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 4n equipped with a torsion-free
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.
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
.The long-awaited analog of strong Lefschetz theorem was published [4] in 1982 :
If we regard the quaternionic vector space
of nonzero quaternions acting by scalar multiplication on
(the group of scalar matrices with nonzero real coefficients), we have the isomorphism An almost quaternionic structure on a smooth manifold
is called the almost quaternionic structure bundle.
naturally admits a bundle metric coming from the quaternionic algebra structure, and, with this metric,
splits into an orthogonal direct sum of vector bundles
is the trivial line bundle through the identity operator, and
is a rank-3 vector bundle corresponding to the purely imaginary quaternions.
corresponds to the pure unit imaginary quaternions.
These are endomorphisms of the tangent spaces that square to −1.
is called the twistor space of the manifold
are (locally defined) almost complex structures.
with an entire 2-sphere of almost complex structures defined on
may admit no global sections (e.g. this is the case with quaternionic projective space
This is in marked contrast to the situation for complex manifolds, which always have a globally defined almost complex structure.
A quaternionic structure on a smooth manifold
which admits a torsion-free affine connection
Such a connection is never unique, and is not considered to be part of the quaternionic structure.
An almost hypercomplex structure corresponds to a global frame of
, or, equivalently, triple of almost complex structures
corresponding to the pure unit imaginary quaternions (or almost complex structures) is called the twistor space of
, there exists a natural complex structure on
can be reconstructed entirely from holomorphic data on
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.