generated by the Euler vector field Ψ. One can easily check that the followings compatibility relations hold

1.3  I_{+}^{a}ξ_{+}^{a} = −ϵ^{abc}ξ_{+}^{c} + δ^{ab}Ψ.

This leads to

Definition 1.4: An almost hypercomplex structure {I^{a}}_{a=1}^{3} on a smooth manifold M is said to be Sp(1)-compatible if there are a smooth action of Sp(1) and a vector field Ξ on M such that (i) Ξ is an infinitesimal automorphism of I^{a} for each a = 1, 2, 3.

(ii) The vector space V spanned by {I^{a}}_{a=1}^{3} is the adjoint representation of Sp(1).

(iii) For all a, b = 1, …, 3 we have I^{a}ξ^{a} = −ϵ^{abc}ξ^{c} + δ^{ab}Ξ, where ξ^{a} are the infinitesimal generators of the Sp(1) action.

We denote an Sp(1) compatible hypercomplex structure by the triple (I^{a}, ξ^{a}, Ξ). One easily checks that the above conditions imply the following facts:

(i) The vector fields Ξ and ξ^{a} commute, that is, [Ξ, ξ^{a}] = 0.

(ii) Ξ is nowhere vanishing on M if and only if the Sp(1) action is locally free.

(iii) All complex structures in the two sphere of complex structures of the hypercomplex structure I^{a} are equivalent.

Assume that the vector field Ξ is nowhere vanishing on M. Then ξ^{1}, ξ^{2}, ξ^{3}, Ξ span a trivial subbundle V_{4} of the tangent bundle T M. Furthermore, since these vector fields give a basis for the Lie algebra 𝔰𝔭(1)⊕ℝ, for each a = 1, 2, 3 there is a nested sequence of foliations F_{1} ⊂ F_{2}^{a} ⊂ F_{4} of M generated by the subbundles V_{1} = span {Ξ}, V_{2}^{a} = span {Ξ, ξ^{a}} and V_{4}, respectively. We refer to this nested sequence of foliations on M as the multifoliate structure associated with the Sp(1) compatible hypercomplex structure. Clearly, we have

Proposition 1.5: The triple (I_{+}^{a}, ξ_{+}^{a}, Ψ) defines an Sp(1) compatible hypercomplex structure on both ℍ^{n} and ℍ^{n} \ {0}.

Remark 1.6: The space ℍ^{n} \ {0} is the Swann ℍ^{∗} bundle over quaternionic projective space ℍℙ^{n−1}, and the spaces of leaves of the foliations F_{1} and F_{2}^{a}, are the 3-Sasakian manifold S^{4n−1} and the twistor space ℂℙ^{2n−1}, respectively. One might thus expect that under suitable conditions, in the general case of a given Sp(1) compatible hypercomplex manifold, the geometry of these foliations persist. That is, that F_{1} describes 3-Sasakian geometry, F_{2} describes twistor geometry with a complex contact structure, and F_{4} describes quaternionic Kähler geometry.

A particularly efficient way of constructing explicit new examples of a given geometry is by the method of symmetry reduction. In the case at hand there are two ways of performing this reduction. The first is related to 3-Sasakian reduction [BGM2] and is tied to a Riemannian metric. The second is Joyce’s [Joy1] hypercomplex reduction and is ostensibly independent of any metric. In our case it will be shown that these two methods give equivalent hypercomplex structures on the Stiefel manifold 𝕍_{n,2}^{ℂ}. We now consider symmetry reduction of the flat hypercomplex structure on ℍ^{n} \ {0} by a one-parameter subgroup of its group of automorphisms GL(n, ℍ). The maximal compact subgroup of GL(n, ℍ) is Sp(n) with maximal torus T^{n}. This torus lies in a U(n) subgroup of Sp(n), and thus fixes a complex structure, say I_{−}^{1}, of the “opposite” hypercomplex structure I_{−}^{a}.

The action of this torus on ℍ^{n} is linear and given by the diagonal representation

1.7  θ̃(t, u) = (e^{2πit_{1}}u_{1}, …, e^{2πit_{n}}u_{n}).

Thus, the corresponding representation of the Lie algebra 𝔱_{n} of T_{n} is given by the multiplication by diagonal matrices diag(p_{1}, …, p_{n}) on the quaternionic vector u = (u_{1}, …, u_{n}) ∈ ℍ^{n}. Thus, we identify p = (p_{1}, …, p_{n}) ∈ ℝ^{n} as an element of the Lie algebra 𝔱_{n}. Let Ξ : 𝔱_{n} → Γ(T (ℍ^{n} \ {0})) be the map associating to each element in 𝔱_{n} the corresponding vector field on ℍ^{n} \ {0}, where Γ(T M) denotes the smooth vector fields on a smooth manifold M. Let e_{i} denote the standard basis of ℝ^{n} ≃ 𝔱_{n}, then we have

1.8  Ξ(e_{i}) = H_{i} = 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}} .

We shall be interested in the vector field Ξ(p) on ℍ^{n} \ {0} corresponding to an element p ∈ 𝔱_{n}. If p_{i} ≠ 0 for all i = 1, …, n the vector field Ξ(p) is nowhere vanishing on ℍ^{n} \ {0}. So we define the subset

1.9  𝔱̂_{n} = {p ∈ 𝔱_{n} : p_{i} ≠ 0 for all i = 1, …, n} ≃ (ℝ^{∗})^{n}.

In [BGM2] we constructed a 3-Sasakian moment map from the hyperkähler moment map μ(p) on ℍ^{n}. This moment map is given by

1.10  μ(p)(u^{∗}, u) = \frac{1}{2}η_{+}(Ξ(p)) = u^{∗} · ipu,

where η_{+} = iη_{+}^{1} + jη_{+}^{2} + kη_{+}^{3}, the 1-forms {η_{+}^{a}}_{a=1}^{3} are dual to the vector fields ξ_{+}^{a}, ∗ denotes the quaternionic adjoint, · is the flat Hermitian inner product on ℍ^{n}, and p is identified with the diagonal matrix diag (p_{1}, …, p_{n}).

Definition 1.11:

N (p) = μ(p)^{−1}(0) ∩ S^{4n−1} ⊂ ℍ^{n} \ {0}.

Our first result is a slight generalization of Proposition 7.5 of [BGM2] whose proof is identical.

Theorem 1.12: For every p ∈ 𝔱̂_{n} the subspace N (p) is a smooth manifold of real dimension 4n − 4 diffeomorphic to the complex Stiefel manifold 𝕍_{n,2}^{ℂ}.

From their definitions one easily sees that the restrictions of the vector fields H_{i} and ξ_{+}^{a} to N (p) are tangent to N (p). These restrictions we denote by H_{i} and ξ^{a}, respectively. The vector fields {H_{i}}_{i=1}^{n} generate a commutative Lie algebra sheaf SF that is a subsheaf of the tangent sheaf T N (p) for all p ∈ 𝔱̂_{n}. This sheaf defines a singular foliation [Mol] on N (p) which we also denote by SF. One easily sees that when p ∈ 𝔱̂_{n} (which we shall hereafter always assume), the vector fields Ξ(p) and ξ^{a} are everywhere linearly independent on N (p). These vector fields give rise to locally free actions of certain Lie group on N (p), and associated foliations. We now give an easy generalization of the theory of connections in principal fiber bundles to the case of foliations described by locally free actions.

Let F be the foliation determined by a locally free action of a Lie group G on a smooth manifold M, and let V denote the vertical subbundle of T M generated by the fundamental vector fields determined by the action of G. We have the exact sequence of G-modules

1.13  0⟶V⟶T M⟶T M/V⟶0.

Definition 1.14: A principal connection in the foliation F is a splitting T M ≃ V ⊕ H as G-modules. Given a chain of Lie subgroups G_{k_{n}} < ⋯ < G_{k_{1}} < G_{k_{0}} = G and an associated multi-foliate structure F_{κ} = F_{k_{n}} ⊂ ⋯ ⊂ F_{k_{1}} ⊂ F_{k_{0}} with k_{n} < ⋯ < k_{1} < k_{0}, we call the sequence of splittings T M ≃ V_{k_{i}} ⊕ H_{k_{i}} satisfying H_{k_{n}} ⊃ ⋯ ⊃ H_{k_{1}} ⊃ H_{k_{0}} a principal connection in the multi-foliate structure F_{κ} or simply a multi-foliate principal connection.

We construct almost hypercomplex structures I^{a}(p) on the Stiefel manifold 𝕍_{n,2}^{ℂ} for each p ∈ 𝔱̂_{n}. In order to minimize notational baggage we use the same notation for objects on the submanifold N (p) and on the Stiefel manifold 𝕍_{n,2}^{ℂ}. There are two cases to consider, namely, Ξ(p) generates an S^{1} action on 𝕍_{n,2}^{ℂ}, or Ξ(p) generates an ℝ action on 𝕍_{n,2}^{ℂ}. We denote the respective groups together with their actions by S^{1}(p) and ℝ(p). We have the chain of subgroups G_{1}(p) < G_{2}^{a}(p) < G_{4}(p) given by G_{1}(p) = S^{1}(p), G_{2}(p) = S^{1}(p) × S_{a}^{1}, G_{4}(p) = S^{1}(p) × SU(2) ≃ U(2)(p) or G_{1}(p) = ℝ(p), G_{2}(p) = ℝ(p) × S_{a}^{1}, G_{4}(p) = ℝ(p) × SU(2), where S_{a}^{1} denotes the circle action generated by ξ^{a}. For each a = 1, 2, 3 these subgroup chains give associated multi-foliate structures

1.15  F_{1}(p) ⊂ F_{2}^{a}(p) ⊂ F_{4}(p)

on N (p). Let g_{0}(p) denote the pushforward to 𝕍_{n,2}^{ℂ} of the restriction to N (p) of the flat metric on ℍ^{n}. The vector fields ξ^{a} are mutually orthogonal and of unit norm on 𝕍_{n,2}^{ℂ} and Ξ(p) is orthogonal to these vector fields. Thus, we get splittings of T 𝕍_{n,2}^{ℂ} and horizontal subbundles H_{1}(p) ⊃ H_{2}^{a}(p) ⊃ H_{4}(p) corresponding to the multifoliate structure 1.15. This allows us to construct almost hypercomplex structures on 𝕍_{n,2}^{ℂ} as follows:

Definition 1.16: Let I^{a}(p) denote the sections of End V_{4}(p) ⊕ End H_{4}(p) defined by

(i) On V_{4}(p), I^{a}(p)ξ^{a} = −ϵ^{abc}ξ^{c} + δ^{ab}Ξ and I^{a}(p)Ξ(p) = −ξ^{a}.

(ii) On H_{4}(p), I^{a}(p) = I_{+}^{a}.

One easily checks that

Proposition 1.17: For each p ∈ 𝔱̂_{n} the endomorphisms I^{a}(p) define an Sp(1)-compatible almost hypercomplex structure on 𝕍_{n,2}^{ℂ}.

We now investigate the integrability of I^{a}(p). First, we have

Proposition 1.18: The chain of horizontal subbundles H_{1}(p) ⊃ H_{2}^{a}(p) ⊃ H_{4}(p) defines a principal connection in the multifoliate structure 1.15.

Proof: The vector field Ξ(p) has unit norm with respect to g_{0}(p) only if p = 1. Thus, we set ρ^{2} = g_{0}(p)(Ξ(p), Ξ(p)). One checks directly that ρ^{2} is basic with respect to the foliation F_{4}(p). Letting η^{0}(p), η^{1}, η^{2}, η^{3} denote the 1-forms on 𝕍_{n,2}^{ℂ} dual to the vector fields Ξ(p), ξ^{1}, ξ^{2}, ξ^{3}, respectively, we define another metric h(p) on 𝕍_{n,2}^{ℂ} by rescaling g_{0}(p) in the leaves of the foliation F_{1}(p) by the factor ρ^{−2}. That is,

1.19  h(p) = π_{h}g(p) + ρ^{−2}π_{v}g(p),

where π_{h} and π_{v} denote the projections onto H_{1}(p) and V_{1}(p), respectively. One now sees that the result follows from the fact that Ξ(p) and ξ^{a} are all Killing vector fields with respect to h(p) for a = 1, 2, 3.

The 1-forms {η^{0}(p), η^{a}} are the connection 1-forms of the connection H_{4}(p) with respect to the standard basis {e^{0}, e^{a}} of the Lie algebra 𝔤_{4}(p) of G_{4}(p).

Theorem 1.20: For each p ∈ 𝔱̂_{n} the almost hypercomplex structures I^{a}(p) are integrable.

Proof: Let {ω^{0}, ω^{a}} denote the components with respect to the standard basis of the curvature of the connection η = η^{0}(p)e^{0} +∑_{a} η^{a}e^{a}. A straightforward but tedious computation shows that the vanishing of the Nijenhuis tensor of I^{a}(p) is equivalent to the two facts which are direct to verify:

(i) The curvature components ω^{a} are, up to a factor of 2, the restriction of the fundamental hyperhermitian forms to the horizontal subspaces H_{4}(p).

(ii) The curvature ω^{0} is type (1, 1) with respect to all of the complex structures in the hypercomplex structure I^{a}(p).

It is straightforward to verify the following

Proposition 1.21: The metric h(p) is hyperhermitian with respect to the hypercomplex structure I^{a}(p) on 𝕍_{n,2}^{ℂ}. Furthermore, the leaves of the foliations F_{1}(p), F_{2}^{a}(p), F_{4}(p) are totally geodesic with respect to h(p). The vector field Ξ(p)+ξ^{a} is holomorphic with respect to the complex structure I^{a}(p) for fixed a; hence, the foliation F_{2}^{a}(p) is holomorphic.

As in [BGM2] the Weyl group W of 𝔰𝔭(n) acts on 𝔱̂_{n} ×𝕍_{n,2}^{ℂ} sending the hypercomplex structure I^{a}(p) to the hypercomplex structure I^{a}(wp) for w ∈ W, and a representative for a Weyl group orbit is obtained by restricting p to the positive Weyl chamber C_{n} ⊂ 𝔱̂_{n}, defined by 0 < p_{1} ≤ ⋯ ≤ p_{n}. Thus, we have proved a more precise version of Theorem A of the introduction.

Theorem 1.22: On 𝕍_{n,2}^{ℂ} two hypercomplex structures I^{a}(p) and I^{a}(p′) are equivalent if they lie on the same W orbit.

Henceforth, we restrict p to lie in the positive Weyl chamber C_{n}. Now if the components of p are commensurable each leaf of the foliation F_{4}(p) is a hypercomplex Hopf surface of the form S^{1} × SU(2)/Γ, where Γ is one of the finite subgroups of SU(2), and each leaf of the foliation F_{2}^{a}(p) is an elliptic curve. If some pair of components of p are incommensurable the leaves of both foliations are noncompact, and the spaces of leaves are not Hausdorff. We say that p ∈ C_{n} is commensurable if all the components of p are commensurable. Otherwise p is called incommensurable. When p is commensurable the space of leaves N (p)/F_{1}(p) is a 3-Sasakian orbifold S(p), and the space of leaves Z(p) = N (p)/F_{2}^{a}(p) is a complex orbifold. In fact, in a future work, we shall prove that the twistor space Z associated to a 3-Sasakian orbifold is always a projective algebraic variety. Notice that the orbifolds S(p) and Z(p) do not depend on the scale of p; hence, if p′ = λp for some λ ∈ ℝ^{+}, the respective hypercomplex structures I^{a}(p′) and I^{a}(p) can only differ along the leaves of the elliptic foliation F_{2}^{a}(p). We shall prove in section 4 that this is indeed the case when p is commensurable (see Proposition 4.9).

We now discuss another isomorphic model of N (p) that will be useful in the next section. This model is related to Joyce’s [Joy2] hypercomplex reduction. We begin again with the flat hypercomplex structure I_{+}^{a} on ℍ^{n} \ {0}. We can easily check that the Euler vector field Ψ is annihilated by η_{+}^{a}. The consequence of this is that the moment map 1.10 is also a moment map with respect to the vector field Ψ + Ξ(p) on ℍ^{n} \ {0}. For each p ∈ C_{n} it will be convenient to consider the one-parameter family of vector fields defined by Ψ(λ) = Ψ + Ξ(λp). Each vector field Ψ(λ) generates an ℝ-action on ℍ^{n} \ {0} which we denote by ℝ(λ), and one can use the corresponding flow to show that the sphere S^{4n−1} is a totally transverse submanifold to this action. (Notice that if λ = 0 this ℝ-action is just the radial flow on ℍ^{n} \ {0}.) Since the vector fields Ξ(p) and Ψ are tangent to the zero set μ(p)^{−1}(0) = μ(λp)^{−1}(0), the action ℝ(λ) restricts to an ℝ-action on μ(p)^{−1}(0). Thus, we have

Proposition 1.23: For each λ ∈ ℝ there is a natural identification μ(p)^{−1}(0)/ℝ(λ) ≃ N (p) given by sending each ℝ(λ) orbit in μ(p)^{−1}(0) to its point of intersection with the unit sphere S^{4n−1}. The inverse map sends x ∈ N (p) to the corresponding ℝ(λ) orbit in μ(p)^{−1}(0).

We shall use this quotient model in the next section to compute the Lie algebra 𝔥(p) of infinitesimal automorphisms of N (p). From this point on we assume that λ > 0. First we show that μ(p)^{−1}(0)/ℝ(λ) has an induced hypercomplex structure given essentially by Joyce’s hypercomplex reduction [Joy2]. To see this we define some vector bundles on μ(p)^{−1}(0). Let L_{λp} denote the subbundle of T μ(p)^{−1}(0) generated by Ξ(λp) + Ψ, and Q its quotient bundle. Let ι : μ(p)^{−1}(0) → ℍ^{n} \ {0} be the natural inclusion, and let N(p) denote the normal bundle. Note that N(p) is spanned by the gradient fields igrad μ^{a}(p). We also denote by N_{λ}(p) the subbundle of ι^{∗}T (ℍ^{n} \ {0}) spanned by the vector fields ξ_{+}^{a} − igrad μ^{a}(λp), and define M_{λp} = L_{λp} ⊕ N_{λ}(p). We have following commutative diagram of exact sequences

1.24  \begin{aligned}& & 0 & & 0 & & 0 \\ & & ↓ & & ↓ & & ↓ \\ 0 & ⟶ & L_{λp} & ⟶ & T μ^{−1}(0) & ⟶ & Q_{λp} & ⟶ & 0 \\ & & ↓ & & ↓ & & ↓ \\ 0 & ⟶ & M_{λp} & ⟶ & ι^{∗}T (ℍ^{n} \ \{0\}) & ⟶ & R_{λp} & ⟶ & 0. \\ & & ↓ & & ↓ & & ↓ \\ 0 & ⟶ & N_{λ}(p) & ⟶^{≃} & N(p) & ⟶ & 0 \\ & & ↓ & & ↓ \\ & & 0 & & 0 \end{aligned}

The hypercomplex structure I_{+}^{a} restricts to give automorphisms of the bundle ι^{∗}T (ℍ^{n} \ {0}). Furthermore, the computations

1.25  I_{+}^{a}Ξ(λp) = −igrad μ^{a}(λp) and I_{+}^{a}Ψ = ξ_{+}^{a}

show that the subbundle M_{λp} is invariant under I_{+}^{a}. Thus we have an induced hypercomplex structure on the quotient R_{πp} and hence, on Q_{λp} which we denote by Ī_{+}^{a}(λp). Notice that on Q_{λp} the tensor field Ī_{+}^{a} depends on λ and p, but only through the combination λp. Using the natural isomorphism between the fibers of Q_{λp} and the tangent spaces of μ(p)^{−1}(0)/ℝ(λ), this provides μ(p)^{−1}(0)/ℝ(λ) with a hypercomplex structure, denoted by ¯ Ī_{+}^{a}(λp).

Theorem 1.26: Under the identifications Q_{λp} ≃ T μ(p)^{−1}(0)/ℝ(λ) ≃ T N (p) described above, the hypercomplex structures Ī_{+}^{a}(λp), ¯ Ī_{+}^{a}(λp), and I^{a}(λp) are equivalent.

Proof: First Proposition 1.23 gives a diffeomorphism μ(p)^{−1}(0)/ℝ(λ) ≃ N (p). We give the equivalence between the first and third hypercomplex structures. The hypercomplex structure {Ī_{+}^{a}}_{a=1}^{3} on Q_{λp} is that obtained by restriction and computing modulo the ideal generated by the vector field Ξ(λp)+Ψ. Definition 1.16 and the fact that the vector bundle H̃ is independent of the scale of p show that the two hypercomplex structures coincide on H̃. Then for a fixed a one easily checks that the endomorphisms Ī_{+}^{a}(p) and I^{a}(p) coincide as well on the subbundle spanned by ξ_{+}^{b}, ξ_{+}^{c} for a, b, c all different. Using the definitions, 1.25, and computing modulo the L_{λp}, we have

1.27  Ī_{+}^{a}Ξ(λp) = −Ī_{+}^{a}Ψ = −ξ_{+}^{a} = I^{a}(λp)Ξ(λp).

A similar computation on ξ_{+}^{a} concludes the proof.

§2. The Infinitesimal Automorphisms of N (p)

In this section we compute the Lie algebra 𝔥(p) of infinitesimal automorphisms of the hypercomplex Stiefel manifolds N (p) and thereby prove Theorem C of the introduction. Our proof crucially uses the scaling behavior in the parameter space C_{n}. The Lie algebra 𝔥(p) will depend on the number of equalities among the components of p ∈ C_{n}. Accordingly, we write p = (p_{1}^{m_{1}}, …, p_{k}^{m_{k}}) for some positive integer k = k(p) where the component p_{i} occurs m_{i} times, and m_{i} is called the multiplicity of p_{i}. We define the multiplicity of p by m = (m_{1}, …, m_{k}). Notice that ∑_{i=1}^{k} m_{i} = n.

Definition 2.1: Let 𝔥(p), 𝔥̄(p), and ¯ 𝔥̄(p) denote the Lie algebras of infinitesimal automorphisms of the hypercomplex structures I^{a}(p), Ī_{+}^{a}(p), and ¯ Ī_{+}^{a}(p), respectively.

Notice that elements in ¯ 𝔥̄(p) are vector fields on μ(p)^{−1}(0)/ℝ(λ), whereas elements in 𝔥̄(p) are equivalence classes of vector fields on μ(p)^{−1}(0) modulo sections of L_{λp}. Then Theorem 1.26 gives the identifications

2.2  𝔥(p) ≃ 𝔥̄(p) ≃ ¯ 𝔥̄(p).

In particular, every vector field X representing an element X̄ ∈ 𝔥̄(p) is projectible to a vector field ¯ X̄ ∈ ¯ 𝔥̄(p). It follows from the vanishing of the first prolongation 𝔤𝔩(n, ℍ)^{(1)} that the Lie algebra 𝔥(p) is finite dimensional.

Definition 2.3: Let 𝔥_{1}(p) denote the Lie algebra of vector fields on μ(p)^{−1}(0) that satisfy L_{X}I_{+}^{a}(p) = 0.

Recall the vector fields H_{j} of equation 1.8. It is easy to check that Ψ, H_{1}, ⋯, H_{n} generate an (n + 1)-dimensional abelian Lie algebra which we denote by 𝔱_{n+1}. Notice that for all p the vector field Ξ(p) lies in the subalgebra 𝔱_{n} of 𝔱_{n+1} spanned by the H_{j}.

Lemma 2.4: For each p ∈ C_{n}, 𝔱_{n+1} is a Lie subalgebra of 𝔥_{1}(p). Moreover, for each λ ∈ ℝ^{+} we have 𝔥_{1}(λp) = 𝔥_{1}(p).

Proof: The first statement follows by a straightforward computation. The second statement follows from the equalities μ(λp)^{−1}(0) = μ(p)^{−1}(0) and I_{+}^{a}(λp) = I_{+}^{a}(p).

Lemma 2.4 allows us to define linear maps π_{λ} : 𝔥_{1}(p) → 𝔥̄(λp) for each λ ∈ ℝ^{+} by sending each X ∈ 𝔥_{1}(p) to its equivalence class modulo Ψ + Ξ(λp).

We have

Proposition 2.5: For every p ∈ C_{n} and each λ ∈ ℝ^{+} there is an exact sequence of Lie algebras

0⟶{Ψ + Ξ(λp)}⟶𝔥_{1}(p)⟶^{π_{λ}} 𝔥̄(λp)⟶0.

This exact sequence splits so there are Lie algebra isomorphisms:

𝔥_{1}(p) ≃ 𝔥̄(p) ⊕ {Ψ + Ξ(p)} ≃ 𝔥(p) ⊕ {Ψ}.

Moreover, we have

(i) The two-dimensional abelian algebra spanned by Ψ and Ξ(p) is central in 𝔥_{1}(p).

(ii) 𝔥(λp) = 𝔥(p) for all λ ∈ ℝ^{+}.

Proof: The one-dimensional vector space spanned by Ψ+Ξ(λp) is clearly the kernel of π_{λ}, and since all vector fields in 𝔥_{1}(p) are projectible this vector space is a Lie algebra ideal. So the first statement will follow from the surjectivity of the map π_{λ}. We prove the surjectivity together with (ii) by considering the equivalence of the two models (N (p), I^{a}(p)) and (Q_{p}, I_{+}^{a}(p)). Let ϕ_{λp}(t) denote the flow generated by the vector field Ψ + Ξ(λp). We see that

μ(λp)^{−1}(0) = μ(p)^{−1}(0) = {ϕ_{λp}(t)(N (λp)) | t ∈ ℝ}.

This gives an isomorphism π_{λ} ∘ ϕ_{λp}(t)_{∗} : T N (λp) → Q_{λp}, and induces the equivalence between I^{a}(λp) and Ī_{+}^{a}(λp). Then we have for any vector field X on N

2.6  L_{π_{λ}∘ϕ_{λp}(t)_{∗}X} Ī_{+}^{a}(λp) = π_{λ}ϕ_{λp}(t)_{∗}L_{X}I^{a}(λp) ∘ (π_{λ}ϕ_{λp}(t)_{∗})^{−1}.

Now choose any splitting of the top horizontal exact sequence in diagram 1.24, and accordingly consider Q_{λp} as a subspace of T μ(p)^{−1}(0). Let X be any vector field on μ(p)^{−1}(0) that commutes with Ξ(p). Then ϕ_{λp}(t)_{∗}X = ψ(t)_{∗}X, where ψ(t) is the flow of the Euler field Ψ. Notice that this vector field is independent of both λ and p. Moreover, I_{+}^{a}(λp) = I_{+}^{a}(p) for all λ, so on the subspace Q_{λp} equation 2.14 lifts to

2.7  L_{ψ(t)_{∗}X}I_{+}^{a}(p) = (Ψ + Ξ(λp)) ⊗ b_{ψ(t)_{∗}X} + ϕ_{λp}(t)_{∗}L_{X}I^{a}(λp) ∘ (ϕ_{λp}(t)_{∗})^{−1},

where b_{ψ(t)_{∗}X} is some 1-form on μ(p)^{−1}(0). Now the left hand side of 2.7 is independent of λ, and the Euler field Ψ is not in the image of ϕ_{λp}(t)_{∗}. It follows that the two terms in the right hand side of equation 2.7 must both be separately independent of λ. In particular, if X ∈ 𝔥(λp) for a fixed λ it must be in 𝔥(λp) for all λ. This proves (ii).

Proving the surjectivity of π_{λ} amounts to proving that the isomorphism

π_{λ} ∘ ϕ_{λp}(t)_{∗} : 𝔥(λp) → 𝔥̄(λp)

factors through 𝔥_{1}(p). From 2.7 we have for X ∈ 𝔥(λp)

2.8  L_{ψ(t)_{∗}X}I_{+}^{a}(p) = (Ψ + Ξ(λp)) ⊗ b_{ψ(t)_{∗}X}

on Q_{λp}. The independence of λ once again implies that b_{ψ(t)_{∗}X} vanishes on Q_{λp}. We claim that it vanishes on all of T μ(p)^{−1}(0). A simple calculation using 2.8 shows that

[ψ(t)_{∗}X, Ψ] = b_{ψ(t)_{∗}X}(Ψ)I_{+}^{a}(p)(Ψ + Ξ(λp)).

Again the left hand side is independent of λ so that b_{ψ(t)_{∗}X}(Ψ) = 0. But since the vector space generated by Ψ is complementary to Q_{λp} we must have b_{ψ(t)_{∗}X} = 0, implying that ψ(t)_{∗}X ∈ 𝔥_{1}(p). Notice that this argument also implies that Ψ is central in 𝔥_{1}(p). Similarly it is not difficult to show that Ξ(p) is central in 𝔥_{1}(p) proving (i).

It is now easy to see that the map ψ(t)_{∗} ∘ (π_{λ} ∘ ϕ_{λp}(t)_{∗})^{−1} : 𝔥̄(λp) → 𝔥_{1}(p) splits the exact sequence, and the string of isomorphisms follows easily.

Proposition 2.9: Every infinitesimal hypercomplex automorphism X ∈ 𝔥(p) commutes with the vector fields ξ^{a}. In particular, every such X is foliate with respect to the foliation F_{4}(p). Furthermore, every infinitesimal hypercomplex automorphism is an infinitesimal isometry with respect to the metric h(p).

Proof: Consider the endomorphism

2.10  I^{a}(λp) − I^{a}(p) = (λ − 1)Ξ(p) ⊗ η̂^{a} − (λ^{−1} − 1)ξ̂^{a} ⊗ η̂^{0}(p).

Now for X ∈ 𝔥(p) (ii) of Proposition 2.5 implies that the Lie derivative of the left hand side of this expression with respect to X vanishes. This gives

2.11  (λ − 1)Ξ(p) ⊗ L_{X}η^{a} − (λ^{−1} − 1)([X, ξ^{a}] ⊗ η^{0} + ξ^{a} ⊗ L_{X}η^{0}) = 0.

Now one easily checks that (L_{X}η^{0})(Ξ(p)) = (L_{X}η^{a})(Ξ(p)) = 0. Thus, evaluating 2.11 on Ξ(p) proves the first statement, and then post-composing 2.11 with η^{0} and η^{a} gives

2.12  L_{X}η^{a} = L_{X}η^{0} = 0.

This in turn implies that the tensor field L_{X}h(p) is horizontal with respect to the foliation F_{4}(p). But for any sections Y, Z of H_{4}(p) we have 2h(p)(I^{a}(p)Y, Z) = dη^{a}(Y, Z). The last statement now follows from 2.12.

Corollary 2.13: The group Aut N (p) of hypercomplex automorphisms of N (p) is compact.

Proof: It follows from Proposition 2.9 Aut N (p) is a subgroup of the isometry group I(N (p), h(p)) which is compact by a well-known result of Myers and Steenrod. But it is immediate that any subgroup of a Lie transformation group that is defined by an invariance condition on a tensor field is a closed subgroup.

Lemma 2.14: If X ∈ 𝔥(p) then Xρ^{2} = 0; hence, X is a Killing vector field with respect to g(p).

Proof: The second statement follows from the first since g(p) = h(p) + (ρ^{2} − 1)(η̂^{0})^{2}, where ρ^{2} is the basic function defined in the previous section. To prove the first statement we consider the curvature form ω^{0} = dη^{0}. From 2.12 this is an invariant of the infinitesimal hypercomplex automorphisms. From this fact a direct computation shows that the 1-form

ϕ = −ρ^{4} \frac{d(ρ^{−2}(Xρ^{2}))}{2Xρ^{2}}

is closed. But it is easy to see that this implies the existence of a smooth function F depending only on ρ^{2} such that Xρ^{2} = F (ρ^{2}). Now by (ii) of Proposition 2.5 we can take X to be independent of λ and ρ^{2} is a polynomial of degree 2 in the components of p. This implies the scaling behavior F (ρ^{2}(λp)) = λ^{2}F (ρ^{2}(p)), and this implies that F (ρ^{2}) = αρ^{2} for some real constant α. Then the 1-parameter group generated by X is a dilatation of ρ^{2}. But on N (p) we have the bound ρ^{2} ≤ |p|^{2}. This gives a contradiction unless α = 0.

This lemma allows us to use the restriction of the flat metric to the surface ι : N (p) → ℍ^{n} \ {0}. On N (p) the normal bundle has a global orthonormal frame (and is thus trivial) given by (n^{0} = Ψ, n^{a} = \frac{−1}{ρ}igrad μ^{a}(p)). The dual coframe is τ^{0} = dr, τ^{a} = \frac{i}{ρ}dμ^{a}(p), where r^{2} = ∑_{i} |u_{i}|^{2}. Thus the metric in the vector bundle ι^{∗}T ℍ^{n} on N (p) obtained by restricting the flat metric to N (p) is g_{0} = g(p) + ∑_{a}(τ^{a})^{2} + (τ^{0})^{2}. We now have

Proposition 2.15: Every X ∈ 𝔥(p) is the restriction to N (p) of a Killing vector field X̂ on S^{4n−1} with respect to the canonical metric.

Proof: By Lemma 2.14 we know that any X ∈ 𝔥(p) is a Killing vector field with respect to the metric g(p) on N (p). Moreover, according to either Theorem 18 or 19 of volume IV of Spivak [Sp], if the metric g_{0}, the second fundamental forms, and the normal fundamental forms are invariant under every X ∈ 𝔥(p), then X is the restriction of an infinitesimal Euclidean motion. But since X itself is already tangent to the sphere, any extension to an infinitesimal Euclidean motion will be in 𝔰𝔬(4n). To show that the fundamental forms are invariant we prove

Lemma 2.16: If X ∈ 𝔥(p) then L_{X}n^{0} = L_{X}n^{a} = L_{X}τ^{0} = L_{X}τ^{a} = 0.

Proof: The invariance of n^{0} and τ^{0} is immediate. To see the other conditions we compute [X, −igrad μ^{a}(p)] = [X, I_{+}^{a}(p)Ξ(p)] = (L_{X}I_{+}^{a})(Ξ(p)) + [X, Ξ(p)] = 0. Then the result follows from Lemma 2.14.

Continuing with the proof of Proposition 2.15 we notice that Lemmas 2.14 and 2.17 imply that L_{X}g_{0} = 0. Thus, if we denote by ∇^{0} the connection in ι^{∗}T ℍ^{n} induced by the flat Levi-Civita connection on ℍ^{n}, then also L_{X}∇^{0} = 0. The second fundamental form is given by the projection P_{N}∇^{0} where P_{N} = ∑_{ν=0}^{3} n^{ν} ⊗ n^{ν}⌋g_{0}. Thus, Lemma 2.16 implies the invariance of the second fundamental form. Similarly, the normal fundamental forms are given by β_{a}^{b} = g_{0}(∇^{0}n^{a}, n^{b}), so again L_{X}β_{a}^{b} = 0 by Lemma 2.16.

To conclude the proof of Theorem C we see that any X ∈ 𝔥(p) generates a unique linear map on ℍ^{n} \ {0} that also satisfies L_{X}I_{+}^{a}(p) = 0. It follows that X is the restriction of an element X̂ ∈ 𝔤𝔩(n, ℍ). But we must also satisfy [X, Ξ(p)] = 0. The proof of the following lemma is standard and left to the reader

Lemma 2.17: The centralizer of Ξ(p) in 𝔤𝔩(n, ℍ) is ⨁_{i=1}^{k} 𝔤𝔩(m_{i}, ℂ).

Now Theorem C follows from 𝔰𝔬(4n) ∩ ⨁_{i=1}^{k} 𝔤𝔩(m_{i}, ℂ) = ⨁_{i=1}^{k} 𝔲(m_{i}).

Finally we remark that 𝔥(p) is clearly an invariant of the hypercomplex structure I^{a}(p). However, since 𝔥(p) depends on p only through the multiplicities m_{i} as an unordered set of natural numbers, it is a very crude invariant. We shall prove in section 4 that, for commensurable p, N (p) and N (p′) are equivalent only if p = p′, whereas Theorem C implies

Corollary 2.18: The Lie algebras 𝔥(p) and 𝔥(p′) are abstractly isomorphic if and only if k(p) = k(p′) and there is a permutation σ in the symmetric group Σ_{k} on k letters such that m′ = σm.

§3. Singularities and Leaf Holonomy

In this section we analyze the holonomy of the singular leaves of the F_{1} and F_{4} foliations on N (p). In the special case when p = (q, p, …, p) such an analysis can be found in [GL]. Here we generalize this study to arbitrary p in order to solve the hypercomplex equivalence problem in the next section.

Definition 3.1: Let q = (q_{1}, …, q_{n}) ∈ C_{n}. If q is commensurable then there is a positive number λ so that q = λp = λ(p_{1}, …, p_{n}) where p ∈ C_{n} with each p_{i} ∈ ℤ and the greatest common divisor gcd(p_{1}, …, p_{n}) = 1. We shall call this p the basic commensurable sequence associated to q. A basic commensurable sequence is called coprime if the p_{i} coordinates are pairwise relatively prime.

Given a basic commensurable sequence p = (p_{1}, …, p_{n}) ∈ C_{n} we consider ordered subsequences p^{(l)} = (p_{i_{1}}, …, p_{i_{l}}) (which are not necessarily basic), and denote the length of the subsequence by l. To each such subsequence p^{(l)} with l ≥ 2 we can associate an embedded submanifold N (p^{(l)}) ⊂ N (p) defined by setting the quaternionic coordinates u_{j} = 0 for all j ≠ i_{1}, …, i_{l}. We only consider subsequences with length l ≥ 2. As the notation indicates one can easily verify

Proposition 3.2: The submanifolds N (p^{(l)}) are hypercomplex Stiefel manifolds of real dimension 4l − 4 with hypercomplex structure I^{a}(p^{(l)}).

The submanifolds N (p^{(2)}) of length 2 will play a special role in what follows. For p^{(2)} = (p_{i}, p_{j}) we denote these submanifolds by H_{ij} and refer to them as vertices. Our next task is to compute the holonomy groups of the singular leaves of the foliations F_{4}(p) and F_{1}(p) when p is a basic commensurable sequence. Recall that these foliations were defined in section one. First we analyze the foliation F_{1}. For a leaf L of F_{1}(p) we denote its holonomy group by G_{1}(L). The circle action on ℍ^{n} generated by the vector field Ξ(p) is given by

3.3  (u_{1}, …, u_{n}) ↦ (e^{2πip_{1}t}u_{1}, …, e^{2πip_{n}t}u_{n}).

Notice that if a leaf L of F_{1} intersects one of the submanifold N (p^{(l)}) then L lies entirely in N (p^{(l)}). Thus, we have

Lemma 3.4: Let L be a leaf of F_{1} lying on the submanifold N (p^{(l)}) corresponding to the subsequence p^{(l)} = (p_{i_{1}}, …, p_{i_{l}}) of length l. Then the holonomy group of L is G_{1}(L) = ℤ_{r_{i_{1}...i_{l}}} where r_{i_{1}...i_{l}} = gcd(p_{i_{1}}, …, p_{i_{l}}).

Next we consider the holonomy groups of the foliation F_{4}. If L is a leaf of F_{4} we denote its holonomy group by G_{4}(L). Since the action of SU(2) on the quaternionic coordinates u is the diagonal action on it components, a leaf of F_{4} is either disjoint from the submanifold N (p^{(l)}) or it lies entirely in it.

Lemma 3.5: The manifolds H_{ij} are Hopf surfaces diffeomorphic to S^{1} × S^{3} and there is a covering map H_{ij} → L_{ij} where L_{ij} is a leaf of the foliation F_{4}(p). Moreover, the holonomy group G_{4}(L_{ij}) is given by

G_{4}(L_{ij}) = \{ \begin{matrix}ℤ_{\frac{p_{i}+p_{j}}{2}} & if p_{i} + p_{j} is even for all i, j; \\ ℤ_{p_{i}+p_{j}} & otherwise. \end{matrix}

The leaf L_{ij} is singular unless p_{i} = p_{j} = 1 and all other sums p_{k} + p_{m} are even. Furthermore, the holonomy group of any singular leaf L of F_{4} that lies in a submanifold N (p^{(l)}) but not in a submanifold of smaller length has the form

G_{4}(L) = \{ \begin{matrix}ℤ_{\frac{g}{2}} & if p_{i} + p_{j} is even for all i, j; \\ ℤ_{g} & otherwise. \end{matrix}

Here g is the gcd of the sequences of length at least l − 1 whose elements are of the form p_{i_{l}} ± p_{j} where j = i_{1}, …, i_{l−1}. In particular, if the sequence p^{(l)} contains both i and j and U_{ij} is a neighborhood of the vertex H_{ij} in N (p^{(l)}), then g ≤ p_{i} + p_{j} for all leaves intersecting U_{ij}.

Proof: Clearly H_{ij} is equal to N (p^{(2)}) where p^{(2)} = (p_{i}, p_{j}) which is, in turn, diffeomorphic to N (1, 1) ≃ U(2). Thus, H_{ij} is a Hopf surface. This last diffeomorphism is given by writing u_{i} = z_{i}^{1} + z_{i}^{2}j and forming the U(2) matrix

Z = ( \begin{matrix}z_{i}^{1} & z_{i}^{2} \\ z_{j}^{1} & z_{j}^{2} \end{matrix} ) .

Now the action of U(2) on N (p^{(2)}) restricts to a U(2) action on H_{ij}. Furthermore, the diffeomorphism above intertwines the action of U(2) on N (p^{(2)}) with its action on N (p). This later action is given by sending the Z ∈ U(2) to the matrix

3.6  ( \begin{matrix}e^{2πip_{i}t} & 0 \\ 0 & e^{2πip_{j}t} \end{matrix} ) Zσ,

where t ∈ [0, 1] and σ ∈ SU(2). By Lemma 3.4 the isotropy subgroup of the circle group U(1)_{p} acting on H_{ij} is ℤ_{r_{ij}}. Thus, the quotient of H_{ij} with respect to this circle action is a 3-Sasakian manifold S^{3}/Γ. We need to determine the group Γ. Since S^{3}/Γ is a homogeneous manifold with respect to the SU(2) action we can use σ ∈ SU(2) to set z_{j}^{2} = 0. The moment map equations for H_{ij} then imply that z_{i}^{1} = z_{j}^{2} = 0. The isotropy subgroup of this set of

points is the torus subgroup T^{2} obtained by setting σ = ( \begin{matrix}e^{2πis} & 0 \\ 0 & e^{−2πis} \end{matrix} ) in 3.6. Then

the isotropy subgroup Γ ⊂ T^{2} is determined by the conditions

s − p_{j}t = 0 and s + p_{i}t = 1.

This implies that Γ = ℤ_{p_{i}+p_{j}}. This is the holonomy group if SU(2) acts effectively on N (p), and then, as in the proof of Proposition 7.16 of [BGM2], this occurs only if p_{k} + p_{l} is odd for some pair (p_{k}, p_{l}) of components of p. If all of the \frac{n(n−1)}{2} sums of pairs of components of p are even, the action of SU(2) is not effective and one has an effective action of the factor group SO(3). In this case Γ = ℤ_{\frac{p_{i}+p_{j}}{2}} .

To determine the holonomy group of singular leaves L that are not covered by a vertex we argue as above and, by using a choice of σ, set z_{i_{l}}^{2} = 0 and then choose s = p_{i_{l}}t. The conditions for fixed points are

3.7  z_{j}^{1} = e^{2πi(p_{i_{l}} −p_{j})t}z_{j}^{1} and z_{j}^{2} = e^{2πi(p_{i_{l}} +p_{j})t}z_{j}^{2}

for all j = i_{1}, …, i_{l−1}. Since u does not lie on a sub-Stiefel of smaller length than l, both z_{j}^{1} and z_{j}^{2} cannot vanish. Thus, at most l − 1 of the z_{j}^{1}’s and z_{j}^{2}’s can vanish while the moment map constraints imply that not all the z_{j}^{2}’s can vanish. It follows that the order of G_{4}(L) is as stated in Lemma 3.5 where g is the gcd of a sequence of elements of the form p_{i_{l}} ± p_{j} of length at least l − 1. To prove the last statement we notice that in a neighborhood of a vertex H_{ij} in N (p^{(l)}) we must have u_{i}, u_{j} ≠ 0. As before we can set z_{j}^{2} = 0 and choose s = p_{j}t. Then since u_{i} ≠ 0, g must divide either |p_{j} − p_{i}| or p_{j} + p_{i}. In either case g ≤ p_{i} + p_{j}.

The proof of Lemma 3.5 implies that there are two types of Stiefel manifolds N (p), namely, those for which SU(2) acts effectively, and those where its ℤ_{2} factor group SO(3) acts effectively. We refer to these two type as type 1 and type 2, respectively. The type also gives the generic fibre of the orbifold fibration N (p) → O(p). For type 1 the generic fibre is S^{1} × SU(2), whereas for type 2 it is S^{1} × SO(3). As illustrated in the proof above type 2 occurs if and only if the sums p_{i} + p_{j} are even for all i, j.

Given a basic commensurable sequence p, consider the case that a component p_{i} has multiplicity m_{i} > 1. Then we let p_{i}^{(m_{i})} denote the subsequence (p_{i}, …, p_{i}) with m_{i} repetitions. We have

Proposition 3.8: If m_{i} ≥ 2 then N (p_{i}^{(m_{i})}) is a homogeneous hypercomplex Stiefel manifold of real dimension 4m_{i} − 4. In this case the holonomy group G_{1}(L) of any leaf L of the foliation F_{1}(p) lying in N (p_{i}^{(m_{i})}) is ℤ_{p_{i}} and the holonomy group G_{4}(L) of any leaf of the foliation F_{4}(p) lying in N (p_{i}^{(m_{i})}) is given by the equation

G_{4}(L) = \{ \begin{matrix}ℤ_{p_{i}} & if p_{k} + p_{j} is even for all k, j; \\ ℤ_{2p_{i}} & otherwise. \end{matrix}

Proof: It suffices to prove the homogeneity statement as the remainder of the proposition follows easily from Lemmas 3.4 and 3.5. The group Aut N (p) of hypercomplex automorphisms of N (p) is given explicitly in Theorem C, and since p contains the subsequence p_{i}^{(m_{i})}, Aut N (p) must contain a subgroup isomorphic to U(m_{i}). Furthermore, one can check that U(m_{i}) acts transitively on N (p_{i}^{(m_{i})}).

Let Σ(p) denote the singular locus of the quaternionic Kähler orbifold O(p). Since the holonomy group of any leaf of F_{4}(p) is a cyclic group, we can stratify O(p) and thus Σ(p) according to the orders of the holonomy groups G_{4}. For convenience we say that a leaf L of F_{4}(p) has order p if its holonomy group G_{4}(L) has order p. Notice that in general a stratum Σ_{p} can be disconnected. However, the stratum consisting of leaves of maximal order is special. We have

Proposition 3.9: Let Σ_{p_{n}} denote the stratum of the singular locus consisting of leaves of maximal order.

(1) If m_{n} = 1 then Σ_{p_{n}} consists of m_{n−1} isolated points and the associated leaves are Hopf surfaces S^{1} × S^{3}/ℤ_{r_{n}} that are covered by vertices. Here r_{n} = p_{n} + p_{n−1} or r_{n} = \frac{p_{n}+p_{n−1}}{2} if N (p) is type 1 or type 2, respectively.

(2) If m_{n} ≥ 2 then Σ_{p_{n}} is a homogeneous Wolf space 𝔾_{m_{n},2}(ℂ). In particular, if m_{n} = 2 then Σ_{p_{n}} = 𝔾_{2,2}(ℂ) = {pt} is an isolated point, and the leaf is a homogeneous hypercomplex Hopf surface S^{1} × S^{3} or S^{1} × SO(3) if N (p) is type 1 or type 2, respectively.

Note that the leaves of maximal order for m_{n} ≥ 2 are all Hopf surfaces, just as the generic non-singular leaf. However, they are not isometric to the generic leaf as they have non-trivial U(1) holonomy.

§4. Holonomy and the Equivalence Problem

In this section we show that an equivalence between hypercomplex structures on 𝕍_{n,2}^{ℂ} implies an equivalence between the various foliations, and then use the holonomy of the singular leaves to solve the equivalence problem for the hypercomplex structures I^{a}(p) on 𝕍_{n,2}^{ℂ} for commensurable p. The equivalence problem for the case of incommensurable p is more involved and is relegated to future work. However, the fact that I^{a}(p) and I^{a}(p′) are inequivalent when p is commensurable and p′ is incommensurable will follow immediately from our considerations below.

Definition 4.1: We say that hypercomplex structures I and Í are equivalent if there is a choice of generators I^{a} and Í^{a} for I and Í, respectively, and an F ∈ Diff(𝕍_{n,2}^{ℂ}) such that

4.2  F_{∗}I^{a} = Í^{a}F_{∗}

for all a = 1, 2, 3. Alternatively, F is a map of Clifford algebras.

Recall the definition of Sp(1)-compatible hypercomplex structures given by Definition 1.4.

Lemma 4.3: Let I and Í be two hypercomplex structures on 𝕍_{n,2}^{ℂ} that are compatible with two given Sp(1) actions. Let A and Á, respectively denote the extension of these actions by the one-parameter groups generated by the vector fields Ξ = I^{a}ξ^{a}, and Ξ́ = Í^{a}ξ́^{a}, respectively. Furthermore, suppose that I and Í are equivalent under a diffeomorphism F ∈ Diff(N). Then Á = F ∘ A.

Proof: Since every element of U(2) lies on a one parameter subgroup, it suffices to work infinitesimally. Let (ξ^{a}, Ξ) and (ξ́^{a}, Ξ́) denote the fundamental vector fields corresponding to the action A and Á, respectively. Let ad_{X} denote the global endomorphism sending any vector field Y to [X, Y]. Then we have

4.4  L_{F_{∗}ξ̂^{a}} Í^{b} = [ad_{F_{∗}ξ̂^{a}}, Í^{b}] = F_{∗}[adξ̂^{a}, I^{b}]F_{∗}^{−1} = 2ϵ^{abc}Í^{c}.

This equation implies that for each a = 1, 2, 3 the vector fields τ^{a} defined by τ^{a} = F_{∗}ξ̂^{a}−ξ́^{a} lie in the Lie algebra 𝔥′ of infinitesimal automorphisms of the hypercomplex structure Í. Thus, by Proposition 2.9, τ^{a} commutes with ξ́^{b} and one easily checks that τ^{a} span the Lie algebra 𝔰𝔲(2), viz. [τ^{a}, τ^{b}] = 2ϵ^{abc}τ^{c}. Moreover, a similar computation to that above shows that F_{∗}Ξ ∈ 𝔥′. So τ^{0} defined by τ^{0} = F_{∗}Ξ − Ξ′ must lie in 𝔥′, and it is direct to check that quadruple (τ^{a}, τ^{0}) satisfies the compatibility condition

4.5  Í^{a}τ^{b} = ϵ^{abc}τ^{c} + δ^{ab}τ^{0}.

But by Proposition 2.9, τ^{0} commutes with ξ́^{a}. So taking the Lie derivative of equation 4.5 with respect to ξ́^{e} we find that τ^{a} = τ^{0} = 0 for all a = 1, 2, 3.

This lemma says that any F ∈ Diff(N) which gives an equivalence of U(2)-compatible hypercomplex structures also gives an equivalence of the corresponding foliations. In fact F gives an equivalence of the nested sequence 1.15. Now let F and F́ denote any of the above foliations on N . Let L be a leaf of F, and let G(L) and Ǵ(F (L)) denote the holonomy groups of L and F (L), respectively. We have the following two immediate consequences of Lemma 4.3:

Corollary 4.6: Let F ∈ Diff(N) be an equivalence of U(2)-compatible hypercomplex structures I and Í. Then F induces a diffeomorphism of the leaves of F with the leaves of F́ and thus, an isomorphism of the holonomy groups G(L) ≃ Ǵ(F (L)). In particular, in accord with the notation of the previous section, if L is a leaf of F_{1} (respectively, F_{4}) then G_{1}(L) ≃ Ǵ_{1}(F (L)) (respectively, G_{4}(L) ≃ Ǵ_{4}(F (L)).

Corollary 4.7: Suppose that F : N (p) → N (p′) is an equivalence of hypercomplex manifolds. Then F induces orbifold diffeomorphisms F_{1}, F_{2}, F_{4} such that the following diagram commutes:

N (p) ⟶^{F} N (p′)
↓  ↓
S(p) ⟶^{F_{1}} S(p′)
↓  ↓
Z(p) ⟶^{F_{2}} Z(p′)
↓  ↓
O(p) ⟶^{F_{4}} O(p′).

To prove Theorem B we need the following two propositions.

Proposition 4.8: Let p, p′ ∈ C_{n} be basic commensurable sequences. Then I^{a}(p) is equivalent to I^{a}(p′) if and only if p = p′.

Proof: The if direction is obvious. Let F : N (p) → N (p′) be a diffeomorphism inducing a hypercomplex equivalence. We assume that N (p) is type 1 (the proof for type 2 is essentially identical). We first show that p_{n} = p′_{n} and then inductively show that this fact forces equality for the remaining p_{i}, p′_{i} coordinates. We begin with two cases:

Case 1: m_{n} ≥ 2 : By Proposition 3.8, N (p_{n}^{m_{n}}) is a homogeneous hypercomplex Stiefel submanifold of N (p). Moreover, the order of every leaf of the foliation F_{4}(p) through this submanifold is maximal, namely 2p_{n}. Let L be any such leaf then, by Proposition 4.6, F (L) is a leaf of the foliation F_{4}(p′) of order 2p_{n}. Thus, 2p_{n} ≤ p′_{n} + p′_{n−1}. Applying the same argument to F^{−1} shows that 2p_{n} = p′_{n} + p′_{n−1}. F (N (p_{n}^{m_{n}})) is a homogeneous submanifold of N (p′) all of whose leaves of the foliation F_{4}(p′) have maximal holonomy. The only way that this can occur is that p′_{n} have multiplicity m_{n}, and that F (N (p_{n}^{m_{n}})) = (N (ṕ_{n}^{m_{n}})). Thus, we have p′_{n} = p_{n} and m_{n} = m′_{n}.

Case 2: m_{n} = 1: By Proposition 3.9 there are m_{n−1} isolated leaves of maximal order p_{n} + p_{n−1}. As above there are precisely m_{n−1} leaves of the foliation F_{4}(p′) of order p_{n} + p_{n−1}. So p_{n} + p_{n−1} ≤ p′_{n} + p′_{n−1}. Again applying the same argument to F^{−1} implies that p_{n} + p_{n−1} = p′_{n} + p′_{n−1}, and that p′_{n−1} must occur with multiplicity m′_{n−1} = m_{n−1}.

Furthermore, each of the m_{n−1} maximal leaves F (L_{j}) of the foliation F_{4}(p′) where j = n − m_{n−1}, …, n is covered by a vertex H′_{jn}. It follows that the submanifold N (p^{m_{n−1},1}) of N (p), where p^{m_{n−1},1} = (p_{n−1}, …, p_{n−1}, p_{n}), is mapped diffeomorphically by F to the submanifold N (ṕ^{m_{n−1},1}) of N (p′). If m_{n−1} ≥ 2 then each of these submanifolds contains a homogeneous Stiefel submanifold by Proposition 3.8, namely N (p_{n−1}^{m_{n−1}}) ⊂ N (p^{m_{n−1},1}) and N (ṕ_{n−1}^{m_{n−1}}) ⊂ N (ṕ^{m_{n−1},1}). Thus, in this case, F (N (p^{m_{n−1}})) = N (ṕ_{n−1}^{m_{n−1}}) and hence that p_{n−1} = p′_{n−1}. This implies that p_{n} = p′_{n} as well.

Now assume that m_{n−1} = 1. We look at leaves of F_{4}(p) of order p_{n} + p_{n−2}. All such leaves must lie on the Stiefel submanifold N (p^{m_{n−2},1,1}) where

p^{m_{n−2},1,1} = (p_{m−2}, …, p_{m−2}, p_{n−1}, p_{n}).

There are several types of leaves on N (p^{m_{n−2},1,1}). There is a unique leaf of maximal order p_{n} + p_{n−1}, and since p_{n} + p_{n−2} cannot divide p_{n} + p_{n−1}, and p_{n} + p_{n−2} ≠ p_{i} + p_{j} for all other i, j, there are precisely m_{n−2} isolated leaves L_{j} (j = n − 1 − m_{n−2}, …, n − 2,) of order p_{n} + p_{n−2} and each is covered by a vertex H_{jn}. Now since F (H_{n−1n}) = H′_{n−1n} we see that F (L_{j}) are leaves of F_{4}(p) that of maximal order in the set of all leaves excluding the leaf L_{n−1n}. By a maximality argument similar to that used above we see that p_{n} + p_{n−2} = p′_{n} + p′_{n−2}, and that F (H_{jn}) = H′_{jn} for all j = n − 1 − m_{n−2}, …, n − 2. Hence, F (N (p^{m_{n−2},1,1})) = N (ṕ^{m_{n−2},1,1}).

If m_{n−2} ≥ 2 then Proposition 3.8 implies that p_{n−2} = p′_{n−2} and thus that p_{n} = p′_{n} and p_{n−1} = p′_{n−1}. If m_{n−2} = 1 then N (p_{n−2}, p_{n−1}, p_{n}) maps diffeomorphically by F to N (p′_{n−2}, p′_{n−1}, p′_{n}) and the vertices H_{n−1n} and H_{n−2n} map diffeomorphically to the vertices H′_{n−1n} and H′_{n−2n}, respectively. Furthermore, N (p_{n−2}, p_{n−1}, p_{n}) contains at least one leaf of order p_{n−1} + p_{n−2} covered by the vertex H_{n−1n−2} (if p_{n−1} + p_{n−2} divides p_{n} + p_{n−1} or p_{n} + p_{n−2} there are many leaves of order p_{n−1} + p_{n−2} that are not covered by vertices but this fact is not relevant to the proof). Applying Proposition 4.7 to the submanifolds N (p_{n−2}, p_{n−1}, p_{n}) and N (p′_{n−2}, p′_{n−1}, p′_{n}) we obtain a diffeomorphism of spaces of leaves, namely O(p_{n−2}, p_{n−1}, p_{n}) ≃ O(p′_{n−2}, p′_{n−1}, p′_{n}). Let S and S′ denote the suborbifolds of O(p_{n−2}, p_{n−1}, p_{n}) and O(p′_{n−2}, p′_{n−1}, p′_{n}), respectively, which are obtained by removing the leaves of orders p_{n} + p_{n−1} and p_{n} + p_{n−2} in the first case, and those of orders p′_{n} + p′_{n−1} and p′_{n} + p′_{n−2}, in the second. By applying the maximality argument to S and S′ we see that p_{n−1} + p_{n−2} = p′_{n−1} + p′_{n−2}. This implies that p_{n} = p′_{n}, p_{n−1} = p′_{n−1}, p_{n−2} = p′_{n−2}.

Thus, in all cases, p_{n} = p′_{n}. Now let j be the greatest positive integer less than n such that p_{j} ≠ p′_{j} but p_{i} = p′_{i} for all i > j. Let q = (p_{j+1}, …, p_{n}) then, by the inductive hypothesis, N (q) is a submanifold of both N (p) and N (p′). Moreover, a leaf L of maximal order in N (p) that is not in N (q) has order p_{n} + p_{j}. Similarly, a leaf of N (p′) that is not in N (q) has maximal order p′_{n} + p′_{j} = p_{n} + p′_{j}. Again, by the maximality argument, p_{j} = p′_{j}.

Next we relate any commensurable sequence to a basic commensurable sequence.

Proposition 4.9: Let p ∈ C_{n} be commensurable. For λ ∈ ℝ^{+} the hypercomplex structures I^{a}(p) and I^{a}(λp) are equivalent if and only if λ = 1.

Proof: Here assume that N (p) is of type 2 (the proof for type 1 is essentially identical). We identify the complex structure on a generic leaf of the foliation F_{2} with a point of the Teichmüller space of a Riemann surface of genus 1. To do this we choose a complex structure in the hypercomplex structure by fixing a = 1 and notice that the hyperhermitian metric h(p) restricted to a leaf of the foliation F_{2} is given by

ds^{2} = (η̂^{0})^{2} + (η̂^{1})^{2}.

The generic leaves of the foliation F_{4} are the form S^{1} × SO(3), and we can parameterize the elliptic curve E by two angles ϕ, ψ. The angle ψ is the azimuthal angle about the a = 1 axis in the group SO(3). Its range is 0 ≤ ψ ≤ 2π. To determine the range of the angle ϕ, consider the circle action generated by Ξ(p) on the coordinates u_{j} of ℍ^{n}. This action is u_{j} ↦ e^{ip_{j}ϕ}u_{j}. Since p is commensurable there is a λ ∈ ℝ^{+} such that p = λs where s is a basic commensurable sequence. Then we see that a complete circle is obtained when ϕ starts at zero and ends at \frac{2π}{λ} . Now the Teichmüller space of a genus 1 Riemann surface is the upper half plane H, and since Ξ(p) is the real part of the holomorphic vector field generating translations on E, we see that the complex structure on E corresponds to the point iλ ∈ H. Now two complex structures on a genus one Riemann surface are equivalent if and only if they differ by an element in the subgroup P SL(2, ℤ) ⊂ P SL(2, ℝ), where we have the natural action of P SL(2, ℝ) on the upper half plane H. It is easy to see that there is no element other than the identity in P SL(2, ℤ) that corresponds to a dilatation on p.

Proof of Theorem B: The first statement is now an immediate consequence of Corollary 4.7, Propositions 4.8 and 4.9, and the fact that Z(λp) = Z(p). Next, by Theorem C, the connected component of the hypercomplex automorphism group in the p = (1, …, 1) case is U(n), and the action is transitive; hence, N (λ1) is U(n)-homogeneous. For all other p’s the inhomogeneity follows from Propositions 2.5, 2.9, and the fact that the group of 3-Sasakian isometries of S(p) cannot act transitively unless p = 1 [BGM2].

We shall now briefly consider the case of incommensurable p. In this case the vector field Ξ(p) integrates to an action of ℝ rather than an S^{1} action, and the generic leaf of the foliation F_{1}(p) is not closed in N (p). Turning to the foliation F_{4}(p), one sees easily from the proof of Lemma 3.5 that the holonomy group G_{4}(L) of any leaf L contained in a vertex is isomorphic to the integers ℤ independent of the value of p_{i} + p_{j}. Hence, our method of distinguishing hypercomplex structures by the orders of leaf holonomy groups is of little use in the incommensurable case. Nevertheless, if the sequence p contains a commensurable subsequence, there will be closed ℝ orbits that are diffeomorphic to S^{1}, and these can be used to distinguish the hypercomplex structures. More generally we consider

Definition 4.10: Let p ∈ C_{n} be an arbitrary sequence. We define the integer rank (or simply rank) of p to be the number of independent constraints over the rational numbers ℚ of the form

∑_{i=1}^{n} a_{i}p_{i} = 0

that are satisfied by the components p_{i} of p with a_{i} ∈ ℤ for i = 1, …, n. The integer rank of p is denoted by rk(p).

Note that 0 ≤ rk(p) ≤ n−1 and if p is commensurable then rk(p) = n−1. We conclude this section by showing that the integer rank rk(p) is an invariant of the hypercomplex structure I^{a}(p). To begin we need to study the closures of the leaves of the foliation F_{1}(p). Let \overline{F_{1}}(p) denote the partition of N (p) by the closures of the leaves of F_{1}(p). It is known [Mol] that \overline{F_{1}}(p) is a singular Riemannian foliation of N (p). There is another important singular Riemannian foliation [Mol] on 𝕍_{n,2}^{ℂ}, namely that given by the sheaf SF discussed in the paragraph following Definition 1.11. Now H_{i} ∈ 𝔥(p) for all p ∈ C_{n}, so that SF ⊃ F_{1}(p) for all p. The leaves of SF are all tori T^{k} of variable dimension ranging from k = 2, …, n. At a generic point of N (p), that is, a point for which none of the quaternionic coordinates u_{i} vanish, any leaf is a maximal torus T^{n} of Aut N (p). The leaves of smallest dimension are T^{2}’s lying on the vertices H_{ij}. Notice from the form of the vector field Ξ(p) every leaf L(p) of the foliation F_{1}(p) is contained in precisely one leaf T^{l} of the singular foliation SF, and associated to each such leaf L(p) there is a unique subsequence p^{(l)} ⊂ p of length l such that T^{l} ⊂ N (p^{(l)}). Generally there are two extreme cases: If p is commensurable then all the leaves of F_{1}(p) are closed and \overline{F_{1}}(p) = F_{1}(p). On the other extreme if the integer rank rk(p) = 0, then p is totally incommensurable and \overline{F_{1}}(p) = SF. More precisely, we have

Lemma 4.11: Let L(p) be a leaf of F_{1}(p) lying on the toral leaf T^{l} of SF, and let p^{(l)} be the unique subsequence of p associated to L(p). Then the closure \overline{L(p)} of the leaf L(p) satisfies the equation

\overline{L(p)} = T^{l−rk(p^{(l)})}.

Furthermore, every leaf closure \overline{L(p)} is an embedded submanifold of N (p).

Proof: Suppose that L(p) lies in T^{l}. If we can show that the closure \overline{L(p)} in N (p) coincides with the closure of L(p) in T^{l}, then the result will follow by known results for tori [Mol]. Consider the sequence of inclusions

L(p) ↪ T^{l} ↪ N (p^{(l)}) ↪ N (p).

Since the right most inclusion is an embedding, it suffices to show that the inclusion T^{n} ↪ N (p) is an embedding, that is, that the maximal tori are closed, embedded submanifolds. But this follows from the form 1.7 of the action of the torus T^{n} on ℍ^{n} and the fact that N (p) ↪ ℍ^{n} is an embedding.

We are now able to prove the following

Proposition 4.12: Let F : N (p) → N (p′) be an equivalence of hypercomplex structures, then rk(p) = rk(p′).

Proof: Lemma 4.11 implies that, for any p ∈ C_{n}, the leaves of \overline{F_{1}(p)} of maximal dimension are tori of dimension n − rk(p). Without loss of generality we can assume that rk(p) ≤ rk(p′). By Lemma 4.3 F is foliation preserving, so F (L) is a leaf of F_{1}(p′). Moreover, F sends closures to closures so that \overline{F (L)} is a leaf of the singular foliation \overline{F_{1}(p′)} of dimension n − rk(p). But the leaves of \overline{F_{1}(p′)} of maximal dimension have dimension n − rk(p′) by Lemma 4.11. So n − rk(p) ≤ n − rk(p′) which implies the equality of the ranks.

So far we have not been able to completely solve the equivalence problem for the incommensurable case, although we do have several invariants including the integer rank rk(p) and the unordered set {m_{i}}_{i=1}^{k(p)} of multiplicities. We shall return to this classification problem in a future work.

§5. Some Hypercomplex Quotients of 𝕍_{n,2}^{ℂ}

In this section we consider certain quotients of the Stiefel manifolds that also carry hypercomplex structures. Our construction here can be seen as a generalization of both the Stiefel manifolds themselves and the hypercomplex structure on the trivial bundle S(p) × S^{1} noticed in [BGM2]. Let k be a positive integer and p ∈ C_{n} an integer multiple of a basic commensurable sequence. If gcd(p_{i}, p_{j}, k) = 1 for all 1 ≤ i < j ≤ n then p is called k-coprime; otherwise such p is called k-composite. Let p be a commensurable sequence, k a positive integer, and let H(p, k) = N (p) ×_{(τ_{p},k)} S^{1}, where the action map (τ_{p}, k) : S^{1} × N (p) × S^{1} ⟶ N (p) × S^{1} is given by θ_{p} on the first factor and by the standard map of degree k on the second. More precisely,

5.1  (τ_{p}, k)(θ, (n, ω)) = (θ_{p}(n), θ^{k}ω).

We set H(p, 0) = S(p) × S^{1}. The following proposition, which is direct to verify, explains why we referred to N (p) and S(p) × S^{1} as the two extreme H(p, k) subfamilies.

Proposition 5.2: For all commensurable p there is a diffeomorphism N (p) ≃ H(p, 1). Furthermore, if p is k-coprime then H(p, k) is a smooth (4n−4)-dimensional manifold with fundamental group π_{1}(H(p, k)) = ℤ_{k}, and if k = lm > 0 then there is a natural projection π(l, k) : H(p, l) ⟶ H(p, k) which is an m-fold covering space map. In particular, the universal covering space of H(p, k) is N (p). Here we are using the convention that ℤ_{1} = 0. If p is k-composite then H(p, k) is an orbifold.

Proof of Theorem D: This follows directly from Theorems 1.20, 1.22 and Proposition 5.2.

Next we have a direct corollary of Theorems B and C:

Corollary 5.3: Let p be k-coprime and q be l-coprime. Then H(p, k) is hypercomplex equivalent to H(q, l) if and only if p = q and k = l. Furthermore, the Lie algebra of infinitesimal hypercomplex automorphisms of H(p, k) is the Lie algebra 𝔥(p) given in Theorem C.

It is natural to ask how the homology of H(p, k) depends on p. For the rest of this section we will restrict our attention to the case when p is coprime. Here the θ_{p} factor in the (τ_{p}, k) action is free so we have a circle bundle S^{1}⟶H(p, k)⟶S(p) where the base space is one of the 3-Sasakian manifolds S(p) described in [BGM2]. While the topological structure of H(p, 1) ≃ N (p) is independent of p, for every n ≥ 3 the S(p) run through infinitely many distinct homotopy types. Our next Theorem computes the integral cohomology ring H^{∗}(H(p, k); ℤ).

Theorem 5.4: Let p be coprime and k a positive integer. Then, as rings,

5.5  H^{∗}(H(p, k), ℤ) ≅ (\frac{ℤ_{k}[x_{2}]}{[x_{2}^{n} = 0]} ⊗ E[y_{2n−3}, z_{2n−1}])/R(H(p, k)),

where the subscripts on x_{2}, y_{2n−3} and z_{2n−1} denote the cohomological dimension of each generator. The relations R(H(p, k)) are given by

d(p, k)x_{2}^{n−1} = d(p, k)x_{2}y_{2n−3} = x_{2}^{2}y_{2n−3} = x_{2}^{n−1}z_{2n−1} = x_{2}y_{2n−3}z_{2n−1} = 0.

Here σ_{n−1}(p) is the (n−1)^{st} elementary symmetric polynomial in the coordinates of p and d(p, k) = gcd(σ_{n−1}(p), k). The conventions here are that d(p, 0) = σ_{n−1}(p) and ℤ_{1} = 0.

Corollary 5.6: As abelian groups

H^{∗}(H(p, k); ℤ) = \{ \begin{matrix}ℤ_{k} & if ∗ = 2, 4, …, 2n − 4, \\ ℤ & if ∗ = 0, 2n − 3, 4n − 4, \\ ℤ_{d(p,k)} & if ∗ = 2n − 2, \\ ℤ ⊕ ℤ_{d(p,k)} & if ∗ = 2n − 1, \\ ℤ_{k} & if ∗ = 2n + 1, 2n + 3, …, 4n − 5, \\ 0 & otherwise. \end{matrix}

Proof of Theorem 5.4: Given the circle bundle S^{1}⟶H(p, k)⟶S(p) there is an associated complex line bundle with first Chern class c_{1}. This Chern class determines the key homomorphism in the associated Gysin sequence. Since the cohomology ring of S(p) is completely known [BGM2:Theorem E], standard techniques using the Gysin sequence and Poincare dualtity gives the result.

Notice that, with our convention that ℤ_{1} = 0, Theorem 54. recovers the classical computation of the torsion free ring H^{∗}(H(p, 1); ℤ) ≃ H^{∗}(N (p); ℤ) ≃ H^{∗}(𝕍_{n,2}^{ℂ}; ℤ) ≃ E(f_{2n−3}, f_{2n−1}). Another immediate corollary of our results is:

Corollary 5.7: For all coprime p, n ≥ 3, and k > 1 there is one cohomological invariant of H(p, k) that depends on p; namely, the integer d(p, k) which is the order of the torsion subgroups of the 2n − 2 and 2n − 1 integral cohomology groups of H(p, k).

While the fundamental group can be used to distinguish the H(p, k) for different values of k it is more interesting to look for distinct homotopy types within a fixed choice of dimension and fundamental group. On one extreme, when the fundamental group vanishes, all the examples in any fixed dimension are homeomorphic. On the other extreme, when the fundamental group is ℤ, there are infinitely many distinct homotopy types for the H(p, 0) in every dimension of the form 4n − 4 for n ≥ 3. Of course, this cohomological computation does not completely determine the homotopy type of the H(p, k) but only gives a lower bound on the number of possibly distinct homotopy types that occur. Still, Corollary 5.7 does permit us to detect some interesting distinctions for fixed k. Moreover, the computations when the fundamental group is finite cyclic are also dependent on n. For example, when n = 3, it is known that the σ_{2}(p_{1}, p_{2}, p_{3}) take on every odd integer value greater than one [BGM2]. This implies the following proposition.

Proposition 5.8: Let k = 2^{r}m with m odd. Then there are at least τ(m) distinct homotopy types for the 8-dimensional hypercomplex manifolds H(p_{1}, p_{2}, p_{3}; k). Here τ(m) is the number of positive divisors of m. Moreover, all these homotopy distinct examples have isomorphic fundamental groups.

On the other hand, since it is always the case that σ_{2}(p_{1}, p_{2}, p_{3}) is odd, Theorem 5.4 cannot be used to distinguish homotopy types in dimension 8 for even values of k when n = 3. However, when n = 4 and k = 2 the examples H(1, 2, 3, 5; 2) and H(1, 3, 5, 7; 2) are not homotopy equivalent. There are many other similar examples for other values of k and n.

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Department of Mathematics and Statistics August 1994 University of New Mexico Revised December 1994 Albuquerque, NM 87131 email: cboyer@math.unm.edu, galicki@math.unm.edu, mann@math.unm.edu