Quaternionic contact 4n + 3-manifolds and their 4n-quotients

We study some types of qc-Einstein manifolds with zero qc-scalar curvature introduced by S. Ivanov and D. Vassilev. Secondly, we shall construct a family of quaternionic Hermitian metrics (ga,{Jα}α=13)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(g_a,\{J_\alpha \}_{\alpha =1}^3)$$\end{document} on the domain Y of the standard quaternion space Hn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {H}}^n$$\end{document} one of which, say (ga,J1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(g_a,J_1)$$\end{document} is a Bochner flat Kähler metric. To do so, we deform conformally the standard quaternionic contact structure on the domain X of the quaternionic Heisenberg Lie groupM\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {M}}}$$\end{document} to obtain quaternionic Hermitian metrics on the quotient Y of X by R3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^3$$\end{document}.


Introduction
We study the quaternionic contact structure [3] (qc-structure for short) on 4n + 3-manifolds X to construct quaternionic Hermitian 4n-manifolds as their quotients. In the previous paper [2], we studied a qc-structure ( , Q) whose Im ℍ-valued (globally defined) 1-form representing satisfies that each distribution defined by d + 2 ∧ = 0 (( , , ) ∼ (1, 2, 3)) has the three-dimensional common kernel on X. ( , , Q) is called a quaternionic CR-structure (cf. [2,Definition 2.1]). It has shown in [1,2] that every positive definite quaternionic CR-structure (X, ( , Q)) induces a 3-Sasaki manifold. Then, X admits a (local) principal Sp(1)-bundle : Sp(1)→X⟶X∕Sp(1) over a quaternionic Kähler orbifold X∕Sp (1) . In particular, according to the results [8,9] of Biquard's connection [3], X is a qc-Einstein manifold with nonzero qc-scalar curvature. For the remaining case of vanishing qc-scalar curvature, there is no nondegenerate quaternionic CR-structure on X since the integrability of quaternionic CR-structure does not hold. Taking into account these results, we shall interpret a qc-Einstein manifold with vanishing qc-scalar curvature in terms of the differentiable equations of the contact forms ( = 1, 2, 3) . Given a quaternionic contact manifold X, let be the distribution on X. If has the three-dimensional kernel, then we call ( , , Q) a strict qc-structure on X.
When X is a qc-Einstein manifold with vanishing qc-scalar curvature, it follows from Lemma 6.4 [8] (also (1) of Proposition 6.3) that the Reeb fields { } =1,2,3 of are Killing and generate a (local) abelian Lie group (that is, [ , ] = 0 ), it is easy to see that = { } =1,2,3 . Thus, a qc-Einstein manifold with vanishing qc-scalar curvature is a strict qc-manifold. Conversely, if X is a strict qc-manifold, then we prove in Proposition 2.5 of Sect. 2 that generates a three-dimensional local abelian Lie group R and if R extends to a global ℝ 3 -action on X, then there is a principal bundle : ℝ 3 →X⟶X∕ℝ 3 over the hyper-Kähler manifold X∕ℝ 3 . (This holds always locally over an appropriate neighborhood of X in case R is a local qc-action.) Since X∕ℝ 3 is hyperKähler (locally in general), using the pullback by , both qc-Ricci tensor and qc-scalar curvature of X vanish by the definition (cf. [8]), so X is a qc-Einstein manifold with vanishing qc-scalar curvature. Thus, a strict quaternionic contact manifold is the same as a qc-Einstein manifold with vanishing qcscalar curvature. Indeed, we owe a lot to the referee who pointed out this equivalence in our earlier draft.
If a Lie group G admits a left invariant strict qc-structure, then G is called a strict qcgroup. An example is the quaternionic Heisenberg nilpotent Lie group M with the standard qc-structure admitting a nontrivial central extension 1→ℝ 3 → M⟶ ℍ n →1 (cf. Sect. 3.2). We construct a family of simply connected strict qc solvable Lie subgroups M(k, ) of M ⋊ T n where k + = n , T n ≤ Sp(n) , (see Sect. 3.3, cf. [4]).

Theorem A If G is a contractible unimodular strict qc-group, then G is isomorphic to M(k, ).
A 4n + 3-dimensional qc-manifold X is uniformizable (or spherical) if X is locally modeled over (PSp(n + 1, 1), S 4n+3 ) . (This is the case W qc = 0 , see [10] also.) The pair (PSp(n + 1, 1), S 4n+3 ) is obtained from projective compactification of the complete simply connected quaternionic hyperbolic space ℍ n+1 ℍ with Isom(ℍ n+1 ℍ ) = PSp(n + 1, 1). Denote by Aut qc (X) the group of qc-transformations of X. If there exists a discrete subgroup Γ ≤ Aut qc (X) acting properly with compact quotient X∕Γ , then X is said to be divisible (cf. Definition 4.3). The following result [13, Theorem 1.1] was proved for the compact case.
Theorem B Let M be a (4n + 3)-dimensional compact uniformizable strict qc-manifold. Then, M is qc-conformal to the quaternionic infranilmanifold M∕Γ (some finite cover of which is a principal T 3 -bundle over the quaternionic flat torus T n ℍ .) The following uniqueness theorem characterizes especially the noncompact case (cf. Theorem 4.4).

=1
a for some function a ∶ X→SO (3) . Then, For the difference between Theorem B and Theorem C, we remark that in Theorem B there is a T 3 -action on X∕Γ which lifts to X an ℝ 3 -action centralizing Γ , while in Theorem C X is divisible by Γ , but the intersection ℝ 3 ∩ Γ is not necessarily uniform in ℝ 3 , which does not imply to induce a T 3 -action on X∕Γ.
The paper is organized as follows In Sect. 2, we give some basic facts on strict qcstructure. The fundamental property of strict qc-manifolds is proved in Proposition 2.5 which produces hyperKähler structures on their ℝ 3 -quotients as mentioned. In Sect. 3, we review quaternionic Heisenberg nilpotent Lie group M where the group structure and qc-structure are explained explicitly. We give a nontrivial strict qc-group as a qc manifold in Theorem 3.3. From another viewpoint, we discuss strict qc manifolds in connection with spherical (uniformizable) qc geometry (PSp(n + 1, 1), S 4n+3 ) in Sect. 4. Theorem 4.4 gives a sufficient condition for a divisible group Γ of the qc-automorphism group Aut qc (X) characterizing that the quotient X∕ℝ 3 may be isometric to ℍ n as the standard hyperKähler manifold. In Sect. 5, we relax the condition strict on in order to get a quaternionic Hermitian structure (ĝ, {Ω ,Ĵ } 3 =1 ) on the quotient domain Y = X∕ℝ 3 of ℍ n . This can be achieved by the conformal change of the Im ℍ-valued one-form 0 which represents the standard qc-structure on M . We can show that one of them, say (ĝ,Ω 1 ,Ĵ 1 ) is a Kähler metric on Y. Moreover, in Sect. 6 a prominent property of this construction is that (Y,Ω 1 ,Ĵ 1 ) admits a Bochner flat Kähler structure. In particular, Y is not locally isometric to any domain of the flat space ℍ n . In Sect 7, we discuss the quaternionic isometry group Isom qH (Y,ĝ, {Ω ,Ĵ } 3 =1 ) . In course of discussion, we obtain a strictly pseudoconvex spherical pseudo-Hermitian structure {̂1, J 1 } on the (4n + 1)-quotient X∕ℝ 2 such that the pseudo-Hermitian transformation group Psh(X∕ℝ 2 ) is isomorphic to ℝ × T 2n . Theorem D is a consequence of the results of Sects. 6 and 7.1.

Strict quaternionic contact manifolds
The hypercomplex structure {J , J , J } on is defined by the following equation There is the reciprocity on : It is easy to see from (2.2)

Proposition 2.3 Suppose that R generates a global abelian group of a strict qc-manifold X.
Then, R acts properly on X as qc-transformations, that is a closed subgroup R ≤ Aut qc (X).
Proof By (1), (2) of Proposition 2.2, it follows t * = , t * •J = J •t * for any t ∈ R ( = 1, 2, 3) . Define a Riemannian metric on X by (We may choose whichever d •J from the reciprocity d 1 •J 1 = d 2 •J 2 = d 3 •J 3 .) Then, note that R ≤ Isom(X, g) ≤ Aut qc (X) . If R is the closure of R in Isom(X, g) , then it acts properly on X. Let be a vector field induced by a one-parameter subgroup of R . Then, there is a sequence of vector fields { (n) } ⊂ such that d 1 ( , A) = lim n→∞ d 1 ( (n) , A) = 0 ( ∀ A ∈ TX) by (1.1). And so ∈ . This implies R = R.
For example, if X is complete with respect to g of (2.5), then R extends to a global action of X. If a strict qc-manifold (X, , , {J } 3 =1 ) admits a global ℝ 3 -action induced by , then ℝ 3 acts properly by Proposition 2.3 and hence freely on X. There is a principal bundle over a 4n-dimensional manifold Y = X∕ℝ 3 : ℝ 3 →X ⟶Y . We will show that Y admits a hyperKähler metric. Since each t ∈ ℝ 3 satisfies J ⋅ t * = t * ⋅ J on by (2) of Proposition 2.2, ℝ 3 induces a well-defined almost complex structure Ĵ on Y such that * ⋅ J =Ĵ ⋅ * ∶ →TY at each point of X. {Ĵ } 3 =1 constitutes a quaternionic structure on Y. Define a 2-form Ω ( = 1, 2, 3) on Y to be Proposition 2.4 The 2-form Ω is a well-defined closed 2-form ( = 1, 2, 3) satisfying the following equality: In summary, we obtain the result implied in Introduction.
) be a strict qc-manifold. Let R be a local abelian group generated by the distribution . If R extends to a global action of ℝ 3 on X, then the quotient manifold Y = X∕ℝ 3 supports a hyperKähler structure (g, {Ω ,Ĵ } 3 =1 ).

Quick review of quaternionic parabolic geometry
We recall parabolic quaternionic group derived from the quaternionic hyperbolic group. The quaternionic hyperbolic space ℍ n+1 ℍ has a (projective) compactification whose boundary is diffeomorphic to S 4n+3 . The isometric action of the quaternionic hyperbolic group Isom (ℍ n+1 ℍ ) = PSp(n + 1, 1) extends to an analytic action on S 4n+3 , which we may call a quaternionic contact action on S 4n+3 . Let ∞ be the point at infinity of S 4n+3 . The standard sphere S 4n+3 with ∞ removed admits a qc-structure isomorphic to the quaternionic Heisenberg Lie group M with Aut qc (M) = M ⋊ (Sp(n) ⋅ Sp(1) × ℝ + ) . Recall the definition of M from [2]. Put t = (t 1 , t 2 , t 3 ), s = (s 1 , s 2 , s 3 ) ∈ ℝ 3 = Im ℍ , and z = t (z 1 , … , z n ), w = t (w 1 , … , w n ) ∈ ℍ n and so on. Then, M is the product ℝ 3 × ℍ n with group law: where ⟨z, w⟩ = tz w is the Hermitian inner product. M is a nilpotent Lie group such that the center is the commutator subgroup [M, M] = ℝ 3 consisting of elements (t, 0). (1) is a normal subgroup of Aut qc (M) acting properly and transitively on M in the manner of (3.1).

The qc-structure of M
The Im ℍ-valued 1-form on M is defined by The hypercomplex structure {J 1 , J 2 , J 3 } on 0 is given as in (2.1). Alternatively if ∶ M→ℍ n is the canonical projection (homomorphism), then * ∶ 0 →Tℍ n is an isomorphism at each point of M for which each J on 0 is defined by the commutative rule: =1 of the right hand side is the standard quaternionic structure {i, j, k} on ℍ n , respectively.
) is a strict qc-manifold for which generates the center ℝ 3 of M , transverse to 0 .
. The remaining follows from Proposition 2.5. Explicitly, if g ℍ is the standard quaternionic euclidean metric on ℍ n , then (3.4), There is a canonical equivariant Riemannian submersion :

Strict qc-group
) be a qc-structure on X. Put It is a subgroup of Aut qc (X) . We apply the similar constriction for Sasaki groups (cf. [4]). Let ∶ ℍ → Sp(k) be a non-trivial homomorphism (k + = n) . Define ℍ(k, ) to be the semidirect product ℍ k ⋊ ℍ which is canonically embedded to the group of hyperkähler isometries ℍ n ⋊ Sp(n) of flat quaternionic space ℍ n . Since ℍ(k, ) acts simply transitively on ℍ n , it is a flat hyperKähler group.
(ii) If the conjugate by the map a ∶ X→Sp(1) represents the matrix (a ) ∶ X→SO (3) ,
Proof Put Aut qc (X) = Aut(X) . Let G = (Aut(X)) 0 be the identity component of the closure of the holonomy image (Aut(X)) in PSp(n + 1, 1) . We first show that (i) G is not compact. Case 1. Suppose G is compact. If G has no fixed point on S 4n+3 , then G has the unique fixed point at the origin 0 in ℍ n+1 ℍ where S 4n+3 = ℍ n+1 ℍ . As in the proof of [13], dev ∶ X→S 4n+3 is shown to be an isometry, which is excluded by the non-compactness of X. So G has the fixed point set F in S 4n+3 . We may assume that Aut(X) acts properly on X by Theorem 4.1, so dev misses F. It reduces to an immersion dev ∶ X→S 4n+3 − F . As Aut(S 4n+3 − F) acts properly on S 4n+3 − F by the result of [14], there is a Riemannian metric on S 4n+3 − F invariant under Aut(S 4n+3 − F) . Since X is divisible, X is complete with respect to the pullback metric, dev ∶ X→S 4n+3 − F is a covering map. On the other hand, if we note that the action of G is linear on S 4n+3 , F must be a subsphere S k (0 ≤ k < 4n + 3) such that the complement S 4n+3 − F is unknotted, that is, homeomorphic to ℝ k+1 × S 4n+2−k . Moreover, it is shown in [13, Lemma 3.1] (also [5, p.77]) that S 4n+3 − F is either one of the following:
(2). Suppose some ( ) ∈ (Γ) has a nontrivial summand in ℝ + of Aut(M) . It follows from (3.6) that (  Taking the norm in ℍ , it follows Hence, u = 1 on X. This implies ( ) ∈ E(M) so that (Γ) ≤ E(M) . As usual, there is the E(M)-invariant Riemannian metric on M . Since X is divisible, X is complete with respect to the pullback metric. Thus, ( , dev) ∶ (Γ, X)→(E(M), M) is an equivariant isometry. As ∶ Aut qc (X)→ Aut qc (M) is an isomorphism, and R is normalized by Aut qc (X) , so does (R) in Aut qc (M) . By the action of (3.1) and the group structure of Aut qc (M) we note (R) = ℝ 3 which is the center of M . This proves (i). In particular, dev * = 0 .

.2), a calculation shows
This gives an isometry of (Y, g, . ◻

Remark 4.5
The new Kähler form Θ and Θ = Θ 1 i + Θ 2 j + Θ 3 k are related to the original forms Ω and Ω as follows. For some constant c > 0,

Quotient quaternionic Hermitian manifolds
For the strict qc-structure ( 0 , 0 , {J } 3 =1 ) on the quaternionic Heisenberg Lie group M , we consider a qc-structure = 1 i + 2 j + 3 k which is qc-conformal to . Take a one-form, say 1 to define a distribution: 1 does not induce a distribution such as . When 1 generates a three-dimensional abelian Lie group R , we shall show that there is an invariant domain X such that the quotient X∕R admits a special kind of quaternionic Hermitian structure.
Choose numbers a 1 , … , a n such that More precisely, this action is defined on M = ℝ 3 × ℍ n as (5.2) 0 < a 1 < a 2 < ⋯ < a n .
) is a quaternionic Hermitian manifold.
As in (5.4), the distribution 1 , Proof For any A ∈ TX , we prove Then, it is easy to see that . ◻