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

Abstract

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 $(g_a,\{J_\alpha \}_{\alpha =1}^3)$ on the domain Y of the standard quaternion space ${\mathbb {H}}^n$ one of which, say $(g_a,J_1)$ 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 group ${{\mathcal {M}}}$ to obtain quaternionic Hermitian metrics on the quotient Y of X by ${\mathbb {R}}^3$.

1 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 $({{\mathsf {D}}},Q)$ whose $\mathrm{Im}\, {\mathbb {H}}$-valued (globally defined) 1-form $\omega$ representing ${{\mathsf {D}}}$ satisfies that each distribution defined by $\displaystyle {\text {d}}\omega _\alpha +2\omega _\beta \wedge \omega _\gamma =0$ $((\alpha ,\beta ,\gamma )\sim (1,2,3))$ has the three-dimensional common kernel on X. $({{\mathsf {D}}},\omega , Q)$ is called a quaternionic CR-structure (cf. [2, Definition 2.1]). It has shown in [1, 2] that every positive definite quaternionic CR-structure $(X,(\omega ,Q))$ induces a 3-Sasaki manifold. Then, X admits a (local) principal $\mathrm{Sp}(1)$-bundle : $\mathrm{Sp}(1){\rightarrow }X{\longrightarrow }X/\mathrm{Sp}(1)$ over a quaternionic Kähler orbifold $X/\mathrm{Sp}(1)$. 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 $\omega _\alpha$ $(\alpha =1,2,3)$. Given a quaternionic contact manifold X, let

$$\begin{aligned} {{\mathsf {E}}}=\{\xi \in TX\ \mid {\text {d}}\omega _1(\xi ,\varvec{v})={\text {d}}\omega _2(\xi ,\varvec{v})= {\text {d}}\omega _3(\xi ,\varvec{v})=0,\, {}^\forall \, \varvec{v}\in TX\} \end{aligned}$$

(1.1)

be the distribution on X. If ${{\mathsf {E}}}$ has the three-dimensional kernel, then we call $({{\mathsf {D}}},\omega ,Q)$ 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 $\{\xi _\alpha \}_{\alpha =1,2,3}$ of $\omega$ are Killing and generate a (local) abelian Lie group (that is, $[\xi _\alpha ,\xi _\beta ]=0$), it is easy to see that ${{\mathsf {E}}}=\{\xi _\alpha \}_{\alpha =1,2,3}$. 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 ${{\mathsf {E}}}$ generates a three-dimensional local abelian Lie group ${{\mathcal {R}}}$ and if ${{\mathcal {R}}}$ extends to a global ${\mathbb {R}}^3$-action on X, then there is a principal bundle : $\displaystyle {\mathbb {R}}^3{\rightarrow }X{\mathop {{\longrightarrow }}\limits ^{{{\mathsf {p}}}}}X/{\mathbb {R}}^3$ over the hyperKähler manifold $X/{\mathbb {R}}^3$. (This holds always locally over an appropriate neighborhood of X in case ${{\mathcal {R}}}$ is a local qc-action.) Since $X/{\mathbb {R}}^3$ is hyperKähler (locally in general), using the pullback by ${{\mathsf {p}}}$, 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 qc-scalar 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 qc-group. An example is the quaternionic Heisenberg nilpotent Lie group ${{\mathcal {M}}}$ with the standard qc-structure admitting a nontrivial central extension $\displaystyle 1{\rightarrow }{\mathbb {R}}^3{\rightarrow }\, {{\mathcal {M}}}{\longrightarrow }\, {\mathbb {H}}^n{\rightarrow }1$ (cf. Sect. 3.2). We construct a family of simply connected strict qc solvable Lie subgroups ${{\mathcal {M}}}(k,\ell )$ of ${{\mathcal {M}}}\rtimes T^n$ where $k+\ell =n$, $T^n\le \mathop {\mathrm{Sp}}\nolimits (n)$, (see Sect. 3.3, cf. [4]).

Theorem A

If G is a contractible unimodular strict qc-group, then G is isomorphic to ${{\mathcal {M}}}(k,\ell )$.

A $4n+3$-dimensional qc-manifold X is uniformizable (or spherical) if X is locally modeled over $(\mathop {\mathrm{PSp}}\nolimits (n+1,1), S^{4n+3})$. (This is the case $W^{qc}=0$, see [10] also.) The pair $(\mathop {\mathrm{PSp}}\nolimits (n+1,1), S^{4n+3})$ is obtained from projective compactification of the complete simply connected quaternionic hyperbolic space ${\mathbb {H}}^{n+1}_{\mathbb {H}}$ with $\mathop {\mathrm{Isom}}\nolimits ({\mathbb {H}}^{n+1}_{\mathbb {H}})=\mathop {\mathrm{PSp}}\nolimits (n+1,1)$.

Denote by $\mathop {\mathrm{Aut}}\nolimits _{qc}(X)$ the group of qc-transformations of X. If there exists a discrete subgroup $\Gamma \le \mathop {\mathrm{Aut}}\nolimits _{qc}(X)$ acting properly with compact quotient $X/\Gamma$, 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 ${\mathcal {M}}/\Gamma$ (some finite cover of which is a principal $T^3$-bundle over the quaternionic flat torus $T^n_{\mathbb {H}}$.)

The following uniqueness theorem characterizes especially the noncompact case (cf. Theorem 4.4).

Theorem C

Let $(X,{{\mathsf {D}}},\omega ,\{J_\alpha \}_{\alpha =1}^3)$ be a noncompact simply connected uniformizable strict qc-manifold. Put ${{\mathsf {E}}}=\{\xi _1,\xi _2,\xi _3\}$. Suppose X is divisible by $\Gamma$.

(1) If $\mathop {\mathrm{Aut}}\nolimits _{qc}(X)$ leaves ${{\mathsf {E}}}$ invariant, then the developing pair reduces to the equivariant immersion:

$$\begin{aligned} (\rho ,\mathop {\mathop {\mathrm{dev}}\nolimits }):(\mathop {\mathrm{Aut}}\nolimits _{qc}(X),X){\rightarrow }(\mathop {\mathrm{Aut}}\nolimits _{qc}({{\mathcal {M}}}),{{\mathcal {M}}}). \end{aligned}$$

In addition,

(2) For any $\gamma \in \Gamma$ and $\alpha =1,2,3$, suppose $\displaystyle \gamma _*\xi _\alpha =\sum \nolimits _{\beta =1}^3a_{\alpha \beta }\xi _\beta$ for some function $a_{\alpha \beta }:X{\rightarrow }\mathrm{SO}(3)$. Then,

• (i) $\mathop {\mathrm{dev}}\nolimits : X{\rightarrow }{{\mathcal {M}}}$ is a qc-diffeomorphism so that ${{\mathcal {R}}}={\mathbb {R}}^3$.

• (ii) There exists a strict qc-structure $({{\mathsf {D}}}, \eta ,\{J_\alpha '\}_{\alpha =1}^3)$ qc-conformal to $(\omega ,\{J_\alpha \}_{\alpha =1}^3)$. The quotient $(X/{\mathbb {R}}^3, \{\Theta _\alpha ,{\hat{J}}_\alpha '\}_{\alpha =1}^3)$ is a hyperKähler manifold isometric to ${\mathbb {H}}^n$.

For the difference between Theorem B and Theorem C, we remark that in Theorem B there is a $T^3$-action on $X/\Gamma$ which lifts to X an ${\mathbb {R}}^3$-action centralizing $\Gamma$, while in Theorem C X is divisible by $\Gamma$, but the intersection ${\mathbb {R}}^3\cap \Gamma$ is not necessarily uniform in ${\mathbb {R}}^3$, which does not imply to induce a $T^3$-action on $X/\Gamma$.

The second part of this paper treats the quaternionic Hermitian quotient in place of the hyperKähler quotient. We construct a noncompact qc-manifold to obtain a quaternionic Hermitian manifold $(Y,\{\hat{\Omega }_\alpha ,{\hat{J}}_\alpha \}_{\alpha =1}^3)$ such that one of $(\hat{\Omega }_\alpha ,{\hat{J}}_\alpha )$’s is Kähler. (See Theorem 6.5, Theorem 7.2, Corollary 7.3.)

Theorem D

There exists a uniformizable noncompact qc-manifold X whose quotient by the ${\mathbb {R}}^3$-action gives a 4n-dimensional quaternionic Hermitian manifold $(Y, {\hat{g}}, \{\hat{\Omega }_\alpha , {\hat{J}}_\alpha \}_{\alpha =1}^3)$. Moreover,

• 1. $(Y, {\hat{g}} , \{\hat{\Omega }_1, {\hat{J}}_1\})$ is a Bochner flat complex Kähler manifold.

• 2. $(Y,{\hat{g}})$ is not Einstein. In particular, Y is not isometric to any domain of the quaternionic euclidean space ${\mathbb {H}}^n$.

• 3. The quaternionic Hermitian isometry group $\displaystyle \mathop {\mathrm{Isom}}\nolimits _{qH}(Y,{\hat{g}}, \{{\hat{J}}_\alpha \}_{\alpha =1}^3)$ is isomorphic to a k-torus $T^k$ for some k where $n+1\le k\le 2n$.

The paper is organized as follows In Sect. 2, we give some basic facts on strict qc-structure. The fundamental property of strict qc-manifolds is proved in Proposition 2.5 which produces hyperKähler structures on their ${\mathbb {R}}^3$-quotients as mentioned. In Sect. 3, we review quaternionic Heisenberg nilpotent Lie group ${{\mathcal {M}}}$ 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 $(\mathop {\mathrm{PSp}}\nolimits (n+1,1), S^{4n+3})$ in Sect. 4. Theorem 4.4 gives a sufficient condition for a divisible group $\Gamma$ of the qc-automorphism group $\mathop {\mathrm{Aut}}\nolimits _{qc}(X)$ characterizing that the quotient $X/{\mathbb {R}}^3$ may be isometric to ${\mathbb {H}}^n$ as the standard hyperKähler manifold. In Sect. 5, we relax the condition strict on ${{\mathsf {E}}}$ in order to get a quaternionic Hermitian structure $({\hat{g}},\{\hat{\Omega }_\alpha , {\hat{J}}_\alpha \}_{\alpha =1}^3)$ on the quotient domain $Y=X/{\mathbb {R}}^3$ of ${\mathbb {H}}^n$. This can be achieved by the conformal change of the $\mathrm{Im}\, {\mathbb {H}}$-valued one-form $\omega _0$ which represents the standard qc-structure on ${{\mathcal {M}}}$. We can show that one of them, say $({\hat{g}},{\hat{\Omega }}_1, {\hat{J}}_1)$ is a Kähler metric on Y. Moreover, in Sect. 6 a prominent property of this construction is that $(Y, {\hat{\Omega }}_1, {\hat{J}}_1)$ admits a Bochner flat Kähler structure. In particular, Y is not locally isometric to any domain of the flat space ${\mathbb {H}}^n$. In Sect 7, we discuss the quaternionic isometry group $\mathop {\mathrm{Isom}}\nolimits _{\!\, qH}(Y, {\hat{g}},\{\hat{\Omega }_\alpha , {\hat{J}}_\alpha \}_{\alpha =1}^3)$. In course of discussion, we obtain a strictly pseudoconvex spherical pseudo-Hermitian structure $\{{\hat{\omega }}_1, J_1\}$ on the $(4n+1)$-quotient $X/{\mathbb {R}}^2$ such that the pseudo-Hermitian transformation group $\mathop {\mathrm{Psh}}\nolimits (X/{\mathbb {R}}^2)$ is isomorphic to ${\mathbb {R}}\times T^{2n}$. Theorem D is a consequence of the results of Sects. 6 and 7.1.

2 Strict quaternionic contact manifolds

The hypercomplex structure $\{J_\alpha ,J_\beta ,J_\gamma \}$ on ${{\mathsf {D}}}$ is defined by the following equation $((\alpha , \beta ,\gamma ) \sim (1,2,3))$:

$$\begin{aligned} {\text {d}}\omega _\alpha (\varvec{u},\varvec{v})={\text {d}}\omega _\beta (J_\gamma \varvec{u},\varvec{v}) \ \, ({}^\forall \,\varvec{u},\varvec{v}\in {{\mathsf {D}}}). \end{aligned}$$

(2.1)

There is the reciprocity on ${{\mathsf {D}}}$:

$$\begin{aligned} \begin{aligned} {\text {d}}\omega _\alpha (J_\alpha \varvec{u},\varvec{v})={\text {d}}\omega _\beta (J_\beta \varvec{u},\varvec{v}) ={\text {d}}\omega _\gamma (J_\gamma \varvec{u},\varvec{v})\ \, ((\alpha , \beta ,\gamma ) \sim (1,2,3)). \end{aligned} \end{aligned}$$

(2.2)

It is easy to see from (2.2)

$$\begin{aligned} {\text {d}}\omega _\alpha (J_\alpha \varvec{u},J_\alpha \varvec{v})={\text {d}}\omega _\alpha (\varvec{u},\varvec{v})\ \, (\alpha =1,2,3). \end{aligned}$$

(2.3)

2.1 Strict qc-manifolds

Let $(X,D,\omega ,\{J_\alpha \}_{\alpha =1}^3)$ be a strict qc-manifold with distribution ${{\mathsf {E}}}$ (cf. (1.1)).

Lemma 2.1

${{\mathsf {E}}}$ generates a three-dimensional local abelian Lie group ${{\mathcal {R}}}$.

Proof

Since ${{\mathsf {E}}}$ is of dimension 3 and transverse to ${{\mathsf {D}}}$, it follows $\omega ({{\mathsf {E}}})=\mathrm{Im}\,{\mathbb {H}}$. There exist vector fields $\{\xi _\alpha \}_{\alpha =1,2,3}\subset {{\mathsf {E}}}$ such that

$$\begin{aligned} \omega _\alpha (\xi _\beta )=\delta _{\alpha \beta }. \end{aligned}$$

(2.4)

(Equivalently $\displaystyle \omega (\xi _\alpha )= \omega _1(\xi _\alpha )i+\omega _2(\xi _\alpha )j+ \omega _3(\xi _\alpha )k=\delta _{1\alpha }i+\delta _{2\alpha }j+\delta _{3\alpha }k$.) By (1.1), $\displaystyle 2{\text {d}}\omega _\alpha (\xi _\beta ,\xi _\gamma )=-\omega _\alpha ( [\xi _\beta ,\xi _\gamma ])=0$ and so $[\xi _\beta ,\xi _\gamma ]\in {{\mathsf {D}}}$. For any $\varvec{v}\in {{\mathsf {D}}}$, $0=2{\text {d}}\omega _\alpha (\xi _\beta ,\varvec{v})=-\omega _\alpha ([\xi _\beta ,\varvec{v}])$ $(\alpha =1,2,3)$ so $[\xi _\beta ,\varvec{v}]\in {{\mathsf {D}}}$ $((\alpha ,\beta ,\gamma )\sim (1,2,3))$. Using the Jacobi identity,

$$\begin{aligned} \begin{aligned} 2{\text {d}}\omega _\alpha ([\xi _\beta ,\xi _\gamma ],\varvec{v})&=-\omega _\alpha ([\,[\xi _\beta ,\xi _\gamma ],\varvec{v}]) =\omega _\alpha ([\,[\xi _\gamma ,\varvec{v}],\xi _\beta ])+\omega _\alpha ([\,[\varvec{v},\xi _\beta ],\xi _\gamma ])\\&=-2{\text {d}}\omega _\alpha ([\xi _\gamma ,\varvec{v}],\xi _\beta )-2{\text {d}}\omega _\alpha ([\varvec{v},\xi _\beta ],\xi _\gamma )=0.\\ \end{aligned} \end{aligned}$$

By the non-degeneracy of ${\text {d}}\omega _\alpha$ on ${{\mathsf {D}}}$, it follows $[\xi _\beta ,\xi _\gamma ]=0$ for any $\beta ,\gamma$. Thus, ${{\mathsf {E}}}=\{\xi _\alpha , \alpha =1,2,3\}$ generates a local abelian Lie group. $\square$

Proposition 2.2

Denote by ${{\mathcal {L}}}_\xi$ the Lie derivative of a vector field $\xi$ on X.

• (1) ${{\mathcal {L}}}_{\xi _\alpha }\omega _\beta =0$, ${{\mathcal {L}}}_{\xi _\alpha }{\text {d}}\omega _\beta =0$    $(\alpha ,\beta =1,2,3)$. In particular, ${{\mathcal {L}}}_{\xi _\alpha }{{\mathsf {D}}}={{\mathsf {D}}}$.

• (2) ${{\mathcal {L}}}_{\xi _\alpha }J_\beta =0$  $(\alpha ,\beta =1,2,3)$.

Proof

First note that $\displaystyle {\mathcal {L}}_{\xi _{\beta }}\omega _\alpha =({\text {d}}\iota _{\xi _{\alpha }}+\iota _{\xi _{\alpha }}d)\omega _\beta = \iota _{\xi _{\alpha }}{\text {d}}\omega _\alpha =0,\ \ {\mathcal {L}}_{\xi _{\alpha }}{\text {d}}\omega _\beta ={\text {d}}{\mathcal {L}}_{\xi _{\alpha }}\omega _\beta =0$ from (2.4), (1.1). For any $\varvec{v}\in {{\mathsf {D}}}$, $0=({\mathcal {L}}_{\xi _{\alpha }}\omega )(\varvec{v}) =-\omega ({\mathcal {L}}_{\xi _{\alpha }}\varvec{v})$ so ${\mathcal {L}}_{\xi _{\alpha }}\varvec{v}\in {{\mathsf {D}}}$. Thus, $({\mathcal {L}}_{\xi _{\alpha }}J_\beta )\varvec{v}={\mathcal {L}}_{\xi _{\alpha }}(J_\beta \varvec{v})- J_\beta ({\mathcal {L}}_{\xi _{\alpha }}\varvec{v})\in {{\mathsf {D}}}$. For ${}^\forall \,\varvec{u},\varvec{v}\in {{\mathsf {D}}}$, $\displaystyle ({\mathcal {L}}_{\xi _{\alpha }}{\text {d}}\omega _\gamma )(J_\alpha \varvec{u},\varvec{v})=0$, which equals

$$\begin{aligned}\begin{aligned} {\mathcal {L}}_{\xi _{\alpha }}({\text {d}}\omega _\gamma (J_\alpha \varvec{u},\varvec{v})) -{\text {d}}\omega _\gamma ({\mathcal {L}}_{\xi _{\alpha }}(J_\alpha \varvec{u}),\varvec{v})-{\text {d}}\omega _\gamma (J_\alpha \varvec{u},{\mathcal {L}}_{\xi _{\alpha }}\varvec{v})=0. \end{aligned}\end{aligned}$$

$$\begin{aligned} \begin{aligned} {\text {d}}\omega _\gamma (({\mathcal {L}}_{\xi _{\alpha }}J_\alpha )\varvec{u},\varvec{v})&={\text {d}}\omega _\gamma ({\mathcal {L}}_{\xi _{\alpha }}(J_\alpha \varvec{u}),\varvec{v})- {\text {d}}\omega _\gamma (J_\alpha {\mathcal {L}}_{\xi _{\alpha }}\varvec{u},\varvec{v})\\&={\mathcal {L}}_{\xi _{\alpha }}({\text {d}}\omega _\gamma (J_\alpha \varvec{u},\varvec{v}))-{\text {d}}\omega _\gamma (J_\alpha \varvec{u},{\mathcal {L}}_{\xi _{\alpha }}\varvec{v}) -{\text {d}}\omega _\gamma (J_\alpha {\mathcal {L}}_{\xi _{\alpha }}\varvec{u},\varvec{v})\\&={\mathcal {L}}_{\xi _{\alpha }}({\text {d}}\omega _\beta (\varvec{u},\varvec{v}))- {\text {d}}\omega _\beta (\varvec{u},{\mathcal {L}}_{\xi _{\alpha }}\varvec{v})-{\text {d}}\omega _\beta ({\mathcal {L}}_{\xi _{\alpha }}\varvec{u},\varvec{v})\ \, (2.1)\\&=({\mathcal {L}}_{\xi _{\alpha }}{\text {d}}\omega _\beta )(\varvec{u},\varvec{v})=0.\\ \end{aligned} \end{aligned}$$

Similarly, it follows $\displaystyle {\text {d}}\omega _\alpha (({\mathcal {L}}_{\xi _{\alpha }}J_\beta )\varvec{u},\varvec{v})=0$. By the non-degeneracy of ${\text {d}}\omega _\alpha$ on ${{\mathsf {D}}}$, it follows ${\mathcal {L}}_{\xi _{\alpha }}J_\alpha =0,\ {\mathcal {L}}_{\xi _{\alpha }}J_\beta =0$.

Let ${{\mathcal {R}}}$ denote the local abelian group obtained from Lemma 2.1.

Proposition 2.3

Suppose that ${{\mathcal {R}}}$ generates a global abelian group of a strict qc-manifold X. Then, ${{\mathcal {R}}}$ acts properly on X as qc-transformations, that is a closed subgroup ${{\mathcal {R}}}\le \mathop {\mathrm{Aut}}\nolimits _{qc}(X)$.

Proof

By (1), (2) of Proposition 2.2, it follows $\displaystyle t^*\omega _\alpha =\omega _\alpha ,\ t_*\circ J_\alpha =J_\alpha \circ t_*$ for any $t\in {{\mathcal {R}}}$ $(\alpha =1,2,3)$. Define a Riemannian metric on X by

$$\begin{aligned} g(A,B)=\sum _{i=1}^{3}\omega _i(A)\cdot \omega _i(B)+{\text {d}}\omega _1(J_1 A,B)\ \ ({}^\forall \, A,B\in TX). \end{aligned}$$

(2.5)

(We may choose whichever ${\text {d}}\omega _\alpha \circ J_\alpha$ from the reciprocity $\displaystyle {\text {d}}\omega _1\circ J_1= {\text {d}}\omega _2\circ J_2={\text {d}}\omega _3\circ J_3$.) Then, note that ${{\mathcal {R}}}\le \mathop {\mathrm{Isom}}\nolimits (X,g)\le \mathop {\mathrm{Aut}}\nolimits _{qc}(X)$. If $\displaystyle {\overline{{{\mathcal {R}}}}}$ is the closure of ${{\mathcal {R}}}$ in $\mathop {\mathrm{Isom}}\nolimits (X,g)$, then it acts properly on X. Let $\varvec{\tau }$ be a vector field induced by a one-parameter subgroup of ${\overline{{{\mathcal {R}}}}}$. Then, there is a sequence of vector fields $\{\xi ^{(n)}\}\subset {{\mathsf {E}}}$ such that $\displaystyle {\text {d}}\omega _1(\varvec{\tau },A)=\lim _{n{\rightarrow }\infty } {\text {d}}\omega _1(\xi ^{(n)},A)=0$ $({}^\forall \, A\in TX)$ by (1.1). And so $\varvec{\tau }\in {{\mathsf {E}}}$. This implies ${\overline{{{\mathcal {R}}}}}={{\mathcal {R}}}$.

For example, if X is complete with respect to g of (2.5), then ${{\mathcal {R}}}$ extends to a global action of X. If a strict qc-manifold $(X,\omega ,{{\mathsf {D}}},\{J_\alpha \}_{\alpha =1}^3)$ admits a global ${\mathbb {R}}^3$-action induced by ${{\mathsf {E}}}$, then ${\mathbb {R}}^3$ acts properly by Proposition 2.3 and hence freely on X. There is a principal bundle over a 4n-dimensional manifold $Y=X/{\mathbb {R}}^3$: $\displaystyle {\mathbb {R}}^3{\rightarrow }X{\mathop {{\longrightarrow }}\limits ^{\pi }}Y$. We will show that Y admits a hyperKähler metric. Since each $t\in {\mathbb {R}}^3$ satisfies $J_\alpha \cdot t_*=t_*\cdot J_\alpha$ on ${{\mathsf {D}}}$ by (2) of Proposition 2.2, ${\mathbb {R}}^3$ induces a well-defined almost complex structure ${\hat{J}}_\alpha$ on Y such that $\displaystyle \pi _*\cdot J_\alpha ={\hat{J}}_\alpha \cdot \pi _*: {{\mathsf {D}}}{\rightarrow }TY$ at each point of X. $\{{\hat{J}}_\alpha \}_{\alpha =1}^3$ constitutes a quaternionic structure on Y. Define a 2-form $\Omega _\alpha$ $(\alpha =1,2,3)$ on Y to be

$$\begin{aligned} \pi ^*\Omega _\alpha ={\text {d}}\omega _\alpha . \end{aligned}$$

(2.6)

Proposition 2.4

The 2-form $\Omega _\alpha$ is a well-defined closed 2-form $(\alpha =1,2,3)$ satisfying the following equality:

$$\begin{aligned} \Omega _1({\hat{J}}_1 {\hat{A}}, {\hat{B}})=\Omega _2({\hat{J}}_2{\hat{A}},{\hat{B}}) =\Omega _3({\hat{J}}_3{\hat{A}},{\hat{B}}) \, \ ({\hat{A}},{\hat{B}}\in TY). \end{aligned}$$

Moreover,

$$\begin{aligned} g({\hat{A}},{\hat{B}})=\Omega _\alpha ({\hat{J}}_\alpha {\hat{A}},{\hat{B}})\ \, (\alpha =1,2,3) \end{aligned}$$

is a hyperKähler metric on $(Y,\{{\hat{J}}_\alpha \}_{\alpha =1,2,3})$.

Proof

Let $A=V+\varvec{u},\, B=W+\varvec{v}\in TX$ $({}^\exists \,V,W\in {{\mathsf {E}}}={\mathsf {R}}^3$, ${}^\exists \,\varvec{u},\varvec{v}\in {{\mathsf {D}}})$. (Similarly, $A'=V'+\varvec{u}',\, B'=W'+\varvec{v}'$.) Suppose $\pi _*A_p=\pi _*A'_q$, $\pi _*B_p=\pi _*B'_q$. Then, $q=tp\in X$ for some $t\in {\mathbb {R}}^3$ in which $A'=t_*T+t_*A$,  $B'=t_*T'+t_*B$ $({}^\exists \, T, T'\in {{\mathsf {R}}}^3_p={{\mathsf {E}}}_p)$. Since each element t leaves $\omega _\alpha$ invariant by (1) of Proposition 2.2, (1.1) shows ${\text {d}}\omega _\alpha (A',B')={\text {d}}\omega _\alpha (A,B)$, thus (2.6) is well-defined. In particular,

$$\begin{aligned} {\text {d}}\omega _\alpha =0 \text{ on } \, Y. \end{aligned}$$

Furthermore, as $V,W\in {{\mathsf {E}}}$, $\displaystyle {\text {d}}\omega _\alpha (A,B)={\text {d}}\omega _\alpha (\varvec{u},\varvec{v})$. Since ${\hat{A}}= \pi _*A=\pi _*\varvec{u},\ {\hat{B}}= \pi _*B=\pi _* \varvec{v}$, it follows $\displaystyle {\text {d}}\omega _\alpha (J_\alpha \varvec{u},\varvec{v})=\pi ^*\Omega _\alpha (J_\alpha \varvec{u},\varvec{v}) =\Omega _\alpha ({\hat{J}}_\alpha {\hat{A}},{\hat{B}})$. Since ${\text {d}}\omega _1(J_1 \varvec{u},\varvec{v})={\text {d}}\omega _2(J_2 \varvec{u},\varvec{v})={\text {d}}\omega _3(J_3 \varvec{u},\varvec{v})$ is positive definite on ${{\mathsf {D}}}$ [cf. (2.2)], we have a positive definite 2-form on Y:

$$\begin{aligned} g({\hat{A}},{\hat{B}})=\Omega _1({\hat{J}}_1 {\hat{A}},{\hat{B}})= \Omega _2({\hat{J}}_2 {\hat{A}},{\hat{B}})=\Omega _3({\hat{J}}_3 {\hat{A}},{\hat{B}})\ \ ({\hat{A}},{\hat{B}}\in TY). \end{aligned}$$

By (2.6), $\displaystyle \Omega _\alpha ({\hat{J}}_\alpha {\hat{A}},{\hat{J}}_\alpha {\hat{B}})= \Omega _\alpha ({\hat{A}},{\hat{B}})$. It follows

$$\begin{aligned} g({\hat{J}}_\alpha {\hat{A}},{\hat{J}}_\alpha {\hat{B}})=g({\hat{A}},{\hat{B}}) \, \text{ on } \, Y\ \, (\alpha =1,2,3). \end{aligned}$$

By the definition, g is a hyperKähler metric on Y. $\square$

In summary, we obtain the result implied in Introduction.

Proposition 2.5

Let $(X,{{\mathsf {D}}}, (\omega ,\{J_\alpha \}_{\alpha =1}^3))$ be a strict qc-manifold. Let ${{\mathcal {R}}}$ be a local abelian group generated by the distribution ${{\mathsf {E}}}$. If ${{\mathcal {R}}}$ extends to a global action of ${\mathbb {R}}^3$ on X, then the quotient manifold $Y=X/{\mathbb {R}}^3$ supports a hyperKähler structure $(g, \{\Omega _\alpha ,{\hat{J}}_{\alpha }\}_{\alpha =1}^3)$.

3 Quaternionic Heisenberg Lie group ${{\mathcal {M}}}$

3.1 Quick review of quaternionic parabolic geometry

We recall parabolic quaternionic group derived from the quaternionic hyperbolic group. The quaternionic hyperbolic space ${\mathbb {H}}^{n+1}_{\mathbb {H}}$ has a (projective) compactification whose boundary is diffeomorphic to $S^{4n+3}$. The isometric action of the quaternionic hyperbolic group $\mathrm{Isom}\,({\mathbb {H}}^{n+1}_{\mathbb {H}})=\mathrm{PSp}(n+1,1)$ extends to an analytic action on $S^{4n+3}$, which we may call a quaternionic contact action on $S^{4n+3}$. Let $\infty$ be the point at infinity of $S^{4n+3}$. The standard sphere $S^{4n+3}$ with $\infty$ removed admits a qc-structure isomorphic to the quaternionic Heisenberg Lie group ${{\mathcal {M}}}$ with $\mathop {\mathrm{Aut}}\nolimits _{qc}({{\mathcal {M}}})={{\mathcal {M}}}\rtimes (\mathop {\mathrm{Sp}}\nolimits (n)\cdot \mathop {\mathrm{Sp}}\nolimits (1)\times {\mathbb {R}}^+)$. Recall the definition of ${\mathcal {M}}$ from [2]. Put $t=(t_1,t_2,t_3), s=(s_1,s_2,s_3)\in {\mathbb {R}}^3=\text{ Im }\,{\mathbb {H}}$, and $z={}^t(z_1,\ldots ,z_n), w={}^t(w_1,\ldots ,w_n)\in {\mathbb {H}}^n$ and so on. Then, ${{\mathcal {M}}}$ is the product ${\mathbb {R}}^3\times {\mathbb {H}}^n$ with group law:

$$\begin{aligned}(t,z)\cdot (s,w)=(t+s-\text{ Im }\langle z,w\rangle , z+w)\end{aligned}$$

where $\langle z,w\rangle ={}^t{\bar{z}}w$ is the Hermitian inner product. ${{\mathcal {M}}}$ is a nilpotent Lie group such that the center is the commutator subgroup $[{\mathcal {M}},{\mathcal {M}}]={\mathbb {R}}^3$ consisting of elements (t, 0).

Each element $h=\big ((t,v), \sqrt{u} A\cdot a\big )\in \mathop {\mathrm{Aut}}\nolimits _{qc}({{\mathcal {M}}})$ acts on ${{\mathcal {M}}}$ as

$$\begin{aligned} h(s,z)=(t+ u a s{\bar{a}} -\text{ Im }\langle v,\sqrt{u} Az{\bar{a}}\rangle , v+\sqrt{u} Az{{\bar{a}}})\ \, ({}^\forall \,(s,z)\in {{\mathcal {M}}}). \end{aligned}$$

(3.1)

In particular, $\displaystyle \mathrm{E}({{\mathcal {M}}})={\mathcal {M}}\rtimes \bigl (\mathrm{Sp}(n)\cdot \mathrm{Sp}(1)\bigr )$ is a normal subgroup of $\mathop {\mathrm{Aut}}\nolimits _{qc}({{\mathcal {M}}})$ acting properly and transitively on ${{\mathcal {M}}}$ in the manner of (3.1).

3.2 The qc-structure of ${{\mathcal {M}}}$

The $\mathrm{Im}\,{\mathbb {H}}$-valued 1-form on ${\mathcal {M}}$ is defined by

$$\begin{aligned} \begin{aligned} \omega _0={\text {d}}t_1i+{\text {d}}t_2j+{\text {d}}t_3k+\mathrm{Im}\langle z,{\text {d}}z\rangle . \end{aligned} \end{aligned}$$

(3.2)

Put $\displaystyle \omega _0=\omega _1i+\omega _2j+\omega _3k$. Let $\displaystyle {{{\mathsf {D}}}}_0= \mathop {\cap }_{i=1}^{3}\mathrm{ker}\, \omega _i=\mathrm{ker}\, \omega _0$ which denotes the codimension 3-subbundle on ${{\mathcal {M}}}$ satisfying $\displaystyle T{{\mathcal {M}}}={{\mathsf {D}}}_0\oplus \langle \frac{\text {d}}{{\text {d}}t_1},\frac{\text {d}}{{\text {d}}t_2}, \frac{\text {d}}{\text {d}t_3}\rangle$. Let $x_\alpha$ $(\alpha =1,\dots ,4n)$ be a real number so that ${\mathbb {R}}^{4n}$ is identified with $\displaystyle {\mathbb {H}}^n=(x_1+ix_2+jx_3+kx_4;\ \ldots \ldots ;x_{4n-3}+ix_{4n-2}+jx_{4n-3}+kx_{4n})$. A direct calculation shows

$$\begin{aligned} \begin{aligned} \omega _1&={\text {d}}t_1+(x_1{\text {d}}x_2-x_2{\text {d}}x_1)+(x_4{\text {d}}x_3-x_3{\text {d}}x_4)+ \cdots \\&\quad \cdots +(x_{4n-3}{\text {d}}x_{4n-2}-x_{4n-2}{\text {d}}x_{4n-3})+(x_{4n}{\text {d}}x_{4n-1}-x_{4n-1}{\text {d}}x_{4n}).\\ \omega _2&=dt_2+(x_1dx_3-x_3dx_1)+(x_2dx_4-x_4{\text {d}}x_2)+ \cdots \\&\quad \cdots +(x_{4n-3}{\text {d}}x_{4n-1}-x_{4n-1}{\text {d}}x_{4n-3})+(x_{4n-2}{\text {d}}x_{4n}-x_{4n}{\text {d}}x_{4n-2}).\\ \omega _3&={\text {d}}t_3+(x_1{\text {d}}x_4-x_4{\text {d}}x_1)+(x_3{\text {d}}x_2-x_2{\text {d}}x_3)+ \cdots \\&\quad \cdots +(x_{4n-3}{\text {d}}x_{4n}-x_{4n}{\text {d}}x_{4n-3})+(x_{4n-1}{\text {d}}x_{4n-2}-x_{4n-2}{\text {d}}x_{4n-1}). \end{aligned} \end{aligned}$$

(3.3)

The hypercomplex structure $\{J_1,J_2,J_3\}$ on $\displaystyle {{{\mathsf {D}}}}_0$ is given as in (2.1). Alternatively if $\pi :{{\mathcal {M}}}{\rightarrow }{\mathbb {H}}^n$ is the canonical projection (homomorphism), then $\pi _*:{{\mathsf {D}}}_0{\rightarrow }T{\mathbb {H}}^n$ is an isomorphism at each point of ${{\mathcal {M}}}$ for which each $J_\alpha$ on ${{\mathsf {D}}}_0$ is defined by the commutative rule:

$$\begin{aligned} \pi _*\circ J_\alpha = I_\alpha \circ \pi _* \end{aligned}$$

(3.4)

where $\{I_\alpha \}_{\alpha =1}^3$ of the right hand side is the standard quaternionic structure $\{i,j,k\}$ on ${\mathbb {H}}^n$, respectively.

Proposition 3.1

$({{\mathcal {M}}}, ({{\mathsf {D}}}_0, \omega _0, \{J_\alpha \}_{\alpha =1}^3))$ is a strict qc-manifold for which

• (1) $\displaystyle {{\mathsf {E}}}_0= \left\langle \frac{\text {d}}{{\text {d}t}_1},\frac{\text {d}}{{\text {d}}t_2}, \frac{\text {d}}{{\text {d}}t_3}\right\rangle$ generates the center ${\mathbb {R}}^3$ of ${{\mathcal {M}}}$, transverse to ${{\mathsf {D}}}_0$.

• (2) There is a principal bundle: $\displaystyle {\mathbb {R}}^3{\rightarrow }{{\mathcal {M}}}{\mathop {{\longrightarrow }}\limits ^{\pi }}{\mathbb {H}}^n$ whose qc-structure $(\omega _0,\{J_\alpha \}_{\alpha =1}^3)$ induces the standard hyperKähler structure on ${\mathbb {H}}^n$.

Proof

It follows $\displaystyle {\text {d}}\omega _\alpha \left( \frac{d}{dt_\beta }\right) =\delta _{\alpha \beta }$, $\displaystyle {\text {d}}\omega _\alpha \left( \frac{d}{dt_\beta },X\right) =0$ $({}^\forall \, X\in T{{\mathcal {M}}})$ by (3.3). And so $\displaystyle {{\mathsf {E}}}_0=\left\langle \frac{\text {d}}{\text {d}t_1},\frac{\text {d}}{\text {d}t_2}, \frac{\text {d}}{\text {d}t_3}\right\rangle$. The remaining follows from Proposition 2.5. Explicitly, if $g_{\mathbb {H}}$ is the standard quaternionic euclidean metric on ${\mathbb {H}}^n$, then (3.4), (3.2) show ${\text {d}}\omega _\alpha (J_\alpha X,Y)=g_{\mathbb {H}}(\pi _*X,\pi _*Y)$ $({}^\forall \, X,Y\in {{\mathsf {D}}}_0)$. $\square$

Remark 3.2

There is a canonical equivariant Riemannian submersion :

(3.5)

where $g_\omega =\mathop {\sum }_{\alpha =1}^{3}\omega _\alpha \cdot \omega _\alpha +{\text {d}}\omega _1\circ J_1$ is a Riemannian metric (cf. (2.5)). Note that this metric is not a 3-Sasaki metric globally defined on ${{\mathcal {M}}}$. Here, $\mathrm{E}({\mathbb {H}}^n)={\mathbb {H}}^n\rtimes \mathrm{Sp}(n)\cdot \mathrm{Sp}(1)$ is the quaternionic isometry group $\mathop {\mathrm{Isom}}\nolimits ({\mathbb {H}}^n,g_{\mathbb {H}})$.

From (3.1) (cf. [2]), we see that any element $h=\big ((t,v),\sqrt{u} A\cdot a\big )\in \mathop {\mathrm{Aut}}\nolimits _{qc}({{\mathcal {M}}})$ satisfies

$$\begin{aligned} h^*\omega _0=u\cdot a\omega _0 {{\bar{a}}} \end{aligned}$$

(3.6)

which thus preserves ${{\mathsf {D}}}_0$. Thus, for $h\in {{\mathcal {M}}}\rtimes \mathrm{Sp}(n)$, it follows

$$\begin{aligned} h^*\omega _0=\omega _0 \text{ on } {{\mathcal {M}}}, \ \, h_*J_\alpha =J_\alpha h_* \text{ on } {{\mathsf {D}}}_0. \end{aligned}$$

(3.7)

In particular, note that $t_*J_\alpha =J_\alpha t_*$ $({}^\forall \, t\in {\mathbb {R}}^3=C({{\mathcal {M}}}))$.

3.3 Strict qc-group

Let $({{\mathsf {D}}},\omega ,\{J_\alpha \}_{\alpha =1}^3$) be a qc-structure on X. Put

$$\begin{aligned} \mathop {\mathrm{Psh}}\nolimits _{qc}(X)=\{h\in \mathop {\mathrm{Diff}}\nolimits (X)\mid h^*\omega =\omega ,\, h_*J_\alpha =J_\alpha h_*|_{{{\mathsf {D}}}},\, \alpha =1,2,3\}. \end{aligned}$$

(3.8)

It is a subgroup of $\mathop {\mathrm{Aut}}\nolimits _{qc}(X)$. We apply the similar constriction for Sasaki groups (cf. [4]). Let $\rho :{\mathbb {H}}^\ell \rightarrow \mathrm{Sp}(k)$ be a non-trivial homomorphism $(k+\ell =n)$. Define ${\mathbb {H}}(k,\ell )$ to be the semidirect product $\displaystyle {\mathbb {H}}^k{\rtimes }_{\rho }^{}{\mathbb {H}}^\ell$ which is canonically embedded to the group of hyperkähler isometries $\displaystyle {\mathbb {H}}^n\rtimes \mathrm{Sp}(n)$ of flat quaternionic space ${\mathbb {H}}^{n}$. Since ${\mathbb {H}}(k,\ell )$ acts simply transitively on ${\mathbb {H}}^n$, it is a flat hyperKähler group. (In fact, in view of [7, Theorem II], every flat hyperKähler Lie group contained in ${\mathbb {H}}^n\rtimes \mathrm{Sp}(n)$ may be conjugate to some ${\mathbb {H}}(k,\ell )$.) Let $\mathop {\mathrm{Psh}}\nolimits _{qc}({{\mathcal {M}}})= {{\mathcal {M}}}\rtimes \mathrm{Sp}(n)$ be the normal subgroup of $E({{\mathcal {M}}})$. Take the pull-back ${{\mathcal {M}}}(k,\ell )$ of ${\mathbb {H}}(k,\ell )$ in the following central extension :

(3.9)

Here, ${{\mathcal {M}}}(n,0)={{\mathcal {M}}}$. Then, ${{\mathcal {M}}}(k,\ell )$ is a simply connected solvable Lie group acting simply transitively by qc-transformations on the strict qc-manifold ${{\mathcal {M}}}$. Thus, ${{\mathcal {M}}}(k,\ell )$ admits a strict qc structure as a Lie group.

Theorem 3.3

Let G be a contractible unimodular strict qc Lie group. Then, G is isomorphic to ${{\mathcal {M}}}(k,\ell )$.

Proof

G is viewed as a strict qc-manifold endowed with a left invariant strict qc-structure $(\omega ,\{J_\alpha \}_{\alpha =1}^3)$. Then, $G\le \mathop {\mathrm{Psh}}\nolimits _{qc}(G)$ by (3.8). If ${\mathbb {R}}^3$ is the abelian group generated by ${{\mathsf {E}}}=\langle \xi _\alpha , \alpha =1,2,3\rangle$, then ${\mathbb {R}}^3\le \mathop {\mathrm{Psh}}\nolimits _{qc}(G)$ by Proposition 2.2. Let $\mathop {\mathrm{Isom}}\nolimits _{hk}(G/{\mathbb {R}}^3)=\{h\in \mathop {\mathrm{Diff}}\nolimits (G/{\mathbb {R}}^3)\mid h^*\Omega _\alpha =\Omega _\alpha , h_*{\hat{J}}_\alpha = {\hat{J}}_\alpha h_*, \alpha =1,2,3\}$ be a subgroup of isometries of the hyperKähler manifold $G/{\mathbb {R}}^3$ as in Proposition 2.5. Denote by $\mathop {\mathrm{Isom}}\nolimits _{h}(G/{\mathbb {R}}^3)=\{h\in \mathop {\mathrm{Diff}}\nolimits (G/{\mathbb {R}}^3)\mid h^*\Omega _1=\Omega _1, h_*{\hat{J}}_1={\hat{J}}_1 h_* \}$ the holomorphic isometry group of $G/{\mathbb {R}}^3$ as a Kähler manifold. Recall that $\mathop {\mathrm{Psh}}\nolimits (G/{\mathbb {R}}^2)=\{h\in \mathop {\mathrm{Diff}}\nolimits (G/{\mathbb {R}}^2)\mid h^*\omega _1=\omega _1, h_*J_1=J_1 h_* \}$ is the group of strictly pseudoconvex pseudo-Hermitian transformations of $(G/{\mathbb {R}}^2,(\omega _1,J_1))$. There is the commutative diagram of central group extensions (cf. [4, Proposition 3.4]):

(3.10)

It follows from [4, Theorem 2] that $\mathop {\mathrm{Isom}}\nolimits _{h}(G/{\mathbb {R}}^3)^0=({\mathbb {C}}^k\rtimes \mathrm{U(k)})\times S_0$ for which $S_0$ is a semisimple Lie group of noncompact type. It acts transitively on the Kähler manifold $G/{\mathbb {R}}^3$ holomorphically isometric to the product ${\mathbb {C}}^k\times D$ where D is the bounded symmetric domain.

Consider (1)  ${\mathbb {R}}^3$ is normal in G. Then, $G/{\mathbb {R}}^3$ is a flat hyperKähler group by Hano’s theorem [7] and so $G/{\mathbb {R}}^3={\mathbb {H}}(k,\ell )$ as above. Then, the pull back of this in (3.10) gives $G={{\mathcal {M}}}(k,\ell )$. Otherwise, (2)  ${\mathbb {R}}^k$ is normal in G where $k=0,1,2$. Case (i)  If ${\mathbb {R}}^2$ is normal in G, then the quotient group ${\hat{G}} =G/{\mathbb {R}}^2$ is a Sasaki group for which $\displaystyle S^1={\mathbb {R}}/{\mathbb {Z}}{\rightarrow }{\bar{G}}={\hat{G}}/{\mathbb {Z}}{\longrightarrow }G/{\mathbb {R}}^3$ is a pseudo-Hermitian (Sasaki) bundle. As in the proof of [4, Theorem 2], $G/{\mathbb {R}}^3={\bar{G}}/S^1$ is a bounded symmetric domain so that $G/{\mathbb {R}}^3={\mathbb {H}}^2_{\mathbb {R}}$ with ${\bar{G}}=\mathop {\mathrm{PSL}}\nolimits (2,{\mathbb {R}})$. It is impossible for $G/{\mathbb {R}}^3$ to admit a quaternionic structure. Case (ii)  If ${\mathbb {R}}$ is normal in G, then put ${\hat{G}}=G/{\mathbb {R}}$. The principal bundle $\displaystyle {\mathbb {R}}^2/{\mathbb {Z}}^2 {\rightarrow }{\bar{G}}={\hat{G}}/{\mathbb {Z}}^2{\longrightarrow }G/{\mathbb {R}}^3$ becomes a principal bundle of homogeneous space $\displaystyle T^2{\rightarrow }{\bar{G}} {\longrightarrow }{\bar{G}}/T^2=G/{\mathbb {R}}^3$. As $\bar{G}/T^2$ is a bounded symmetric domain, ${\bar{G}}$ is locally isomorphic to $\mathop {\mathrm{PSL}}\nolimits (2,{\mathbb {R}})\times \mathop {\mathrm{PSL}}\nolimits (2,{\mathbb {R}})$, which is impossible since $\bar{G}/T^2=\mathop {\mathrm{PSL}}\nolimits (2,{\mathbb {R}})/S^1\times \mathop {\mathrm{PSL}}\nolimits (2,{\mathbb {R}})/S^1 ={\mathbb {H}}^2_{\mathbb {R}}\times {\mathbb {H}}^2_{\mathbb {R}}$ has a positive scalar curvature which is not hyperKähler. Finally, Case (iii) ${\bar{G}}=G/{\mathbb {Z}}^3$ and $T^3={\mathbb {R}}^3/{\mathbb {Z}}^3$ such that $G/{\mathbb {R}}^3={\bar{G}}/T^3$ is a bounded symmetric domain. Thus, ${\bar{G}}$ is locally isomorphic to $\mathop {\mathrm{PSL}}\nolimits (2,{\mathbb {R}})\times \mathop {\mathrm{PSL}}\nolimits (2,{\mathbb {R}})\times \mathop {\mathrm{PSL}}\nolimits (2,{\mathbb {R}})$ and $G/{\mathbb {R}}^3={\mathbb {H}}^2_{\mathbb {R}}\times {\mathbb {H}}^2_{\mathbb {R}}\times {\mathbb {H}}^2_{\mathbb {R}}$, which cannot be a quaternionic manifold. $\square$

4 Spherical qc-manifolds

The following theorem is a supporting example to Proposition 2.5 which is implied by Schoen’s result [14]. (Compare [6, 12] for the proofs of the quaternionic case.)

Theorem 4.1

Let $(X,{{\mathsf {D}}},\{J_\alpha \}_{\alpha =1}^3)$ be a noncompact quaternionic contact manifold. If $\mathop {\mathrm{Aut}}\nolimits _{qc}(X)$ does not act properly on X, then X admits the spherical qc-structure qc-conformal to the quaternionic Heisenberg Lie group ${{\mathcal {M}}}$.

Let $(\tilde{\omega },{{\mathsf {D}}}_0, \{I_\alpha \}_{\alpha =1}^3)$ be the standard qCR-structure on $S^{4n+3}$ such that $\ker \,\tilde{\omega }={{\mathsf {D}}}_0$ (cf. [2]).

Definition 4.2

A qc-manifold $(M,{{\mathsf {D}}}, (\omega ,\{J_\alpha \}_{\alpha =1}^3))$ is spherical (or uniformizable) with respect to $(\mathop {\mathrm{PSp}}\nolimits (n+1,1), S^{4n+3})$ if there exists a $\rho$-equivariant developing immersion from the universal covering manifold X of M:

$$\begin{aligned} (\rho ,\mathop {\mathop {\mathrm{dev}}\nolimits }):(\mathop {\mathrm{Aut}}\nolimits _{qc}(X),X){\rightarrow }(\mathop {\mathrm{PSp}}\nolimits (n+1,1), S^{4n+3}) \end{aligned}$$

such that

• (i)   $\displaystyle \mathop {\mathrm{dev}}\nolimits ^*\tilde{\omega }=\lambda \cdot \omega \cdot {\bar{\lambda }}=u\cdot a\omega {\bar{a}}$ for some map $\lambda =\sqrt{u}\cdot a:X{\rightarrow }{\mathbb {R}}^+\times {\mathop {\mathrm{Sp}}\nolimits }(1)$ $(u\in {\mathbb {R}}^+, a\in {\mathop {\mathrm{Sp}}\nolimits }(1))$.

• (ii)   If the conjugate by the map $a:X{\rightarrow }{\mathop {\mathrm{Sp}}\nolimits }(1)$ represents the matrix $(a_{\alpha \beta }): X{\rightarrow }\mathrm{SO}(3)$, then $\mathop {\mathrm{dev}}\nolimits _*\circ J_\alpha =\sum _{\beta }a_{\alpha \beta }I_\beta \circ \mathop {\mathrm{dev}}\nolimits _*: {{\mathsf {D}}}{\rightarrow }{{\mathsf {D}}}_0$.

• (iii)   $\rho :\mathop {\mathop {\mathrm{Aut}}\nolimits }_{qc}(X){\rightarrow }\mathop {\mathrm{PSp}}\nolimits (n+1,1)$ is the holonomy homomorphism such that $\displaystyle \mathop {\mathop {\mathrm{dev}}\nolimits }\circ h=\rho (h)\circ \mathop {\mathop {\mathrm{dev}}\nolimits }$   $({}^\forall \, h\in {\mathop {\mathrm{Aut}}\nolimits }_{qc}(X))$.

Definition 4.3

A qc-manifold X is divisible if there exists a discrete subgroup $\Gamma \le \mathop {\mathrm{Aut}}\nolimits _{qc}(X)$ which acts properly discontinuously with compact quotient.

Theorem 4.4

Let $(X,{{\mathsf {D}}},\omega ,\{J_\alpha \}_{\alpha =1}^3)$ be a simply connected noncompact uniformizable strict qc-manifold. Put ${{\mathsf {E}}}=\{\xi _\alpha ,\alpha =1,2,3\}$. Suppose X is divisible by $\Gamma$.

(1) If $\mathop {\mathrm{Aut}}\nolimits _{qc}(X)$ leaves ${{\mathsf {E}}}$ invariant, then the developing pair reduces to the equivariant immersion:

$$\begin{aligned} (\rho ,\mathop {\mathop {\mathrm{dev}}\nolimits }):(\mathop {\mathrm{Aut}}\nolimits _{qc}(X),X){\rightarrow }(\mathop {\mathrm{Aut}}\nolimits _{qc}({{\mathcal {M}}}),{{\mathcal {M}}}). \end{aligned}$$

(2) For any $\gamma \in \Gamma$ and $\alpha =1,2,3$, suppose $\gamma _*\xi _\alpha =\sum _{\beta =1}^3a_{\alpha \beta }\xi _\beta$ for some function $a_{\alpha \beta }:X{\rightarrow }\mathrm{SO}(3)$. Then,

• (i) $\mathop {\mathrm{dev}}\nolimits : X{\rightarrow }{{\mathcal {M}}}$ is a qc-diffeomorphism so that ${{\mathcal {R}}}={\mathbb {R}}^3$.

• (ii) There exists a strict qc-structure $\displaystyle (\eta ,\{J_\alpha '\}_{\alpha =1}^3,{\mathbb {R}}^3)$ qc-conformal to $(\omega ,\{J_\alpha \}_{\alpha =1}^3)$. The quotient $(X/{\mathbb {R}}^3, \{\Theta _\alpha ,{\hat{J}}_\alpha '\}_{\alpha =1}^3)$ is a hyperKähler manifold isometric to ${\mathbb {H}}^n$.

The method of proof is based on that of [13] by taking into account the results of [14] (cf. [6, 12]).

Proof

Put $\mathop {\mathrm{Aut}}\nolimits _{qc}(X)=\mathop {\mathrm{Aut}}\nolimits (X)$. Let $\displaystyle G={\overline{\rho (\mathop {\mathrm{Aut}}\nolimits (X))}}^0$ be the identity component of the closure of the holonomy image $\displaystyle \rho (\mathop {\mathrm{Aut}}\nolimits (X))$ in $\mathop {\mathrm{PSp}}\nolimits (n+1,1)$. 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 $\varvec{0}$ in ${\mathbb {H}}^{n+1}_{\mathbb {H}}$ where $S^{4n+3}=\partial {\mathbb {H}}^{n+1}_{\mathbb {H}}$. As in the proof of [13], $\mathop {\mathrm{dev}}\nolimits :X{\rightarrow }S^{4n+3}$ 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 $\mathop {\mathrm{Aut}}\nolimits (X)$ acts properly on X by Theorem 4.1, so $\mathop {\mathrm{dev}}\nolimits$ misses F. It reduces to an immersion $\mathop {\mathrm{dev}}\nolimits :X{\rightarrow }S^{4n+3}-F$. As $\mathop {\mathrm{Aut}}\nolimits (S^{4n+3}-F)$ acts properly on $S^{4n+3}-F$ by the result of [14], there is a Riemannian metric on $S^{4n+3}-F$ invariant under $\mathop {\mathrm{Aut}}\nolimits (S^{4n+3}-F)$. Since X is divisible, X is complete with respect to the pullback metric, $\mathop {\mathrm{dev}}\nolimits : X{\rightarrow }S^{4n+3}-F$ 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 $S^{k}$ $(0\le k<4n+3)$ such that the complement $S^{4n+3}-F$ is unknotted, that is, homeomorphic to ${\mathbb {R}}^{k+1}\times S^{4n+2-k}$. Moreover, it is shown in [13, Lemma 3.1] (also [5, p.77]) that $S^{4n+3}-F$ is either one of the following:

• (1) $S^{4n+3}-S^{m-1}$ where $F=S^{m-1}=\partial {\mathbb {H}}^{m}_{{\mathbb {R}}}$ $(1\le m\le n+1)$. ${\mathop {\mathrm{Aut}}\nolimits }(S^{4n+3}-S^{m-1})=P(\mathrm{O}(m,1)\cdot \mathrm{Sp}(1)\times \mathrm{Sp}(n-m+1))$.

• (2) $S^{4n+3}-S^{2m-1}$ where $F=S^{2m-1}=\partial {\mathbb {H}}^{m}_{{\mathbb {C}}}$ $(1\le m\le n+1)$. ${\mathop {\mathrm{Aut}}\nolimits }(S^{4n+3}-S^{2m-1})= P(\mathrm{U}(m,1)\cdot \mathrm{U}(1)\times \mathrm{Sp}(n-m+1))$.

• (3) $S^{4n+3}-S^{4m-1}$ where $F=S^{4m-1}=\partial {\mathbb {H}}^{m}_{{\mathbb {H}}}$ $(1\le m\le n)$. ${\mathop {\mathrm{Aut}}\nolimits }(S^{4n+3}-S^{4m-1})=\mathrm{Sp}(m,1)\cdot \mathrm{Sp}(n-m+1)$.

• (4) $S^{4n+3}-S^{2}$ where $S^2=\partial {\mathbb {H}}_{\mathrm{Im}}$. ${\mathop {\mathrm{Aut}}\nolimits }(S^{4n+3}-S^{2})=\mathrm{SL}(2,{\mathbb {C}})\cdot \mathrm{Sp}(n)$. (This case reduces to (3).)

In particular, $S^{4n+3}-F$ is simply connected in each case. Hence, $\mathop {\mathrm{dev}}\nolimits : X{\rightarrow }S^{4n+3}-F$ is diffeomorphic, so $\rho$ is an isomorphism. However, this case does not occur since $\mathop {\mathrm{Aut}}\nolimits (X)^0\cong G=\rho (\mathop {\mathrm{Aut}}\nolimits (X))=\mathop {\mathrm{Aut}}\nolimits (S^{4n+3}-F)^0$ which is noncompact (in fact $\mathop {\mathrm{Aut}}\nolimits (S^{4n+3}-F)^0$ contains a noncompact subgroup $\mathrm{O}(m,1),\,\mathrm{U}(m,1)$ $(1\le m\le n+1)$, $\mathop {\mathrm{SL}}\nolimits (2,{\mathbb {C}})$ or $\mathop {\mathrm{Sp}}\nolimits (m,1)$ $(1\le m\le n)$, respectively.) As a consequence, the case G is compact does not occur.

Case 2. Suppose G is noncompact. Then, either G has a common fixed point $\{\infty \}$ in $S^{4n+3}$ or G leaves invariant a totally geodesic subspace of ${\mathbb {H}}^{n+1}_{\mathbb {H}}$. (See [5, Theorem 4.4.1].) In the latter case, G (possibly $\mathop {\mathrm{PSp}}\nolimits (n+1,1)$) is either one of the identity component of the above groups (1), (2), (3), (4). On the other hand, let ${{\mathcal {R}}}$ be the local abelian group generated by ${{\mathsf {E}}}$ as before. The holonomy homomorphism $\rho$ maps ${{\mathcal {R}}}$ into $\mathop {\mathrm{PSp}}\nolimits (n+1,1)$. By Liouville’s theorem, $\rho ({{\mathcal {R}}})$ extends globally to a subgroup of $\mathop {\mathrm{PSp}}\nolimits (n+1,1)$ on $S^{4n+3}$. As $\mathop {\mathrm{Aut}}\nolimits _{qc}(X)$ leaves ${{\mathsf {E}}}$ invariant, $\mathop {\mathrm{Aut}}\nolimits (X)^0$ normalizes ${{\mathcal {R}}}$. Thus, G normalizes $\overline{\rho ({{\mathcal {R}}})}$ also. In particular, the radical of G is nontrivial, so G is not semisimple. On the other hand, G is semisimple except for the case $m=1$ of (1) such that ${\mathop {\mathrm{Aut}}\nolimits }(S^{4n+3}-S^{0})^0=\mathrm{SO}(1,1)^0\times \mathrm{Sp}(1)\cdot \mathrm{Sp}(n) \cong {\mathbb {R}}^+\times \mathrm{Sp}(n)\cdot \mathrm{Sp}(1)$. For this case, G has exactly two fixed points $\{\varvec{0},\infty \}$. Noting from Theorem 4.1, as in the argument of (i), $\displaystyle \mathop {\mathrm{dev}}\nolimits : X{\rightarrow }S^{4n+3}-\{\varvec{0},\infty \}= S^{4n+2}\times {\mathbb {R}}^+$ is a diffeomorphism so that $\rho$ maps the radical of $\mathop {\mathrm{Aut}}\nolimits (X)^0$ isomorphically onto the radical ${\mathbb {R}}^+$. Since the radical contains $\rho ({{\mathcal {R}}})$ of dimension three, it is impossible. As a consequence, by the non-ellipticity of elements in $\mathop {\mathrm{PSp}}\nolimits (n+1,1)$, G has a unique common fixed point $\{\infty \}$. Noting $\mathop {\mathrm{Aut}}\nolimits (X)$ acts properly on X, $\mathop {\mathrm{dev}}\nolimits$ misses $\{\infty \}$. This proves (1).

(2). Suppose some $\rho (\gamma )\in \rho (\Gamma )$ has a nontrivial summand in ${\mathbb {R}}^+$ of $\mathop {\mathrm{Aut}}\nolimits ({{\mathcal {M}}})$. It follows from (3.6) that $\displaystyle \rho (\gamma )^*\omega _0= v\cdot b\omega _0\bar{b}$ where $\lambda =\sqrt{v}\cdot b\in \mathop {\mathrm{Sp}}\nolimits (1)\times {\mathbb {R}}^+$. On the other hand, by our hypothesis (2), $\gamma _*\xi _\alpha =\sum _{\beta =1}^3a_{\alpha \beta }\xi _\beta$ for some function $a_{\alpha \beta }:X{\rightarrow }\mathrm{SO}(3)$. Put $\mathop {\mathrm{dev}}\nolimits _*\xi _\alpha =\tilde{\xi }_\alpha$ on ${{\mathcal {M}}}$. As $\mathop {\mathrm{dev}}\nolimits \circ \gamma =\rho (\gamma )\circ \mathop {\mathrm{dev}}\nolimits$, letting $(i,j,k)\sim (i_\alpha ,i_\beta ,i_\gamma )$, it follows

$$\begin{aligned}\rho (\gamma )_*(\tilde{\xi }_\alpha i_\alpha )=a(\tilde{\xi }_\alpha i_\alpha ){\bar{a}} \end{aligned}$$

where the conjugate of a represents the matrix $(a_{\alpha \beta })\in \mathrm{SO}(3)$. Calculate

$$\begin{aligned} \begin{aligned} \omega _0(\rho (\gamma )_*\tilde{\xi }_\alpha i_\alpha )&=a\omega _0(\tilde{\xi }_\alpha i_\alpha ){\bar{a}}\\ \rho (\gamma )^*\omega _0(\tilde{\xi }_\alpha i_\alpha )&=u\cdot b\omega _0(\tilde{\xi }_\alpha i_\alpha ){\bar{b}}. \end{aligned}\end{aligned}$$

(4.1)

Taking the norm in ${\mathbb {H}}$, it follows

$$\begin{aligned} \big |\omega _0(\tilde{\xi }_\alpha i_\alpha )\big |= \big |a\omega _0(\tilde{\xi }_\alpha i_\alpha ){\bar{a}}\big | =\big |u\cdot b\omega _0(\tilde{\xi }_\alpha i_\alpha )\bar{b}\big |=u\big |\omega _0(\tilde{\xi }_\alpha i_\alpha )\big |. \end{aligned}$$

Hence, $u=1$ on X. This implies $\rho (\gamma )\in \mathrm{E}({{\mathcal {M}}})$ so that $\rho (\Gamma )\le \mathrm{E}({{\mathcal {M}}})$. As usual, there is the $\mathrm{E}({{\mathcal {M}}})$-invariant Riemannian metric on ${{\mathcal {M}}}$. Since X is divisible, X is complete with respect to the pullback metric. Thus, $(\rho ,\mathop {\mathrm{dev}}\nolimits ):(\Gamma ,X){\rightarrow }(\mathrm{E}({{\mathcal {M}}}),{{\mathcal {M}}})$ is an equivariant isometry. As $\rho :\mathop {\mathrm{Aut}}\nolimits _{qc}(X){\rightarrow }\mathop {\mathrm{Aut}}\nolimits _{qc}({{\mathcal {M}}})$ is an isomorphism, and ${{\mathcal {R}}}$ is normalized by $\mathop {\mathrm{Aut}}\nolimits _{qc}(X)$, so does $\rho ({{\mathcal {R}}})$ in $\mathop {\mathrm{Aut}}\nolimits _{qc}({{\mathcal {M}}})$. By the action of (3.1) and the group structure of $\mathop {\mathrm{Aut}}\nolimits _{qc}({{\mathcal {M}}})$ we note $\rho ({{\mathcal {R}}})={\mathbb {R}}^3$ which is the center of ${{\mathcal {M}}}$. This proves (i). In particular, $\displaystyle \mathop {\mathrm{dev}}\nolimits _*{{\mathsf {E}}}={{\mathsf {E}}}_0$.

(ii). Let $(\omega _0,\{J_\alpha \}_{\alpha =1}^3)$ be the standard (spherical) qc-structure on ${{\mathcal {M}}}$ where $\omega _0=\omega _1i+\omega _2j+\omega _3k$. By the definition, it satisfies $\displaystyle \mathop {\mathrm{dev}}\nolimits ^*\omega _0=u\cdot a\omega {\bar{a}}$ for some $u\in {\mathbb {R}}^+, a\in {\mathop {\mathrm{Sp}}\nolimits }(1)$. When a represents $(a_{\alpha \beta })\in \mathrm{SO}(3)$ as before, it follows $\mathop {\mathrm{dev}}\nolimits _*J_\alpha =\sum _{\beta }a_{\alpha \beta }J_\beta \mathop {\mathrm{dev}}\nolimits _*$ (cf. Definition 4.2). Put

$$\begin{aligned} \begin{aligned} \eta&=\mathop {\mathrm{dev}}\nolimits ^*\omega _0 \, (\eta _\alpha =\mathop {\mathrm{dev}}\nolimits ^*\omega _\alpha ), \ J_\alpha '=\mathop {\sum }_{\beta =1}^3a_{\beta \alpha }J_\beta \,(\alpha =1,2,3). \end{aligned} \end{aligned}$$

(4.2)

We check $({{\mathsf {D}}}, \eta ,\{J_\alpha '\}_{\alpha =1}^3)$ is a qc-structure qc-conformal to $({{\mathsf {D}}}_0,\omega , \{J_\alpha \}_{\alpha =1}^3)$. For this, let $X,Y\in {{\mathsf {D}}}$ so that $\mathop {\mathrm{dev}}\nolimits _*X,\mathop {\mathrm{dev}}\nolimits _*Y\in {{\mathsf {D}}}_0$. Note that $J_\gamma \mathop {\mathrm{dev}}\nolimits _*=\mathop {\mathrm{dev}}\nolimits _*\sum _{\mu }a_{\mu \gamma }J_\mu$ as above. Let $\displaystyle {{\tilde{J}}}_\gamma =(d\eta _\beta )^{-1}\circ d\eta _\alpha$ as well as $\displaystyle J_\alpha =({\text {d}}\omega _\beta )^{-1}\circ {\text {d}}\omega _\alpha$. By calculation,

$$\begin{aligned} \begin{aligned} {\text {d}}\omega _\beta (J_\gamma (\mathop {\mathrm{dev}}\nolimits _*\varvec{u}), \mathop {\mathrm{dev}}\nolimits _*\varvec{v})&={\text {d}}\omega _\alpha (\mathop {\mathrm{dev}}\nolimits _*\varvec{u}, \mathop {\mathrm{dev}}\nolimits _*\varvec{v})= d\eta _\alpha (\varvec{u},\varvec{v})\\ ={\text {d}}\omega _\beta \bigl (\mathop {\mathrm{dev}}\nolimits _*({\sum }_{\mu }a_{\mu \gamma }J_\mu \varvec{u}&), \mathop {\mathrm{dev}}\nolimits _*\varvec{v}\bigr ) =d\eta _\beta (J'_\gamma \varvec{u}, \varvec{v}). \end{aligned} \end{aligned}$$

Noting $d\eta _\beta ({{\tilde{J}}}_\gamma \varvec{u},\varvec{v})=d\eta _\alpha (\varvec{u},\varvec{v})$, the non-degeneracy of $d\eta _\beta$ implies ${{\tilde{J}}}_\gamma =J'_\gamma$. The equations $\mathop {\mathrm{dev}}\nolimits ^*\omega _0=\eta$, $\mathop {\mathrm{dev}}\nolimits _*{{\mathsf {E}}}={{\mathsf {E}}}_0$ imply $d\eta ({{\mathsf {E}}},A)=0$ $({}^\forall \, A\in TX)$. Thus, $({{\mathsf {D}}},\eta ,\{J_\alpha '\}_{\alpha =1}^3)$ is a strict qc-structure. Noting that ${{\mathsf {R}}}$ acts properly and freely on X, we have a smooth manifold $Y=X/{{\mathsf {R}}}$. As $\mathop {\mathrm{dev}}\nolimits :X{\rightarrow }{{\mathcal {M}}}$ is $\rho$-equivariant, $\mathop {\mathrm{dev}}\nolimits$ induces a diffeomorphism $\hat{\mathop {\mathrm{dev}}\nolimits }: Y{\rightarrow }{\mathbb {H}}^n$ with the commutative diagram:

Applying Proposition 2.5 (cf. (2.6)), the 2-form $\Theta _\alpha$ on Y is defined by

$$\begin{aligned} \pi ^*\Theta _\alpha =d\eta _\alpha \ \, (\alpha =1,2,3). \end{aligned}$$

(4.3)

A quaternionic structure $\{{\hat{J}}'_\alpha \}_{\alpha =1}^3$ on Y is also induced by

$$\begin{aligned} {\hat{J}}'_\alpha \pi _*=\pi _*J'_\alpha . \end{aligned}$$

(4.4)

Using the reciprocity $\displaystyle J'_\gamma =(d\eta _\beta )^{-1}\circ d\eta _\alpha$, it follows for any ${\hat{A}},{\hat{B}}\in TY$

$$\begin{aligned} \Theta _\alpha ({\hat{J}}'_\alpha {\hat{A}}, {\hat{J}}'_\alpha {\hat{B}})=\Theta _\alpha ({\hat{A}},{\hat{B}})\ \, (\alpha =1,2,3). \end{aligned}$$

Thus, $\Theta _\alpha$ is a Kähler form on $(Y,\{J_\alpha '\}_{\alpha =1}^3)$. The quaternionic Hermitian metric

$$\begin{aligned} g({\hat{A}},{\hat{B}})=\Theta _\alpha ({\hat{J}}'_\alpha {\hat{A}},{\hat{B}})= \Theta _\beta ({\hat{J}}'_\beta {\hat{A}},{\hat{B}})= \Theta _\gamma ({\hat{J}}'_\gamma {\hat{A}},{\hat{B}}) \end{aligned}$$

(4.5)

is a hyperKähler metric on $(Y,\{{\hat{J}}_\alpha '\}_{\alpha =1}^3)$. Let $({\mathbb {H}}^n, g_{\mathbb {H}}, \{I_\alpha \}_{\alpha =1}^3)$ be the standard euclidean metric as in Remark 3.2. Noting ${\text {d}}\omega _\alpha \circ J_\alpha =\pi ^*g_{\mathbb {H}}$ and (4.2), a calculation shows

$$\begin{aligned} \begin{aligned} \hat{\mathop {\mathrm{dev}}\nolimits ^*}g_0=g, \hat{\mathop {\mathrm{dev}}\nolimits }_*\circ {\hat{J}}_\alpha '=I_\alpha \circ {\hat{\mathop {\mathrm{dev}}\nolimits }}_*. \end{aligned} \end{aligned}$$

(4.6)

This gives an isometry of $(Y, g, \{{\hat{J}}_\alpha '\}_{\alpha =1}^3)$ onto $({\mathbb {H}}^n, g_{\mathbb {H}}, \{I_\alpha \}_{\alpha =1}^3)$. $\square$

Remark 4.5

The new Kähler form $\Theta _\alpha$ and $\Theta = \Theta _1 i+\Theta _2 j+\Theta _3 k$ are related to the original forms $\Omega$ and $\Omega _\alpha$ as follows. For some constant $c>0$,

$$\begin{aligned} \Theta =c\cdot a\Omega {\bar{a}},\, \Theta _\alpha =c\cdot \sum _{\beta =1}^3a_{\alpha \beta }\Omega _\beta . \end{aligned}$$

In fact, as we put $\displaystyle \eta =\mathop {\mathrm{dev}}\nolimits ^*\omega _0=u\cdot a\omega \bar{a}$, it follows $\displaystyle {\text {d}}\eta =u\cdot a\cdot {\text {d}}\omega \cdot \bar{a}\,|_{{\mathsf {D}}}$ so that $\displaystyle \Theta ={\hat{u}}\cdot \alpha \Omega \bar{\alpha }$ where ${\hat{u}},\alpha$ are induced functions on Y. As usual, $\displaystyle \Theta ^2= {\hat{u}}^2 \Omega ^2$ which shows that ${\hat{u}}$ is a constant $c>0$. We have $\Theta _\alpha =c \sum _{\beta =1}^3\alpha _{\alpha \beta }\Omega _\beta$. $(\pi ^*\alpha =a,\ \pi ^*\alpha _{\alpha \beta }=a_{\alpha \beta })$.

5 Quotient quaternionic Hermitian manifolds

For the strict qc-structure $({{\mathsf {D}}}_0, \omega _0, \{J_\alpha \}_{\alpha =1}^3)$ on the quaternionic Heisenberg Lie group ${{\mathcal {M}}}$, we consider a qc-structure $\eta =\eta _1i+\eta _2 j+\eta _3 k$ which is qc-conformal to $\omega$. Take a one-form, say $\eta _1$ to define a distribution:

$$\begin{aligned} {{\mathsf {E}}}_1=\{\xi \mid d\eta _1(\xi ,A)=0,\ \, {}^\forall \, A\in TX\}. \end{aligned}$$

(5.1)

${{\mathsf {E}}}_1$ does not induce a distribution such as ${{\mathsf {E}}}$. When ${{\mathsf {E}}}_1$ generates a three-dimensional abelian Lie group ${{\mathcal {R}}}$, we shall show that there is an invariant domain X such that the quotient $X/{{\mathcal {R}}}$ admits a special kind of quaternionic Hermitian structure.

Choose numbers $a_1,\ldots ,a_n$ such that

$$\begin{aligned} 0<a_1<a_2<\cdots <a_n. \end{aligned}$$

(5.2)

Let $A_t$ be the diagonal matrix

$$\begin{aligned} \mathrm{diag}(\,e^{ia_1t},\,e^{ia_2t},\,\ldots \, e^{ia_nt}\,) \in T^n\le \mathop {\mathrm{Sp}}\nolimits (n). \end{aligned}$$

(5.3)

Define a homomorphism $\displaystyle \rho _1:{\mathbb {R}}^3{\rightarrow }({\mathbb {R}}^3,0)\times \mathrm{Sp}(n)\le {{\mathcal {M}}}\rtimes \mathop {\mathrm{Sp}}\nolimits (n)$ to be

$$\begin{aligned} \begin{aligned} \rho _1(t_1)&=\bigl (((t_1,0,0),0),\, A_{t_1}\bigr ),\\ \rho _1(t_2)&=\bigl (((0,t_2,0),0),\, I\bigr ),\ \ \rho _1(t_3)=\bigl (((0,0,t_3),0),\, I\bigr ). \\ \end{aligned} \end{aligned}$$

(5.4)

More precisely, this action is defined on ${{\mathcal {M}}}={\mathbb {R}}^3\times {\mathbb {H}}^n$ as

$$\begin{aligned} \begin{aligned} \rho _1(t_1)\bigl ((s_1,s_2,s_3), z_1,\dots ,z_n\bigr )&=\bigl (((s_1+t_1), s_2,s_3 ),\\&\qquad e^{ia_1t_1}z_1,\dots , e^{ia_nt_1}z_n \bigr ),\\ \rho _1(t_2)\bigl ((s_1,s_2,s_3), z_1,\dots ,z_n\bigr )&=\bigl ((s_1,(s_2+t_2),s_3), z_1,\dots ,z_n\bigr ),\\ \rho _1(t_3)\bigl ((s_1,s_2,s_3),z_1,\dots ,z_n\bigr )&=\bigl ((s_1,s_2,(s_3+t_3)),z_1,\dots ,z_n\bigr ). \end{aligned} \end{aligned}$$

(5.5)

In view of (5.4), the group $\langle \rho _1(t_1),\rho _1(t_2),\rho _1(t_3)\rangle$ forms a 3-dimensional abelian Lie group $\rho _1({\mathbb {R}}^3)$. If $\xi _1$ is the vector field induced by $\{\rho _1(t_1)\}_{t_1\in {\mathbb {R}}}$, then it follows

$$\begin{aligned} \begin{aligned} \xi _1&=\frac{\rho _1(t_1)}{{\text {d}}t_1}|_{t_1=0}=\frac{\text {d}}{{\text {d}}t_1}+ (a_1iz_1,\dots , a_n i z_n) =\ \frac{\text {d}}{{\text {d}}t_1}+\\&\sum _{k=1}^n a_k(-x_{4k-2}\frac{{\text {d}}}{{\text {d}}x_{4k-3}}+x_{4k-3}\frac{{\text {d}}}{{\text {d}}x_{4k-2}} - x_{4k} \frac{{\text {d}}}{{\text {d}}x_{4k-1}}+x_{4k-1}\frac{{\text {d}}}{{\text {d}}x_{4k}}).\\ \end{aligned}\end{aligned}$$

(5.6)

Letting $\displaystyle z_k=u_k+v_k j= (x_{4k-3}+ix_{4k-2})+(x_{4k-1}+ix_{4k})j\in {\mathbb {H}}$  $(k=1,\cdots ,n)$, a calculation using (3.3) shows that

$$\begin{aligned} \begin{aligned} \omega _1((\xi _1)_p)&=1+\sum _{k=1}^n a_k(x_{4k-3}^2+x_{4k-2}^2-x_{4k}^2-x_{4k-1}^2)\\&=1+\sum _{k=1}^n a_k(|u_k|^2-|v_k|^2)\ \, \text{ at }\ p=((s_1,s_2,s_3), z_1,\dots ,z_n). \end{aligned}\end{aligned}$$

(5.7)

The singular set $S=\{p\in {{\mathcal {M}}}\mid \omega _1((\xi _1)_p)=0\}$ is not entirely equal to ${{\mathcal {M}}}$ from (5.7). Denote the domain of ${{\mathcal {M}}}$ by

$$\begin{aligned} X=\{p\in {{\mathcal {M}}}\mid \omega _1((\xi _1)_p)\ne 0\} \end{aligned}$$

(5.8)

(if necessary taking the component containing the origin $(0,0)\in {{\mathcal {M}}}$). Since $\rho _1({\mathbb {R}}^3)$ leaves S invariant, so does X. Put $\displaystyle Y=X/\rho _1({\mathbb {R}}^3)$. Then, there is a commutative diagram of principal bundles.

(5.9)

The image $\pi _1S=\{\pi _1(p)\in {\mathbb {H}}^n\mid \sum _{k=1}^n a_k(|u_k|^2-|v_k|^2)=-1\}$ is a real hypersurface in ${\mathbb {R}}^{4n}={\mathbb {H}}^n$. Noting $Y={\mathbb {H}}^n-\pi _1S$, it follows each component of Y is simply connected in ${\mathbb {H}}^n$.

5.1 Conformal change of $\omega _0$

Let $\omega _0=\omega _1i+\omega _2j+\omega _3k$ be the qc-form on ${{\mathcal {M}}}$ (cf. Sect. 3.2). We introduce new 1-forms on X:

$$\begin{aligned} \eta _\alpha =\frac{1}{\omega _1(\xi _1)}\omega _\alpha \, (\alpha =1,2,3). \end{aligned}$$

(5.10)

Put $\displaystyle \eta =\eta _1i+\eta _2j+\eta _3k$ on X. Since $\eta$ is conformal to $\omega _0$, it follows $\ker \, \eta =\ker \, \omega _0={{\mathsf {D}}}_0$ on X. As $\rho _1({\mathbb {R}}^3)$ leaves $\omega _\alpha$ invariant, so does $\eta _\alpha$. The hypercomplex structure of $\eta$, $\displaystyle \{{{\tilde{J}}}_\gamma =(d\eta _\beta |\mathsf{D_0})^{-1}\circ (d\eta _{\alpha }|\mathsf{D_0})\}_{\gamma =1}^3$ coincides with $\{J_\gamma \}_{\gamma =1}^3$ of $\omega _0$ on ${{\mathsf {D}}}_0$. Noting $\displaystyle \rho _1(t)_*J_\alpha =J_\alpha \rho _1(t)_*$ $(t\in {\mathbb {R}}^3)$ from (3.7), $\{J_\alpha \}$ induces a quaternionic structure $\{{\hat{J}}_\alpha \}_{\alpha =1}^3$ on Y such that

$$\begin{aligned} \pi _{1*} J_\alpha (\varvec{v})={\hat{J}}_\alpha \pi _{1*} (\varvec{v})\ \, ({}^\forall \, \varvec{v}\in {{\mathsf {D}}}_0). \end{aligned}$$

(5.11)

Proposition 5.1

$(Y,\{{\hat{J}}_\alpha \}_{\alpha =1}^3)$ is a quaternionic Hermitian manifold.

Proof

Define $\hat{\Omega }_\alpha$ $(\alpha =1,2,3)$ to be

$$\begin{aligned} \hat{\Omega }_\alpha (\pi _{1*}\varvec{u},\pi _{1*}\varvec{v})=d\eta _\alpha (\varvec{u},\varvec{v}) ({}^\forall \, \varvec{u}, \varvec{v}\in {{\mathsf {D}}}_0). \end{aligned}$$

(5.12)

Since $\eta _\alpha$ is $\rho _1({\mathbb {R}}^3)$-invariant and the distribution by $\rho _1({\mathbb {R}}^3)$ is transverse to ${{\mathsf {D}}}_0$, $\hat{\Omega }_\alpha$ is well-defined on Y (cf. Lemma 5.2). Put

$$\begin{aligned} {\hat{g}}(\hat{\varvec{u}},\hat{\varvec{v}})=\hat{\Omega }_\alpha ({\hat{J}}_\alpha \hat{\varvec{u}}, \hat{\varvec{v}}) \, ({}^\forall \, \hat{\varvec{u}}, \hat{\varvec{v}}\in TY). \end{aligned}$$

(5.13)

As $\hat{\Omega }_\alpha$ is invariant under ${\hat{J}}_\alpha$, it follows $\displaystyle {\hat{g}}({\hat{J}}_\alpha \hat{\varvec{u} }, {\hat{J}}_\alpha \hat{\varvec{v}})= {\hat{g}}(\hat{\varvec{u}}, \hat{\varvec{v}})$. Thus, ${\hat{g}}$ is a quaternionic Hermitian metric on $(Y,\{{\hat{J}}_\alpha \}_{\alpha =1}^3)$.

As in (5.4), the distribution $\displaystyle \left\{ \xi _1,\frac{\text {d}}{{\text {d}}t_2},\frac{\text {d}}{{\text {d}}t_3}\right\}$ generates $\rho _1({\mathbb {R}}^3)\le {{\mathcal {M}}}\rtimes \mathop {\mathrm{Sp}}\nolimits (n)$. Note from (3.3) that

$$\begin{aligned} \eta _1(\xi _1)=1, \eta _1(\frac{\text {d}}{{\text {d}}t_2})=0, \, \eta _1(\frac{\text {d}}{{\text {d}}t_3})=0. \end{aligned}$$

(5.14)

Lemma 5.2

$\displaystyle {{\mathsf {E}}}_1=\langle \xi _1, \frac{\text {d}}{{\text {d}}t_2},\frac{\text {d}}{{\text {d}}t_3}\rangle$.

Proof

For any $A\in TX$, we prove $\displaystyle {\text {d}}\eta _1(\xi _1, A)=0,\, \ {\text {d}}\eta _1(\frac{\text {d}}{{\text {d}}t_\beta }, A)=0$ $(\beta =2, 3)$. If $A\in {{\mathsf {D}}}_0$, then $[\xi _1,A]\in {{\mathsf {D}}}_0$. Since the distribution $\displaystyle \langle \xi _1, \frac{\text {d}}{{\text {d}}t_2},\frac{\text {d}}{{\text {d}}t_3}\rangle$ generates ${\mathbb {R}}^3$, $\displaystyle [\xi _1,\frac{\text {d}}{{\text {d}}t_\beta }]=0$ $(\beta =2,3)$. Then, it is easy to see that $\displaystyle d\eta _1(\xi _1,A)=d\eta _1(\xi _1, \frac{\text {d}}{{\text {d}}t_\beta })=0$. As $\displaystyle \frac{\text {d}}{{\text {d}}t_\beta }$ $(\beta =2,3)$ are induced from the central subgroup $(0,{\mathbb {R}}^2)$ of ${\mathbb {R}}^3$, (5.14) shows $\displaystyle {\text {d}}\eta _1(\frac{\text {d}}{{\text {d}}t_\beta }, B)=0$, $\displaystyle d\eta _1(\frac{\text {d}}{{\text {d}}t_\beta },\frac{\text {d}}{{\text {d}}t_\gamma })=0$ $({}^\forall \, B\in {{\mathsf {D}}}_0)$. $\square$

Lemma 5.3

Let $\hat{\Omega }_1$ be the 2-form on Y as in (5.12). Then, $d\hat{\Omega }_1=0$.

Proof

It suffices to show

$$\begin{aligned} \pi _1^*\hat{\Omega }_1={\text {d}}\eta _1\quad \text{ on } Y. \end{aligned}$$

(5.15)

$\pi _1^*\hat{\Omega }_1={\text {d}}\eta _1$ on ${{\mathsf {D}}}_0$ from (5.12). For any $\xi \in {{\mathsf {E}}}_1$ and $A\in TX$, $\displaystyle {\text {d}}\eta _1(\xi , A)=0$ by Lemma 5.2. As ${{\mathsf {E}}}_1\oplus {{\mathsf {D}}}_0=TX$, it follows $\pi _1^*\hat{\Omega }_1={\text {d}}\eta _1$ on X. $\square$

6 Pseudo-Hermitian structure

6.1 Heisenberg Lie group ${{\mathcal {N}}}$

Let ${{\mathcal {N}}}$ be the $4n+1$-dimensional Heisenberg Lie group which has a central group extension $\displaystyle 1{\rightarrow }{\mathbb {R}}{\rightarrow }\, {{\mathcal {N}}}{\mathop {{\longrightarrow }}\limits ^{p}}{\mathbb {C}}^{2n}{\rightarrow }1$. A pseudo-Hermitian structure $(\omega _{{\mathcal {N}}},J_{{\mathcal {N}}})$ consists of a contact form

$$\begin{aligned} \omega _{{\mathcal {N}}}={\text {d}}t+\mathfrak {I}\langle (z,w),({\text {d}}z,{\text {d}}w)\rangle =dt+\mathfrak {I}({}^t\bar{z}dz+{}^t{\bar{w}}dw) \ \ \, ((z,w)\in {\mathbb {C}}^{2n})\end{aligned}$$

together with a complex structure $J_{{\mathcal {N}}}$ on $\mathrm{ker}\, \omega _{{\mathcal {N}}}$ which is isomorphic to the standard complex structure on ${\mathbb {C}}^{2n}$ at each point of ${{\mathcal {N}}}$ (cf. [11]). As ${\mathbb {R}}^2=\{(0,t_2,t_3)\}$ is a central subgroup of ${\mathbb {R}}^3=C({{\mathcal {M}}})$, there is a quotient nilpotent Lie group ${{\mathcal {M}}}/{\mathbb {R}}^2$ with central group extension $\displaystyle 1{\rightarrow }{\mathbb {R}}{\rightarrow }\, {{\mathcal {M}}}/{\mathbb {R}}^2{\mathop {{\longrightarrow }}\limits ^{\hat{\pi }}}{\mathbb {H}}^n{\rightarrow }1$. We shall find an explicit isomorphism to identify ${{\mathcal {M}}}/{\mathbb {R}}^2$ with ${{\mathcal {N}}}$. For our use, let $z+wj\in {\mathbb {H}}^n$ such that $z,w\in {\mathbb {C}}^n$. Then, ${{\mathcal {M}}}/{\mathbb {R}}^2$ is the product $\displaystyle {\mathbb {R}}\times {\mathbb {H}}^n=\mathrm{Im}\, {\mathbb {H}}\,( \mathrm{mod}\, {\mathbb {R}}j+{\mathbb {R}}k)\times {\mathbb {H}}^n$ with group law:

$$\begin{aligned} \begin{aligned} \bigl (a,z+wj\bigr )\cdot \bigl (b,z'+w'j\bigr )&=\bigl (a+b-\mathfrak {I}({}^t\bar{z}z'+{}^tw{\bar{w}}'),z+z'+(w+w')j\bigr ). \end{aligned} \end{aligned}$$

Define a diffeomorphism $\varphi : {{\mathcal {M}}}/{\mathbb {R}}^2={\mathbb {R}}\times {\mathbb {H}}^n{\rightarrow }\, {{\mathcal {N}}}={\mathbb {R}}\times {\mathbb {C}}^{2n}$ to be

$$\begin{aligned} \varphi (a,(z+wj))=(a,(z,{\bar{w}})). \end{aligned}$$

(6.1)

As we see that $\displaystyle \varphi ((a,z+wj)\cdot (b,z'+w'j)) =\bigl (a,(z,{\bar{w}})\bigr )\cdot \bigl (b,(z',{\bar{w}}')\bigr )$,

Lemma 6.1

$\varphi$ is a Lie group isomorphism of ${{\mathcal {M}}}/{\mathbb {R}}^2$ onto ${{\mathcal {N}}}$.

Consider the projection

(6.2)

for which the subbundle $\displaystyle \langle \frac{\text {d}}{{\text {d}}t_2}, \frac{\text {d}}{{\text {d}}t_3}\rangle$ is tangent to the fiber ${\mathbb {R}}^2$. For $\omega _0=\omega _1i+\omega _2j+\omega _3k$, noting (3.3), $\omega _1$ induces a 1-form $\hat{\omega }_1$ on ${{\mathcal {M}}}/{\mathbb {R}}^2$ such that

$$\begin{aligned} p_1^*{\hat{\omega }}_1=\omega _1\ \, \text{ on }\ {{\mathcal {M}}}. \end{aligned}$$

(6.3)

As $\displaystyle {\hat{\omega }}_1={\text {d}}t_1+\mathfrak {I}\langle (z,{\bar{w}}),({\text {d}}z,{\text {d}}\bar{w})\rangle$ from (3.2) $(z+wj\in {\mathbb {H}}^n)$, (6.1) shows

$$\begin{aligned} \varphi ^*\omega _{{\mathcal {N}}}=\hat{\omega }_1\ \text{ on }\ {{\mathcal {M}}}/{\mathbb {R}}^2. \end{aligned}$$

(6.4)

Let ${\hat{\varphi }}:{\mathbb {H}}^n{\rightarrow }{\mathbb {C}}^{2n}$ be a diffeomorphism defined by

$$\begin{aligned} {\hat{\varphi }}(z+wj)=(z,{\bar{w}}), \end{aligned}$$

(6.5)

with the following commutative diagram from (6.1):

(6.6)

Take the standard complex structure $J_1$ on ${\mathbb {H}}^n$ such that $\displaystyle J_1 u=u{\bar{i}}$. As $\displaystyle u{\bar{i}}=(z+wj)\bar{i}=({\bar{i}}z+iwj)$, it follows $\displaystyle \hat{\varphi }(u\bar{i})={\bar{i}}\hat{\varphi }(u)$. If we take the anti-complex structure $J_{\mathbb {C}}'$ on ${\mathbb {C}}^{2n}$ such as $\displaystyle J_{\mathbb {C}}'(v)={\bar{i}} v$, it follows

$$\begin{aligned} \hat{\varphi }_*\circ J_1=J_{\mathbb {C}}'\circ \hat{\varphi }_*: T{\mathbb {H}}^n{\rightarrow }T{\mathbb {C}}^{2n}. \end{aligned}$$

(6.7)

As in (3.4) of Sect. 3.2, the almost complex structure $J_1$ on ${{\mathsf {D}}}_0$ induces an almost complex structure $J_1$ on $p_{1*}{{\mathsf {D}}}_0={\ker }\,\hat{\omega }_1$ such that

$$\begin{aligned} \hat{\pi }_{*}\circ J_1=J_1\circ \hat{\pi }_{*}\, : \ p_{1*}{{\mathsf {D}}}_0\,{\rightarrow }T{\mathbb {H}}^n. \end{aligned}$$

(6.8)

Similarly, if $J'_{{\mathcal {N}}}$ is the anti-complex structure on ${\ker }\,\omega _{{\mathcal {N}}}$ of ${{\mathcal {N}}}$, it follows

$$\begin{aligned} p_{*}\circ J'_{{\mathcal {N}}}=J'_{\mathbb {C}}\circ p_{*} : \mathrm{ker}\, \omega _{{\mathcal {N}}}{\rightarrow }T{\mathbb {C}}^{2n}. \end{aligned}$$

(6.9)

Then, (6.7), (6.8) and (6.9) imply

$$\begin{aligned} \varphi _*\circ J_1=J'_{{\mathcal {N}}}\circ \varphi _* \text{ on } {\ker }\,\hat{\omega }_1\, (=p_{1*}{{\mathsf {D}}}_0). \end{aligned}$$

(6.10)

In particular, $J_1$ is a complex structure on $p_{1*}{{\mathsf {D}}}_0={\ker }\,\hat{\omega }_1$. We have the principal bundle induced from (5.9):

(6.11)

There is a CR-structure $({\ker }\,\hat{\omega }_1,J_1)$ on $X/{\mathbb {R}}^2\subset {{\mathcal {M}}}/{\mathbb {R}}^2$ as above. Let ${\hat{J}}_1$ be the almost complex structure on Y as in (5.11), that is $\displaystyle {\hat{\pi }}_{1*}J_1={\hat{J}}_1{\hat{\pi }}_{1*} : {\ker }\,\hat{\omega }_1{\rightarrow }TY$.

Lemma 6.2

${\hat{J}}_1$ is integrable on Y.

Proof

Let $\displaystyle {\ker }\,\hat{\omega }_1\otimes {\mathbb {C}}=P^{1,0}\oplus P^{0,1}$ be the eigenspace decomposition for $J_1$. We have an isomorphism $\displaystyle \hat{\pi }_{1*}:{\ker }\,\hat{\omega }_1\otimes {\mathbb {C}}{\longrightarrow }TY\otimes {\mathbb {C}}=Q^{1,0}\oplus Q^{0,1}$ (the eigenspace decomposition for ${\hat{J}}_1$, respectively). Since $J_1$ is integrable, $\displaystyle [\varvec{u},\varvec{v}]\in P^{1,0}$ for $\varvec{u},\varvec{v}\in P^{1,0}$. Then, $\hat{\pi }_{1*}([\varvec{u},\varvec{v}])=[\hat{\pi }_{1*}(\varvec{u}),\hat{\pi }_{1*}(\varvec{v})]$. It follows $\displaystyle [\hat{\pi }_{1*}(\varvec{u}),\hat{\pi }_{1*}(\varvec{v})]\in Q^{1,0}$. Thus, ${\hat{J}}_1$ is integrable. $\square$

Combining this lemma with Lemma 5.3, we obtain

Proposition 6.3

$(Y,\{\hat{\Omega }_1,{\hat{J}}_1\})$ is a Kähler manifold.

6.2 Bochner flat structure on $(Y,{\hat{J}}_1)$

Put $\displaystyle e^{iat}z=(e^{ia_1t}z_1,\dots ,e^{ia_nt}z_n)$ for short, similarly for $e^{-iat}w$ for $a=(a_1,\ldots ,a_n)$ satisfying (5.2). Let ${\mathbb {R}}$ act on ${{\mathcal {N}}}$ by

$$\begin{aligned} \rho (t)(s,(z,w))=(t+s, (e^{iat}z, e^{-iat}w)) \end{aligned}$$

(6.12)

such that $\rho ({\mathbb {R}})\le {\mathbb {R}}\times \mathrm{U}(2n)\le {{\mathcal {N}}}\rtimes \mathrm{U}(2n)\le \mathop {\mathrm{Aut}}\nolimits _{CR}({{\mathcal {N}}})$. There induces another principal bundle $\displaystyle \rho ({\mathbb {R}}){\rightarrow }\, {{\mathcal {N}}}{\mathop {{\longrightarrow }}\limits ^{q}}{\mathbb {C}}^{2n}$. By (6.1) and (5.5),

$$\begin{aligned}\begin{aligned} \varphi \bigl (\rho _1(t)(s,(z+wj))\bigr )&= \varphi \bigl (s+t, (e^{iat}z+e^{iat}wj)\bigr )\\&=\bigl (s+t, (e^{iat}z, e^{-iat}{\bar{w}})\bigr ) =\rho (t)\varphi \bigl (s,(z+wj)\bigr ), \end{aligned}\end{aligned}$$

there is the bundle isomorphism (cf. (5.9)):

(6.13)

Let $\xi$ be the vector field induced by $\rho ({\mathbb {R}})$ on ${{\mathcal {N}}}$. Put $\displaystyle p_{1*}\xi _1={\hat{\xi }}_1$ from (5.6), (6.2), which is the Reeb field of $({{\mathcal {M}}}/{\mathbb {R}}^2,(\hat{\omega }_1,J_1))$. We have $\displaystyle \varphi _*{\hat{\xi }}_1=\xi$. At $p=((s_1,s_2,s_3), z_1,\dots ,z_n)\in {{\mathcal {M}}}$ with $z_k=u_k+v_k j$, (6.4), (6.3) and (5.7) imply

$$\begin{aligned} \omega _{{\mathcal {N}}}(\xi ) =\hat{\omega }_1({\hat{\xi }}_1)= \omega _1(\xi _1)=1+\sum _{k=1}^n a_k(|u_k|^2-|v_k|^2). \end{aligned}$$

(6.14)

Corresponding to $X/{\mathbb {R}}^2=\{x\in p_1(X)\mid \hat{\omega }_1((\hat{\xi }_1)_x)\ne 0\}$, the bundle isomorphism $\varphi$ maps $X/{\mathbb {R}}^2$ onto the domain $\displaystyle {{\mathcal {N}}}_1=\{z\in {{\mathcal {N}}}\mid \omega _{{\mathcal {N}}}(\xi _z)\ne 0\}$ of ${{\mathcal {N}}}$. As in (5.10), define the contact forms to be

$$\begin{aligned} \begin{aligned} \hat{\eta }_1&=\frac{1}{{\hat{\omega }}_1(\hat{\xi }_1)}\hat{\omega }_1 \, \text{ on }\ X/{\mathbb {R}}^2, \quad \eta _{{{\mathcal {N}}}_1}=\frac{1}{\omega _{{\mathcal {N}}}(\xi )}\omega _{{\mathcal {N}}}\ \, \text{ on } {{\mathcal {N}}}_1. \end{aligned} \end{aligned}$$

(6.15)

In particular, $p_{1*}{{\mathsf {D}}}_0={\ker }\,\hat{\eta }_1$. Noting (6.14), (6.4) shows that

$$\begin{aligned} \varphi ^*{\eta _{{{\mathcal {N}}}_1}}=\hat{\eta }_1. \end{aligned}$$

Put $J_{{{\mathcal {N}}}_1}=J_{{{\mathcal {N}}}}|_{{{\mathcal {N}}}_1}$. Since $(\eta _{{{\mathcal {N}}}_1},J_{{{\mathcal {N}}}_1})$ represents a spherical pseudo-Hermitian structure on ${{\mathcal {N}}}_1$, this equation together with (6.10) implies (cf. [11])

Proposition 6.4

The pseudo-Hermitian structure $(X/{\mathbb {R}}^2,\hat{\eta }_1, J_1)$ is anti-holomorphically isomorphic to the spherical CR-structure $({{\mathcal {N}}}_1,{\eta _{{{\mathcal {N}}}_1}}, J_{{{\mathcal {N}}}_1})$.

The projection $\displaystyle \hat{\pi }_1:X/{\mathbb {R}}^2{\rightarrow }Y$ of (6.11) induces the following from (5.15), (5.11):

$$\begin{aligned} \begin{aligned} \hat{\pi }_1^*\hat{\Omega }_1&=d\hat{\eta }_1 \, \text{ on } \,X/{\mathbb {R}}^2,\\ {\hat{J}}_\alpha \circ \hat{\pi }_{1*}&=\hat{\pi }_{1*}\circ J_\alpha \, \text{ on } p_{1*}{{\mathsf {D}}}_0\ \, (\alpha =1,2,3). \end{aligned} \end{aligned}$$

(6.16)

Theorem 6.5

$(Y,{\hat{g}}, \{\hat{\Omega }_1, {\hat{J}}_1\})$ is an anti-holomorphic Bochner flat Kähler manifold for the quaternionic Hermitian manifold $(Y, {\hat{g}},\{\hat{\Omega }_\alpha , {\hat{J}}_\alpha \}_{\alpha =1}^3)$.

Proof

Since $d\hat{\eta }_1=\hat{\pi }_1^*\hat\Omega _1$ from $X/{\mathbb {R}}^2$ by (6.16), $\displaystyle \rho _1({\mathbb {R}}){\rightarrow }X/{\mathbb {R}}^2{\mathop {{\longrightarrow }}\limits ^{\hat{\pi }_1}}Y$ is a pseudo-Hermitian bundle with the Reeb field $\hat{\xi }_1$, the Bochner curvature tensor of the Kähler manifold $(Y, \{\hat{\Omega }_1, {\hat{J}}_1\})$ coincides with the Chern–Moser curvature tensor on $(X/{\mathbb {R}}^2, \hat{\eta }_1,J_1)$ by the pull-back of $\hat{\pi }_1$ in (6.16) (cf. [16]). As $X/{\mathbb {R}}^2$ is spherical CR by Proposition 6.4, the Chern–Moser curvature tensor is zero and thus $(Y,\{\hat{\Omega }_1,{\hat{J}}_1\})$ is a Bochner flat manifold. $\square$

7 Pseudo-Hermitian group $\mathop {\mathrm{Psh}}\nolimits (X/{\mathbb {R}}^2)$

We determine the holomorphic isometry group $\displaystyle \mathop {\mathrm{Isom}}\nolimits _{\!\, h}(Y)=\{f\in \mathop {\mathrm{Diff}}\nolimits (Y)\mid f^*\hat{\Omega }_1=\hat{\Omega }_1, \, f_*\circ {\hat{J}}_1={\hat{J}}_1\circ f_*\}$ of the Kähler manifold $(Y,(\hat{\Omega }_1,{\hat{J}}_1))$. In order to do so, consider the pseudo-Hermitian group of the pseudo-Hermitian manifold $(X/{\mathbb {R}}^2,\hat{\eta }_1, J_1)$ (cf. [11]):

$$\begin{aligned} \mathop {\mathrm{Psh}}\nolimits (X/{\mathbb {R}}^2)=\{{{\tilde{f}}}\in \mathop {\mathrm{Diff}}\nolimits (X/{\mathbb {R}}^2) \mid {{\tilde{f}}}^*\hat{\eta }_1={\hat{\eta }}_1,\ {{\tilde{f}}}_*\circ J_1=J_1\circ {{\tilde{f}}}_*\}. \end{aligned}$$

As $H^1(Y;{\mathbb {R}})=0$ (see the remark below (5.9)), it associates the exact sequence from [4, Proposition 3.4]:

(7.1)

Since $\rho _1({\mathbb {R}})$ induces the Reeb field ${\hat{\xi }}_1$ of $\hat{\eta }_1$, $\mathop {\mathrm{Psh}}\nolimits (X/{\mathbb {R}}^2)$ itself is the centralizer of $\rho _1({\mathbb {R}})$ in $\mathop {\mathrm{Psh}}\nolimits (X/{\mathbb {R}}^2)$ (cf. [4, 11]). If we recall the representation of $\rho ({\mathbb {R}})$ from (6.12):

$$\begin{aligned} \rho (t)=\bigl (t,(e^{ia_1t},\dots ,e^{ia_nt}, e^{-ia_1t},\dots ,e^{-ia_nt})\bigr )\in {\mathbb {R}}\times \mathrm{U}(2n)\le {{\mathcal {N}}}\rtimes \mathrm{U}(2n), \end{aligned}$$

under the equivariant diffeomorphism $\varphi$ of (6.13), the condition (5.2) implies that

$$\begin{aligned} \mathop {\mathrm{Psh}}\nolimits (X/{\mathbb {R}}^2)={\mathbb {R}}\times T^n\times T^n \le {\mathbb {R}}\times \mathrm{U}(2n). \end{aligned}$$

(7.2)

We have from (7.1) that

Proposition 7.1

$\mathop {\mathrm{Isom}}\nolimits _{\!\, h}(Y)=T^{2n}$.

Theorem 7.2

The quaternionic Hermitian manifold $(Y,{\hat{g}},\{{\hat{J}}_\alpha \}_{\alpha =1}^3)$ is not Einstein. Moreover, it is never holomorphically isometric to any domain of the quaternionic euclidean space ${\mathbb {H}}^n$.

Proof

When the Bochner flat manifold $(Y,{\hat{g}},\{\Omega _1,{\hat{J}}_1\})$ is Einstein, it is of constant holomorphic curvature by Tachibana’s result [15]. Thus, Y is locally holomorphically isometric to the flat space ${\mathbb {C}}^{2n}$. We may assume that the origin $\varvec{0}$ belongs to Y (cf. (5.7)). Then, the stabilizer at $\varvec{0}$ is the maximal compact subgroup isomorphic to $\mathop {\mathrm{U}}\nolimits (2n)$. Since $T^{2n}$ is the full holomorphic isometry group of $(Y,{\hat{g}},{\hat{J}}_1)$ by Proposition 7.1, it is impossible. $(Y,{\hat{g}},{\hat{J}}_1)$ is not holomorphically isometric to any domain of $({\mathbb {H}}^n,g_{\mathbb {H}})$ with the standard euclidean metric $g_{\mathbb {H}}$. $\square$

7.1 Quaternionic Hermitian isometry group of $(Y,{\hat{g}})$

The quaternionic Hermitian isometry group of the quaternionic Hermitian manifold $(Y, {\hat{g}},\{\hat{\Omega }_\alpha ,{\hat{J}}_\alpha \}_{\alpha =1}^3)$ may be denoted naturally by the following:

$$\begin{aligned} \mathrm{Isom}_{\,\!qH}(Y)=\bigl \{ {\hat{h}}\in \mathop {\mathrm{Diff}}\nolimits (Y) \ \big |\ {{\hat{h}}}^*{\hat{\Omega }}_\alpha = \sum _{\beta =1}^{3}\hat{\Omega }_\beta {\hat{a}}_{\beta \alpha },\ {\hat{h}}_* \circ {\hat{J}}_\alpha =\sum _{\beta =1}^{3}{\hat{a}}_{\alpha \beta }{\hat{J}}_\beta \circ {\hat{h}}_*\bigr \}, \end{aligned}$$

where $\displaystyle ({\hat{a}}_{\alpha \beta })_{\alpha ,\beta =1,2,3}:Y{\longrightarrow }\mathrm{SO}(3)$ are smooth maps.

For the abelian group $\displaystyle \rho _1({\mathbb {R}}^3)$ defined by (5.4), let $N_{\mathop {\mathrm{Aut}}\nolimits ({{\mathcal {M}}})}(\rho _1({\mathbb {R}}^3))$ be the normalizer of $\rho _1({\mathbb {R}}^3)$ in $\mathop {\mathrm{Aut}}\nolimits ({{\mathcal {M}}})$. By the formula (5.3) of $A_t$, the normalizer of $\{A_{t}\}$ in $\mathop {\mathrm{Sp}}\nolimits (n)$ is isomorphic to $T^n$. Since the only subgroup $S^1=\langle e^{i\theta }\rangle$ of $\mathop {\mathrm{Sp}}\nolimits (1)$ normalizes $\rho _1({\mathbb {R}}^3)$ in view of the actions (3.1) and (5.4), it follows that

$$\begin{aligned} \begin{aligned} N_{\mathop {\mathrm{Aut}}\nolimits ({{\mathcal {M}}})}(\rho _1({\mathbb {R}}^3))&=({\mathbb {R}}^3,0)\times T^n\cdot S^1 \le {{\mathcal {M}}}\rtimes \mathop {\mathrm{Sp}}\nolimits (n)\cdot \mathop {\mathrm{Sp}}\nolimits (1) \end{aligned}\end{aligned}$$

(7.3)

where $\displaystyle T^n\cdot S^1=T^n\times _{\{\pm 1\}} S^1$. Recall every element of $\mathop {\mathrm{Aut}}\nolimits (X)$ extends to an element of $\mathop {\mathrm{Aut}}\nolimits ({{\mathcal {M}}})$. Since each element of $N_{\mathop {\mathrm{Aut}}\nolimits ({{\mathcal {M}}})}(\rho _1({\mathbb {R}}^3))$ preserves both $\omega _1$ and $\xi _1$ from (5.8), it follows $\displaystyle N_{\mathop {\mathrm{Aut}}\nolimits (X)}(\rho _1({\mathbb {R}}^3))=N_{\mathop {\mathrm{Aut}}\nolimits ({{\mathcal {M}}})}(\rho _1({\mathbb {R}}^3))$.

Let $\eta =\eta _1i+\eta _2j+\eta _3k$ be as before (cf. (5.10)). Note $h^*\eta =a\cdot \eta \cdot {\bar{a}}$ for $h=((t,0),A\cdot a)\in N_{\mathop {\mathrm{Aut}}\nolimits (X)}(\rho _1({\mathbb {R}}^3))$. By (7.3), the projection $\pi _1:X{\rightarrow }Y$ of (5.9) induces an element ${\hat{h}}:Y{\rightarrow }Y$. Since $\pi _1^*\hat{\Omega }=d\eta$ for $\hat{\Omega }=\hat{\Omega }_1i+\hat{\Omega }_2j+\hat{\Omega }_3k$, we have ${\hat{h}}^*\hat{\Omega }= a\cdot \hat{\Omega }\cdot {\bar{a}}$ for $a\in S^1$. Thus, it assigns an element ${\hat{h}}\in \mathop {\mathrm{Isom}}\nolimits _{\!\,qH}(Y, {\hat{g}},\{\hat{\Omega }_\alpha ,{\hat{J}}_\alpha \}_{\alpha =1}^3) =\mathop {\mathrm{Isom}}\nolimits _{\!\, qH}(Y)$. Letting $a=(a_{\alpha \beta })_{\alpha ,\beta =2,3}\in \mathrm{SO}(2)$, $\mathop {\mathrm{Isom}}\nolimits _{\!\, qH}(Y)$ can be described as

$$\begin{aligned} \begin{aligned} \bigl \{{\hat{h}}\in \mathop {\mathrm{Diff}}\nolimits (Y)\mid&\ {\hat{h}}^*{\hat{\Omega }}_1=\hat{\Omega }_1, {\hat{h}}_*\circ {\hat{J}}_1 = {\hat{J}}_1\circ {\hat{h}}_*, \\&\quad {\hat{h}}^*\hat{\Omega }_\alpha = \sum _{\beta =2,3}\hat{\Omega }_\beta a_{\beta \alpha }, \quad {\hat{h}}_* \circ {\hat{J}}_\alpha \, =\, \sum _{\beta =2,3} a_{\alpha \beta }{\hat{J}}_\beta \circ {\hat{h}}_*\ \bigr \}. \end{aligned} \end{aligned}$$

(7.4)

Setting $\displaystyle \tilde{\phi }(h)={\hat{h}}$, (7.3) gives an exact sequence:

(7.5)

If $\iota : \mathop {\mathrm{Isom}}\nolimits _{\!\,qH}(Y){\rightarrow }\mathop {\mathrm{Isom}}\nolimits _{h}(Y)=T^{2n}$ (cf. Proposition 7.1) is the natural inclusion (that is, forgetting the quaternionic structure but leaving the holomorphic structure as it is), then under the equivariant diffeomorphism $\hat{\varphi }$ of (6.5), it follows $\displaystyle \iota (T^{n}\cdot S^1)= \{(z_1,\ldots ,z_n,{\bar{z}}_1,\ldots , \bar{z}_n)\times (e^{-i\theta },\ldots ,e^{-i\theta })\}\le T^{2n}$ where $z_i\in S^1$ $(i=1,\ldots ,n)$. We obtain

Corollary 7.3

The quaternionic Hermitian isometry group $\mathop {\mathrm{Isom}}\nolimits _{\!\,qH}(Y)^0$ is isomorphic to the torus $T^k$ for some k $(n+1\le k\le 2n)$.

Remark 7.4

Since the forms $\hat{\Omega }_2, \hat{\Omega }_3$ are not Kähler, the equation $\displaystyle \pi _{1}^*\hat{\Omega }_\alpha =d\eta _\alpha$ does not hold on X (only on ${{\mathsf {D}}}_0$), the method of [4, Propositions 3.4, 3.1, 3.2] cannot be applied to show the surjectivity of $\tilde{\phi }$ in (7.5).