Quaternionic manifold

In differential geometry, a quaternionic manifold is a quaternionic analog of a complex manifold. The definition is more complicated and technical than the one for complex manifolds due in part to the noncommutativity of the quaternions and in part to the lack of a suitable calculus of holomorphic functions for quaternions. The most succinct definition uses the language of "G"-structures on a manifold. Specifically, a quaternionic "n-"manifold can be defined as a smooth manifold of real dimension 4"n" equipped with a torsion-free formula_1-structure. More naïve, but straightforward, definitions lead to a dearth of examples, and exclude spaces like quaternionic projective space which should clearly be considered as quaternionic manifolds.
Definitions.
The enhanced quaternionic general linear group.
If we regard the quaternionic vector space formula_2 as a right formula_3-module, we can identify the algebra of right formula_3-linear maps with the algebra of formula_5 quaternionic matrices acting on formula_6 "from the left". The invertible right formula_3-linear maps then form a subgroup formula_8 of formula_9. We can enhance this group with the group formula_10 of nonzero quaternions acting by scalar multiplication on formula_6 "from the right". Since this scalar multiplication is formula_12-linear (but "not" formula_3-linear) we have another embedding of formula_10 into formula_9. The group formula_1 is then defined as the product of these subgroups in formula_9. Since the intersection of the subgroups formula_8 and formula_10 in formula_9 is their mutual center formula_21 (the group of scalar matrices with nonzero real coefficients), we have the isomorphism
Almost quaternionic structure.
An almost quaternionic structure on a smooth manifold formula_23 is just a formula_1-structure on formula_23. Equivalently, it can be defined as a subbundle formula_26 of the endomorphism bundle formula_27 such that each fiber formula_28 is isomorphic (as a real algebra) to the quaternion algebra formula_3. The subbundle formula_26 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 formula_26 naturally admits a bundle metric coming from the quaternionic algebra structure, and, with this metric, formula_26 splits into an orthogonal direct sum of vector bundles
formula_33
where formula_34 is the trivial line bundle through the identity operator, and formula_35 is a rank-3 vector bundle corresponding to the purely imaginary quaternions. Neither the bundles formula_26 or formula_35 are necessarily trivial.
The unit sphere bundle
formula_38
inside formula_35 corresponds to the pure unit imaginary quaternions. These are endomorphisms of the tangent spaces that square to −1. The bundle formula_40 is called the twistor space of the manifold formula_23, and its properties are described in more detail below. Local sections of formula_40 are (locally defined) almost complex structures. There exists a neighborhood formula_43 of every point formula_44 in an almost quaternionic manifold formula_23 with an entire 2-sphere of almost complex structures defined on formula_43. One can always find formula_47 such that 
Note, however, that none of these operators may be extendable to all of formula_23. That is, the bundle formula_40 may admit no "global" sections (e.g. this is the case with quaternionic projective space formula_51). This is in marked contrast to the situation for complex manifolds, which always have a globally defined almost complex structure.
Quaternionic structure.
A quaternionic structure on a smooth manifold formula_23 is an almost quaternionic structure formula_53 which admits a torsion-free affine connection formula_54 preserving formula_53. Such a connection is never unique, and is not considered to be part of the quaternionic structure. A quaternionic manifold is a smooth manifold formula_23 together with a quaternionic structure on formula_23.
Special cases and additional structures.
Hypercomplex manifolds.
A hypercomplex manifold is a quaternionic manifold with a torsion-free formula_8-structure. The reduction of the structure group to formula_8 is possible if and only if the almost quaternionic structure bundle formula_60 is trivial (i.e. isomorphic to formula_61). An almost hypercomplex structure corresponds to a global frame of formula_26, or, equivalently, triple of almost complex structures formula_63, and formula_64 such that
A hypercomplex structure is an almost hypercomplex structure such that each of formula_63, and formula_64 are integrable. 
Quaternionic Kähler manifolds.
A quaternionic Kähler manifold is a quaternionic manifold with a torsion-free formula_68-structure.
Hyperkähler manifolds.
A hyperkähler manifold is a quaternionic manifold with a torsion-free formula_69-structure. A hyperkähler manifold is simultaneously a hypercomplex manifold and a quaternionic Kähler manifold.
Twistor space.
Given a quaternionic formula_70-manifold formula_23, the unit 2-sphere subbundle formula_72 corresponding to the pure unit imaginary quaternions (or almost complex structures) is called the twistor space of formula_23. It turns out that, when formula_74, there exists a natural complex structure on formula_40 such that the fibers of the projection formula_76 are isomorphic to formula_77. When formula_78, the space formula_40 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 formula_23 can be reconstructed entirely from holomorphic data on formula_40.
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 formula_6.