Quaternion geometries on the twistor space of the six-sphere

Abstract

We explicitly describe all ${ SO }(7)$-invariant almost quaternion-Hermitian structures on the twistor space of the six-sphere and determine the types of their intrinsic torsion.

1 Introduction

Recently, Moroianu, Pilca and Semmelmann [5] found that the twistor space $M={ SO }(7)/{{ U }(3)}$ of the six-sphere $S^6$ admits a homogeneous almost quaternion-Hermitian structure. This arose as part of their striking result that $M$ is the only such homogeneous space with non-zero Euler characteristic that is neither quaternionic Kähler (the quaternionic symmetric spaces of Wolf [9]) nor $S^2\times S^2$.

In this paper, we show that there is exactly a one-dimensional family of invariant almost quaternion-Hermitian structures on $M$, with fixed volume, and determine the types of their intrinsic torsion. We will see that the family contains inequivalent structures and includes the symmetric Kähler metric of the quadric ${{{\mathrm{{\widetilde{Gr}}}}}}_2({\mathbb{R }}^6)={ SO }(8)/{{ SO }(2){ SO }(6)}$. Each member of the family will be shown to have almost quaternion-Hermitian type $\Lambda ^3_0E(S^3H+H)$ with the first component non-zero, confirming that they are not quaternionic Kähler; one member of the family has pure type $\Lambda ^3_0ES^3H$, and this is the first known example of such a geometry. However, the structure singled out by this almost quaternionic-Hermitian intrinsic torsion is not the Kähler metric of the quadric nor the squashed Einstein metric in the canonical variation.

2 Invariant forms

The subgroup ${ U }(3)$ of ${ SO }(7)$ arises from a choice of identification of ${\mathbb{R }}^7$ as ${\mathbb{R }}\oplus {\mathbb{C }}^3$. Regarding ${ U }(3)$ as ${ U }(1){ SU }(3)$, we may write ${\mathbb{C }}^3={\mathbb{R }}^6=[\![{L\lambda ^{1,0}}]\!]$, meaning that ${\mathbb{R }}^6\otimes {\mathbb{C }}=L\lambda ^{1,0}+\overline{L\lambda ^{1,0}} \cong L\lambda ^{1,0} + L^{-1}\lambda ^{0,1}$, where $L={\mathbb{C }}$ and $\lambda ^{1,0}={\mathbb{C }}^3$ as the standard representations of ${ U }(1)$ and ${ SU }(3)$, respectively. We thus have ${ U }(3)\leqslant { SO }(6)\leqslant { SO }(7)$, so $M={ SO }(7)/{{ U }(3)}$ fibres over $S^6={ SO }(7)/{{ SO }(6)}$ with fibre ${ SO }(6)/{{ U }(3)}$, the almost complex structures on $T_xS^6$. Thus, $M$ is the (Riemannian) twistor space of $S^6$.

Since $\lambda ^{3,0}=\Lambda ^3\lambda ^{1,0}={\mathbb{C }}$ is trivial, we have $\lambda ^{2,0}\cong \lambda ^{0,1}$ as ${ SU }(3)$-modules. The Lie algebra of ${ SO }(7)$ now decomposes as

$$\begin{aligned} \mathfrak{so }(7)&= \Lambda ^2{\mathbb{R }}^7 =\Lambda ^2({\mathbb{R }}+[\![{L\lambda ^{1,0}}]\!]) =[\![{L\lambda ^{1,0}}]\!]+[\![{L^2\lambda ^{2,0}}]\!]+[{\lambda ^{1,1}}] \\&\cong [\![{L\lambda ^{1,0}}]\!]+[\![{L^2\lambda ^{0,1}}]\!]+\mathfrak{u }(1)+\mathfrak{su }(3). \end{aligned}$$

Here, $[{\lambda ^{1,1}}]$ is the real module whose complexification is $\lambda ^{1,1}=\lambda ^{1,0}\otimes \lambda ^{0,1}$; it splits in to two irreducible modules $[{\lambda ^{1,1}_0}]\cong \mathfrak{su }(3)$ and ${\mathbb{R }}=\mathfrak{u }(1)$.

We thus have that the complexified tangent space of $M={ SO }(7)/{{ U }(3)}$ is the bundle associated with

$$\begin{aligned} T \otimes {\mathbb{C }}&= \left( [\![{L\lambda ^{1,0}}]\!] + [\![{L^2\lambda ^{0,1}}]\!]\right) \otimes \mathbb{C }\nonumber \\&= L\lambda ^{1,0} + L^{-1}\lambda ^{0,1} + L^2\lambda ^{0,1} + L^{-2}\lambda ^{1,0} \nonumber \\&= (L^{1/2}\lambda ^{0,1} + L^{-1/2}\lambda ^{1,0})(L^{3/2}+L^{-3/2}). \end{aligned}$$

(2.1)

This allows us to write $T \otimes {\mathbb{C }} = EH$, where $E=L^{1/2}\lambda ^{0,1} + L^{-1/2}\lambda ^{1,0}$ and $H=L^{3/2}+L^{-3/2}$ are representations of ${ U }(1)_2\times { SU }(3)$ as a subgroup of ${ U }(1)_L{ SU }(3)\times { U }(1)_R\leqslant { Sp }(3)\times { Sp }(1)$. Here, ${ U }(1)_2$ is a double cover of ${ U }(1)$ and is included in ${ U }(1)_L\times { U }(1)_R$ via the map $e^{i\theta }\mapsto (e^{-i\theta }\!,e^{3i\theta })$. In this way, we see that $M={ SO }(7)/{{ U }(3)}$ carries an invariant ${ Sp }(3){ Sp }(1)$-structure, where ${ Sp }(3){ Sp }(1)=({ Sp }(3)\times { Sp }(1))/\{\pm (1,1)\}$. This is the $G$-structure description of an almost quaternion-Hermitian structure.

Geometrically, an almost quaternion-Hermitian structure is specified by a Riemannian metric $g$ and a three-dimensional subbundle $\mathcal{G }$ of ${{\mathrm{End}}}(\text{ TM })$ which locally has a basis $I,\,J,\,K$ satisfying the quaternion identities

$$\begin{aligned} I^2 = -1 = J^2,\quad \quad IJ = K = -JI \end{aligned}$$

and the compatibility conditions

$$\begin{aligned} g(I\cdot ,I\cdot ) = g(\cdot ,\cdot ) = g(J\cdot ,J\cdot ). \end{aligned}$$

There are then local two-forms

$$\begin{aligned} \omega _I(X,Y)&= g(X,IY),\quad \omega _J(X,Y) = g(X,JY),\\&\omega _K(X,Y) = g(X,KY) \end{aligned}$$

and with the local form $\omega _c=\omega _J +i\omega _K$ of type (2,0) with respect to $I$. Since they are non-degenerate, the local forms $\omega _I,\,\omega _J,\,\omega _K$ are sufficient to determine the local almost complex structures $I,\,J$ and $K$ and the metric $g$.

Equation (2.1) show us that $T$ has two inequivalent irreducible summands $[\![{L\lambda ^{1,0}}]\!]$ and $[\![{L^2\lambda ^{0,1}}]\!]$ and so there are two invariant forms $\omega _0$ and ${\tilde{\omega }}_0$ spanning $\Omega ^2(M)^{{ SO }(7)}$. However, we have that

$$\begin{aligned} \Lambda ^2T&= \Lambda ^2[\![{L\lambda ^{1,0}}]\!] + \Lambda ^2[\![{L^2\lambda ^{0,1}}]\!] + [\![{L\lambda ^{1,0}}]\!]\wedge [\![{L^2\lambda ^{0,1}}]\!] \nonumber \\&= ({\mathbb{R }}\omega _0 + [{\lambda ^{1,1}_0}] + [\![{L^2\lambda ^{0,1}}]\!]) + ({\mathbb{R }}{\tilde{\omega }}_0 + [{\lambda ^{1,1}_0}] + [\![{L^4\lambda ^{1,0}}]\!]) \nonumber \\&\quad + ([\![{L^3}]\!] + [\![{L^3}]\!][{\lambda ^{1,1}_0}] + [\![{L\lambda ^{1,0}}]\!] + [\![{L\sigma ^{0,2}}]\!]), \end{aligned}$$

(2.2)

where $\sigma ^{0,2} = S^2\lambda ^{0,1}$. There is thus an addition two-dimensional subspace $[\![{L^3}]\!]$ preserved by the ${ SU }(3)$-action. This space is spanned by local ${ SU }(3)$-invariant forms $\omega _J$ and $\omega _K$ that are mixed under the ${ U }(1)$-action, so that $\omega _c=\omega _J+i\omega _K$ is a basis element of $L^3$. We may now consider the triple of forms

$$\begin{aligned} \omega _I = \lambda \omega _0 + \mu \tilde{\omega }_0,\quad \omega _J\quad \text{ and }\quad \omega _K \end{aligned}$$

(2.3)

which will be seen to result in an almost quaternion-Hermitian structure when

$$\begin{aligned} 20\lambda ^3\mu ^3(\omega _0)^3(\tilde{\omega }_0)^3 = (\omega _J)^6. \end{aligned}$$

(2.4)

This equation is necessary, as each two-form in the triple must define the same volume element.

We note that for an almost quaternion-Hermitian structure the four-form $\Omega =\omega _I^2+\omega _J^2+\omega _K^2$ is globally defined. For an invariant structure, this form must lie in $\Omega ^4(M)^{{ SO }(7)}$ which in our particular case is four-dimensional. Indeed, the complete decomposition of $\Lambda ^4T$ in to irreducible ${ U }(3)$-modules is

$$\begin{aligned} \Lambda ^4T&= [\![{L^6}]\!] +2[\![{L^3}]\!] + 4{\mathbb{R }} +[\![{L^7\lambda ^{1,0}}]\!] + 3[\![{L^4\lambda ^{1,0}}]\!] + 5[\![{L\lambda ^{1,0}}]\!]\\&\quad + 4[\![{L^2\lambda ^{0,1}}]\!] + 2[\![{L^5\lambda ^{0,1}}]\!] + 2[\![{L^2\sigma ^{2,0}}]\!] + 2[\![{L\sigma ^{0,2}}]\!] + [\![{L^4\sigma ^{0,2}}]\!] \\&\quad + [\![{L^3\sigma ^{3,0}}]\!] + [\![{\sigma ^{3,0}}]\!] + [\![{L^3\sigma ^{0,3}}]\!] + [\![{L^6\lambda ^{1,1}_0}]\!] + 4[\![{L^3\lambda ^{1,1}_0}]\!] + 6[{\lambda ^{1,1}_0}] \\&\quad + [\![{L^4\sigma ^{2,1}_0}]\!] + 2[\![{L^2\sigma ^{2,1}_0}]\!] + [\![{L^2\sigma ^{1,2}_0}]\!] + [\![{\sigma ^{2,2}_0}]\!]. \end{aligned}$$

Now, the four-forms $\omega _0^2,\,{\tilde{\omega }}_0^2,\,\omega _0\wedge \tilde{\omega }_0$ and $\omega _J^2+\omega _K^2$ are invariant and linearly independent, so they provide a basis for $\Omega ^4(M)^{{ SO }(7)}$. It follows, Lemma 4.1 below, that any invariant almost hyperHermitian structure on $M$ is described via the forms of (2.3).

3 Intrinsic torsion

Given an invariant almost Hermitian structure on $M$, there is a unique ${ Sp }(3){ Sp }(1)$-connection $\nabla$ characterised by the condition that the pointwise norm of its torsion is the least possible. More precisely, $\nabla$ is related to the Levi-Civita connection by

$$\begin{aligned} \nabla = {\nabla }^\mathrm{LC}+ \xi , \end{aligned}$$

where $\xi$ is the intrinsic torsion given [4] by

$$\begin{aligned} \xi _X Y = - \tfrac{1}{4} \sum _{A=I,J,K} A ({\nabla }^\mathrm{LC}_X A) Y + \tfrac{1}{2} \sum _{A=I,J,K} \lambda _A(X) AY, \end{aligned}$$

with

$$\begin{aligned} 6\lambda _I(X) = g({\nabla }^\mathrm{LC}_X\omega _J,\omega _K), \end{aligned}$$

etc. The tensor $\xi$ takes values in

$$\begin{aligned} {\mathcal{Q }} = T^* \otimes (\mathfrak{sp }(3) + \mathfrak{sp }(1))^\bot \subset T^* \otimes \Lambda ^2 T^* \end{aligned}$$

where $\mathfrak{sp }(3)=[{S^2E}]$ and $\mathfrak{sp }(1)=[{S^2H}]$ are the Lie algebras of ${ Sp }(3)$ and ${ Sp }(1)$. Under the action of ${ Sp }(3){ Sp }(1)$, the space ${\mathcal{Q }}\otimes \mathbb{C }$ decomposes as

$$\begin{aligned} {\mathcal{Q }} \otimes {\mathbb{C }} = (\Lambda ^3_0E + K + E)(S^3H + H) \end{aligned}$$

with $\Lambda ^3_0E$ and $K$ irreducible ${ Sp }(3)$-modules satisfying $\Lambda ^3E = \Lambda ^3_0E + E$ and $E\otimes S^2E =S^3E + K + E$. The space $\mathcal{Q }$ thus has six irreducible summands under ${ Sp }(3){ Sp }(1)$.

For an invariant structure on $M={ SO }(7)/{ U }(3)$, the intrinsic torsion lies in a ${ U }(3)$-invariant submodule of $\mathcal{Q }$. As $\mathfrak{sp }(3)=[{S^2(L^{1/2}\lambda ^{0,1})}] = [\![{L\sigma ^{0,2}}]\!] + [{\lambda ^{1,1}_0}] + {\mathbb{R }}$ and $\mathfrak{sp }(1) = [{S^2(L^{3/2})}]=[\![{L^3}]\!]+{\mathbb{R }}$, Eq. (2.2) implies that

$$\begin{aligned} (\mathfrak{sp }(3)+\mathfrak{sp }(1))^\bot \cong [{\lambda ^{1,1}_0}] + [\![{L^2\lambda ^{0,1}}]\!] + [\![{L^4\lambda ^{1,0}}]\!] + [\![{L^3}]\!][{\lambda ^{1,1}_0}] + [\![{L\lambda ^{1,0}}]\!]. \end{aligned}$$

Comparing with Eq. (2.1), we see that $(\mathfrak{sp }(3)+\mathfrak{sp }(1))^\bot$ contains a unique copy of each of the irreducible summands of $T$, so ${\mathcal{Q }}^{{ U }(3)}$ is two-dimensional. As $\Lambda ^3(A+B) \cong \Lambda ^3A +\Lambda ^2A\otimes B + A\otimes \Lambda ^2B + \Lambda ^3B$, we find that

$$\begin{aligned} \varLambda ^3_0E = (L^{3/2} + L^{-3/2}) + (L^{1/2}\sigma ^{2,0}+L^{-1/2}\sigma ^{0,2}). \end{aligned}$$

The first summand is a copy of $H$ and is also a submodule of $S^3H=L^{9/2}+L^{3/2}+L^{-3/2}+L^{-9/2}$. This shows that $[{\Lambda ^3_0ES^3H}]^{{ U }(3)}$ and $[{\Lambda ^3_0EH}]^{{ U }(3)}$ are each one-dimensional, and so we have

$$\begin{aligned} \xi \in {\mathcal{Q }}^{{ U }(3)} \subset [{\Lambda ^3_0\, ES^3H}] + [{\Lambda ^3_0\, EH}]. \end{aligned}$$

(3.1)

4 Explicit structures

We now wish to determine the components of $\xi$ in each of the summands of (3.1). An invariant almost Hermitian structure on $M$ may be described by two-forms as in (2.3). As $\omega _J$ and $\omega _K$ are only invariant under ${ SU }(3)$, they do not define global forms on $M$. However, we do get two such invariant forms on the total space of the circle bundle $N={ SO }(7)/{ SU }(3)\rightarrow M={ SO }(7)/{ U }(3)$.

Let 0, 1, 2, 3, 1$^{\prime }$, 2$^{\prime }$, 3$^{\prime }$ be an orthonormal basis for ${\mathbb{R }}^7={\mathbb{R }}+{\mathbb{C }}^3$, with $0 \in {\mathbb{R }}$ and $i1=1^{\prime }$, etc. Writing 12 for $1 \wedge 2$, a standard basis for $[\![{L\lambda ^{1,0}}]\!]\subset \mathfrak{so }(7)$ is given by

$$\begin{aligned} A = 01,\quad B = 02,\quad C = 03,\quad A^{\prime } = 01^{\prime },\quad B^{\prime } = 02^{\prime },\quad C^{\prime } = 03^{\prime } \end{aligned}$$

and a corresponding basis for $[\![{L^2\lambda ^{0,1}}]\!]$ is

$$\begin{aligned} P&= 23-2^{\prime }3^{\prime },\quad \, Q = 31-3^{\prime }1^{\prime },\quad \, R = 12-1^{\prime }2^{\prime }, \\ P^{\prime }&= 23^{\prime }-32^{\prime },\quad Q^{\prime } = 31^{\prime }-13^{\prime },\quad R^{\prime } = 12^{\prime }-21^{\prime }. \end{aligned}$$

We put $E = 11^{\prime }+22^{\prime }+33^{\prime }$, and note that, this is a generator of the central $\mathfrak{u }(1)$ in $\mathfrak{u }(3)$. Then $\{{E,A,\dots ,R^{\prime }}\}$ is a basis for ${\mathfrak{n }} = T_{{{\mathrm{Id}}}{ SU }(3)}N$ and $\{{A,\dots ,R^{\prime }}\}$ is a basis for ${\mathfrak{m }} = T_{{{\mathrm{Id}}}{ U }(3)}M$. We use lower case letters to denote the corresponding dual bases of ${\mathfrak{n }}^*$ and ${\mathfrak{m }}^*$. These give left-invariant one-forms on ${ SO }(7)$, with $da(X,Y)=- a([X,Y])$ for $X,Y\in \mathfrak{so }(7)$, etc. We write

$$\begin{aligned} d_Na = (\text{ da })|_{\Lambda ^2\mathfrak{n }} \quad \text{ and }\quad d_Ma = (\text{ da })|_{\Lambda ^2\mathfrak{m }} \end{aligned}$$

at ${{\mathrm{Id}}}\in { SO }(7)$. For a left-invariant form $\alpha \in \Omega ^k({ SO }(7))$, we have at ${{\mathrm{Id}}}\in { SO }(7)$ that $d\alpha =d_N\alpha$ if $\alpha$ is right ${ SU }(3)$-invariant and $d\alpha = d_M\alpha$ if $\alpha$ is right ${ U }(3)$-invariant. For our choice of bases, we have

$$\begin{aligned} d_Ma&= - b\wedge r + c\wedge q - b^{\prime } \wedge r^{\prime } + c^{\prime }\wedge q^{\prime }, \quad d_Mp = - \tfrac{1}{2}(b\wedge c - b^{\prime } \wedge c^{\prime }),\\ d_Ma^{\prime }&= - b\wedge r^{\prime } + c\wedge q^{\prime } + b^{\prime } \wedge r - c^{\prime }\wedge q, \quad d_Mp^{\prime } = - \tfrac{1}{2}(b\wedge c^{\prime } + b^{\prime } \wedge c) \end{aligned}$$

with the other derivatives obtained by applying the cyclic permutation $(a,a^{\prime },p,p^{\prime })\rightarrow (b,b^{\prime },q,q^{\prime })\rightarrow (c,c^{\prime },r,r^{\prime })\rightarrow (a,a^{\prime },p,p^{\prime })$. We use $\mathop { \large \mathfrak S }$ to denote sums over this group of permutations.

The two-form $\omega _I$ of (2.3) is

$$\begin{aligned} \omega _I&= \lambda (a^{\prime }\wedge a + b^{\prime }\wedge b + c^{\prime }\wedge c) + \mu (p^{\prime }\wedge p + q^{\prime }\wedge q + r^{\prime }\wedge r) \\&= {\mathop { \large \mathfrak S }}(\lambda a^{\prime } \wedge a + \mu p^{\prime } \wedge p). \end{aligned}$$

On $N$, we have the forms ${\hat{\omega }}_J$ and ${\hat{\omega }}_K$ given by

$$\begin{aligned} \hat{\omega }_J + i \hat{\omega }_K = {\mathop { \large \mathfrak S }} \left( (p+ip^{\prime })\wedge (a+ia^{\prime })\right) . \end{aligned}$$

Choosing a local section $s$ of $\pi :N \rightarrow M$ such that $s({{\mathrm{Id}}}{ U }(3))={{\mathrm{Id}}}{ SU }(3)$ and $s^*e = 0$, we then obtain local two-forms

$$\begin{aligned} \omega _J = s^*\hat{\omega }_J,\quad \omega _K = s^*\hat{\omega }_K \end{aligned}$$

completing the triple of (2.3). The corresponding metric on $M$ is

$$\begin{aligned} g = {\mathop { \large \mathfrak S }}(\lambda (a^2 + {a^{\prime }}^2) + \mu (p^2 + {p^{\prime }}^2)) \end{aligned}$$

(4.1)

and condition (2.4) is simply

$$\begin{aligned} \lambda \mu = 1. \end{aligned}$$

(4.2)

These are the only invariant metrics on $M$ with normalised volume form, since TM (2.1) has exactly two irreducible summands.

At ${{\mathrm{Id}}}{ U }(3)$, the almost complex structures satisfy

$$\begin{aligned} {IA}&= A^{\prime },\quad IP = P^{\prime },\quad J\tfrac{1}{\sqrt{\lambda }}A = \tfrac{1}{\sqrt{\mu }}P,\quad J\tfrac{1}{\sqrt{\lambda }}A^{\prime } = -\tfrac{1}{\sqrt{\mu }}P^{\prime },\\&\quad \quad \quad K\tfrac{1}{\sqrt{\lambda }}A = \tfrac{1}{\sqrt{\mu }}P^{\prime },\quad K\tfrac{1}{\sqrt{\lambda }}A^{\prime } = \tfrac{1}{\sqrt{\mu }}P. \end{aligned}$$

These act on forms via $Ia = -a(I\cdot )$, so with the normalisation condition (4.2), we have $Ja=\mu p,\,Jp=-\lambda a$, etc.

Lemma 4.1

These describe all invariant almost quaternion-Hermitian structures on $M$ with normalised volume form.

Proof

We have noted above that (4.1) gives all the invariant metrics. Now, the local almost complex structures, or equivalently their Hermitian two-forms, associated with the almost quaternion-Hermitian structure span a ${ U }(3)$-invariant subspace $V$ of $\Lambda ^2T$ of dimension 3. Counting dimensions in the decomposition (2.2) shows that $V$ is a subspace of ${\mathbb{R }}\omega _0+{\mathbb{R }}{\tilde{\omega }}_0 + [\![{L^3}]\!]$. In particular, $V\cap [\![{L^3}]\!]$ is at least one-dimensional; ${ U }(3)$-invariance implies that $[\![{L^3}]\!]\leqslant V$. As $\omega _J$ and $\omega _K$ are $g$-orthogonal of the same length for each normalised $g$ in (4.1), we see that $J$ and $K$ are local almost complex structures belonging to the almost quaternion-Hermitian geometry. Finally, $I = JK$ is specified too.

Lemma 4.2

For the choices of $\omega _I,\,\omega _J$ and $\omega _K$ above normalised by (4.2) we have at the base point ${{\mathrm{Id}}}{ U }(3)\in M$ that

$$\begin{aligned} \text{ Id }\omega _I&= \text{ Id }_M\omega _I = (\tfrac{1}{2}\mu - 2\lambda )\Phi ,\\ \text{ Jd }\omega _J&= 2\lambda \Phi - \tfrac{1}{2}\mu ^3\Psi ,\quad \text{ Kd }\omega _K = 2\lambda \Phi + \tfrac{1}{2}\mu ^3\Psi , \end{aligned}$$

where

$$\begin{aligned} \Phi&= {\mathop { \large \mathfrak S }} (a\wedge b\wedge r - a^{\prime } \wedge b^{\prime } \wedge r + a \wedge b^{\prime } \wedge r^{\prime } + a^{\prime } \wedge b \wedge r^{\prime }),\\ \Psi&= {\mathop { \large \mathfrak S }}(p \wedge q \wedge r - 3 p \wedge q^{\prime } \wedge r^{\prime }) \end{aligned}$$

and $A d\omega _A(\cdot ,\cdot ,\cdot ) = - d\omega _A(A\cdot ,A\cdot ,A\cdot )$, for $A = I, J, K$.

Proof

As $\omega _I$ is ${ U }(3)$-invariant, we have $Id\omega _I=Id_M\omega _I$ which equals

$$\begin{aligned} (2\lambda -\tfrac{1}{2}\mu )I{\mathop { \large \mathfrak S }} (a\wedge b^{\prime } \wedge r + a^{\prime }\wedge b \wedge r - a \wedge b \wedge r^{\prime } + a^{\prime } \wedge b^{\prime } \wedge r^{\prime }) \end{aligned}$$

and gives the first claimed formula valid at any point of $M$.

For our choice of section $s$, we have at ${{\mathrm{Id}}}{ U }(3)$ that $Jd\, \omega _J = J s^*d_N{\tilde{\omega }}_J = Jd_M{\tilde{\omega }}_J$ which is

$$\begin{aligned} J{\mathop { \large \mathfrak S }}\left( -\tfrac{1}{2} a\wedge b\wedge c \!+\! \tfrac{3}{2} a \wedge b^{\prime }\wedge c^{\prime } \!+\!2(a\wedge q\wedge r \!-\! a\wedge q^{\prime }\wedge r^{\prime } \!+\! a^{\prime }\wedge q\wedge r^{\prime } + a^{\prime }\wedge q^{\prime }\wedge r)\right) . \end{aligned}$$

Combined with the description of $J$, we thus get the claimed formula. The computation for $Kd \omega _K$ is similar.

To compute the intrinsic torsion, we use the “minimal description” of [4] which relies on computing the forms $\beta _I=Jd \omega _J+Kd \omega _K$, etc., and the contractions $\Lambda _A\beta _B$ of $\beta _B$ with $\omega _A$. For our structures, we have at the base point

$$\begin{aligned} \beta _I = 4\lambda \Phi ,\quad \beta _J = \tfrac{1}{2}(\mu \Phi + \mu ^3\Psi ),\quad \beta _K = \tfrac{1}{2}(\mu \Phi - \mu ^3\Psi ) \end{aligned}$$

and all contractions $\Lambda _A\beta _B=0$. This confirms that the intrinsic torsion $\xi$ has no components in $[{E(S^3H+H)}]$.

Theorem 4.3

The component of $\xi$ in $[{\Lambda ^3_0\text{ ES }^3H}]$ is always non-zero, so the almost quaternion-Hermitian is never quaternionic. The component of $\xi$ in $[{\Lambda ^3_0\text{ EH }}]$ is zero if and only if $2\lambda = \mu$.

Proof

Since we have shown in §3 that $\xi$ has no component in $[{K(S^3H + H)}]$ and we saw above that each one form $\Lambda _A\beta _B$ is zero, at the base point, the results of [4] show that the $\Lambda ^3_0ES^3H$-component of $\xi$ corresponds to

$$\begin{aligned} \psi ^{(3)} := \tfrac{1}{12}(\beta _I+\beta _J+\beta _K) = \tfrac{1}{12}(4\lambda +\mu )\Phi \end{aligned}$$

which is always non-zero under condition (4.2). The component in $\Lambda ^3_0EH$ is determined by

$$\begin{aligned} \psi ^{(3)}_I := \tfrac{1}{8}(-\beta _I + 2(3 + \mathcal L _I)\psi ^{(3)}), \end{aligned}$$

where $\mathcal{L }_I = I_{(12)} + I_{(13)} + I_{(23)}$, with $I_{(12)}\alpha = \alpha (I\cdot ,I\cdot ,\cdot )$, etc. Now $\mathcal{L }_I\Phi = \Phi$, so

$$\begin{aligned} \psi ^{(3)}_I = \tfrac{1}{12}(\mu -2\lambda )\Phi \end{aligned}$$

and the result follows.

Corollary 4.4

The invariant almost quaternion-Hermitian structures on $M$ are not quaternionic integrable, and their quaternionic twistor spaces are not complex.

Proof

This follows directly from the following two facts [7]: (1) The underlying quaternionic structure is integrable if and only if the intrinsic torsion $\xi$ has no $S^3H$ component, i.e., it lies in $(\Lambda ^3_0E+K+E)H$. (2) The quaternionic twistor space is complex if and only if the underlying quaternionic structure is integrable. But, we have shown the $\Lambda ^3_0\, ES^3H$-component of $\xi$ is non-zero, so the result follows.

The almost Hermitian structure $(g,\omega _I)$ is easily seen to be integrable: $d_M(a+ia^{\prime })=-(b-ib^{\prime }) \wedge (r+ir^{\prime })+ (c-ic^{\prime })\wedge (q+iq^{\prime }) \in \Lambda ^{1,1}_I,\,d_M(p+ip^{\prime })=-\tfrac{1}{2}(b+ib^{\prime })\wedge (c+ic^{\prime }) \in \Lambda ^{2,0}_I$. In addition, from Lemma 4.2, we see that $d\omega _I$ is orthogonal to $\omega _I\wedge \Lambda ^1$. It follows that $d\omega _I$ is primitive.

Now, recall that Gray and Hervella [3] showed that the intrinsic torsion of an almost Hermitian structure $(g,\omega )$ lies in

$$\begin{aligned} \mathcal W = \mathcal W _1 + \mathcal W _2 + \mathcal W _3 + \mathcal W _4 = [\![{\Lambda ^{3,0}}]\!] + [\![{U^{3,0}}]\!] + [\![{\Lambda ^{2,1}_0}]\!] + [\![{\Lambda ^{1,0}}]\!], \end{aligned}$$

with $U^{3,0}$ irreducible: the $\mathcal{W }_1 + \mathcal{W }_2$-part is determined by the Nijenhuis tensor; the $\mathcal{W }_1 +\mathcal{W }_3 + \mathcal{W }_4$-part by $d\omega$. We now have from Lemma 4.2:

Proposition 4.5

The Hermitian structure $(g,\omega _I,I)$ is of Gray-Hervella type $\mathcal{W }_3$, except when $4\lambda =\mu$, when it is Kähler. Furthermore, the Kähler metric is symmetric.

Note that the Kähler parameters do not correspond to the parameters in Theorem 4.3 that give $\xi \in [{\Lambda ^3_0\,ES^3H}]$.

Proof

It remains to prove the last assertion. As in [8], note that ${ SO }(7)/{ U }(3)\cong { SO }(8)/{ U }(6)\cong { SO }(8)/{ SO }(2){ SO }(6)$, which is the quadric. The latter is isotropy irreducible and carries a unique ${ SO }(8)$-invariant metric with fixed volume, which is Hermitian symmetric so Kähler. However, we have seen that there is a unique Kähler metric with the same volume invariant under the smaller group ${ SO }(7)$, so these Kähler metrics must agree.

Remark 4.6

Each ${ SO }(7)$-invariant metric $g$ on $M$ is given by (4.1) and so is a Riemannian submersion over $\mathbb{C }\mathrm{P}(3)$ with fibre $S^6$. The standard theory of the canonical variation [2] tell us that precisely two of these metrics are Einstein. One is the symmetric case $4\lambda =\mu$. The other is when $8\lambda =3\mu$, as verified by Musso [6] in slightly different notation. Again these particular parameters are not those for which $\xi$ is special.

Remark 4.7

It can be shown that the local almost Hermitian structures $(g,\omega _J,J)$ and $(g,\omega _K,K)$ above are each of strict Gray-Hervella type $\mathcal{W }_1 + \mathcal{W }_3$ at the base point, unless $4\lambda = 3\mu$, when they have type $\mathcal{W }_1$. In particular, the Nijenhuis tensors $N_J$ and $N_K$ are skew-symmetric at the base point and equal to $\tfrac{1}{6}(4\lambda +\mu )(3\Phi \mp \mu ^2\Psi )$ at ${{\mathrm{Id}}}{ U }(3)$. In [4] we showed how $N_I$ is determined by $\text{ Jd }\omega _J - \text{ Kd }\omega _K$. In this case, we have the interesting situation that this latter tensor is non-zero, even though $N_I$ vanishes. Using [1], one can prove that the obstruction to quaternionic integrability is proportional to $N_I+N_J+N_K = (4\lambda +\mu )\Phi$, confirming that this is non-zero and the results of Corollary 4.4.