Marcel Berger's 1955 paper^[1] on the classification of Riemannian holonomy groups first raised the issue of the existence of non-symmetric manifolds with holonomy Sp(n)·Sp(1).Interesting results were proved in the mid-1960s in pioneering work by Edmond Bonan^[2] and Kraines^[3] who have independently proven that any such manifold admits a parallel 4-form ${\displaystyle \Omega }$ .The long-awaited analog of strong Lefschetz theorem was published ^[4] in 1982 : ${\displaystyle \Omega ^{n-k}\wedge \bigwedge ^{2k}T^{*}M=\bigwedge ^{4n-2k}T^{*}M.}$ 

The enhanced quaternionic general linear group edit

If we regard the quaternionic vector space ${\displaystyle \mathbb {H} ^{n}\cong \mathbb {R} ^{4n}}$  as a right ${\displaystyle \mathbb {H} }$ -module, we can identify the algebra of right ${\displaystyle \mathbb {H} }$ -linear maps with the algebra of ${\displaystyle n\times n}$  quaternionic matrices acting on ${\displaystyle \mathbb {H} ^{n}}$  from the left. The invertible right ${\displaystyle \mathbb {H} }$ -linear maps then form a subgroup ${\displaystyle \operatorname {GL} (n,\mathbb {H} )}$  of ${\displaystyle \operatorname {GL} (4n,\mathbb {R} )}$ . We can enhance this group with the group ${\displaystyle \mathbb {H} ^{\times }}$  of nonzero quaternions acting by scalar multiplication on ${\displaystyle \mathbb {H} ^{n}}$  from the right. Since this scalar multiplication is ${\displaystyle \mathbb {R} }$ -linear (but not ${\displaystyle \mathbb {H} }$ -linear) we have another embedding of ${\displaystyle \mathbb {H} ^{\times }}$  into ${\displaystyle \operatorname {GL} (4n,\mathbb {R} )}$ . The group ${\displaystyle \operatorname {GL} (n,\mathbb {H} )\cdot \mathbb {H} ^{\times }}$  is then defined as the product of these subgroups in ${\displaystyle \operatorname {GL} (4n,\mathbb {R} )}$ . Since the intersection of the subgroups ${\displaystyle \operatorname {GL} (n,\mathbb {H} )}$  and ${\displaystyle \mathbb {H} ^{\times }}$  in ${\displaystyle \operatorname {GL} (4n,\mathbb {R} )}$  is their mutual center ${\displaystyle \mathbb {R} ^{\times }}$  (the group of scalar matrices with nonzero real coefficients), we have the isomorphism

${\displaystyle \operatorname {GL} (n,\mathbb {H} )\cdot \mathbb {H} ^{\times }\cong (\operatorname {GL} (n,\mathbb {H} )\times \mathbb {H} ^{\times })/\mathbb {R} ^{\times }.}$ 

Almost quaternionic structure edit

An almost quaternionic structure on a smooth manifold ${\displaystyle M}$  is just a ${\displaystyle \operatorname {GL} (n,\mathbb {H} )\cdot \mathbb {H} ^{\times }}$ -structure on ${\displaystyle M}$ . Equivalently, it can be defined as a subbundle ${\displaystyle H}$  of the endomorphism bundle ${\displaystyle \operatorname {End} (TM)}$  such that each fiber ${\displaystyle H_{x}}$  is isomorphic (as a real algebra) to the quaternion algebra ${\displaystyle \mathbb {H} }$ . The subbundle ${\displaystyle H}$  is called the almost quaternionic structure bundle. A manifold equipped with an almost quaternionic structure is called an almost quaternionic manifold.

The quaternion structure bundle ${\displaystyle H}$  naturally admits a bundle metric coming from the quaternionic algebra structure, and, with this metric, ${\displaystyle H}$  splits into an orthogonal direct sum of vector bundles ${\displaystyle H=L\oplus E}$  where ${\displaystyle L}$  is the trivial line bundle through the identity operator, and ${\displaystyle E}$  is a rank-3 vector bundle corresponding to the purely imaginary quaternions. Neither the bundles ${\displaystyle H}$  or ${\displaystyle E}$  are necessarily trivial.

The unit sphere bundle ${\displaystyle Z=S(E)}$  inside ${\displaystyle E}$  corresponds to the pure unit imaginary quaternions. These are endomorphisms of the tangent spaces that square to −1. The bundle ${\displaystyle Z}$  is called the twistor space of the manifold ${\displaystyle M}$ , and its properties are described in more detail below. Local sections of ${\displaystyle Z}$  are (locally defined) almost complex structures. There exists a neighborhood ${\displaystyle U}$  of every point ${\displaystyle x}$  in an almost quaternionic manifold ${\displaystyle M}$  with an entire 2-sphere of almost complex structures defined on ${\displaystyle U}$ . One can always find ${\displaystyle I,J,K\in \Gamma (Z|_{U})}$  such that

${\displaystyle I^{2}=J^{2}=K^{2}=IJK=-1}$ 

Note, however, that none of these operators may be extendable to all of ${\displaystyle M}$ . That is, the bundle ${\displaystyle Z}$  may admit no global sections (e.g. this is the case with quaternionic projective space ${\displaystyle \mathbb {HP} ^{n}}$ ). This is in marked contrast to the situation for complex manifolds, which always have a globally defined almost complex structure.

Quaternionic structure edit

A quaternionic structure on a smooth manifold ${\displaystyle M}$  is an almost quaternionic structure ${\displaystyle Q}$  which admits a torsion-free affine connection ${\displaystyle \nabla }$  preserving ${\displaystyle Q}$ . Such a connection is never unique, and is not considered to be part of the quaternionic structure. A quaternionic manifold is a smooth manifold ${\displaystyle M}$  together with a quaternionic structure on ${\displaystyle M}$ .

Hypercomplex manifolds edit

A hypercomplex manifold is a quaternionic manifold with a torsion-free ${\displaystyle \operatorname {GL} (n,\mathbb {H} )}$ -structure. The reduction of the structure group to ${\displaystyle \operatorname {GL} (n,\mathbb {H} )}$  is possible if and only if the almost quaternionic structure bundle ${\displaystyle H\subset \operatorname {End} (TM)}$  is trivial (i.e. isomorphic to ${\displaystyle M\times \mathbb {H} }$ ). An almost hypercomplex structure corresponds to a global frame of ${\displaystyle H}$ , or, equivalently, triple of almost complex structures ${\displaystyle I,J}$ , and ${\displaystyle K}$  such that

${\displaystyle I^{2}=J^{2}=K^{2}=IJK=-1}$ 

A hypercomplex structure is an almost hypercomplex structure such that each of ${\displaystyle I,J}$ , and ${\displaystyle K}$  are integrable.

Quaternionic Kähler manifolds edit

A quaternionic Kähler manifold is a quaternionic manifold with a torsion-free ${\displaystyle \operatorname {Sp} (n)\cdot \operatorname {Sp} (1)}$ -structure.

Hyperkähler manifolds edit

A hyperkähler manifold is a quaternionic manifold with a torsion-free ${\displaystyle \operatorname {Sp} (n)}$ -structure. A hyperkähler manifold is simultaneously a hypercomplex manifold and a quaternionic Kähler manifold.

Given a quaternionic ${\displaystyle n}$ -manifold ${\displaystyle M}$ , the unit 2-sphere subbundle ${\displaystyle Z=S(E)}$  corresponding to the pure unit imaginary quaternions (or almost complex structures) is called the twistor space of ${\displaystyle M}$ . It turns out that, when ${\displaystyle n\geq 2}$ , there exists a natural complex structure on ${\displaystyle Z}$  such that the fibers of the projection ${\displaystyle Z\to M}$  are isomorphic to ${\displaystyle \mathbb {CP} ^{1}}$ . When ${\displaystyle n=1}$ , the space ${\displaystyle Z}$  admits a natural almost complex structure, but this structure is integrable only if the manifold is self-dual. It turns out that the quaternionic geometry on ${\displaystyle M}$  can be reconstructed entirely from holomorphic data on ${\displaystyle Z}$ .

The twistor space theory gives a method of translating problems on quaternionic manifolds into problems on complex manifolds, which are much better understood, and amenable to methods from algebraic geometry. Unfortunately, the twistor space of a quaternionic manifold can be quite complicated, even for simple spaces like ${\displaystyle \mathbb {H} ^{n}}$ .

• Besse, Arthur L. (1987). Einstein Manifolds. Berlin: Springer-Verlag. ISBN 3-540-15279-2.
• Joyce, Dominic (2000). Compact Manifolds with Special Holonomy. Oxford University Press. ISBN 0-19-850601-5.