Quaternionic manifold
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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 In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
of real dimension 4''n'' equipped with a torsion-free \operatorname(n, \mathbb)\cdot\mathbb^\times-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 \mathbb^n\cong\R^ as a right \mathbb-module, we can identify the algebra of right \mathbb-linear maps with the algebra of n\times n quaternionic matrices acting on \mathbb^n ''from the left''. The invertible right \mathbb-linear maps then form a subgroup \operatorname(n, \mathbb) of \operatorname(4n, \R). We can enhance this group with the group \mathbb^\times of nonzero quaternions acting by scalar multiplication on \mathbb^n ''from the right''. Since this scalar multiplication is \R-linear (but ''not'' \mathbb-linear) we have another embedding of \mathbb^\times into \operatorname(4n, \R). The group \operatorname(n, \mathbb)\cdot\mathbb^\times is then defined as the product of these subgroups in \operatorname(4n, \R). Since the intersection of the subgroups \operatorname(n, \mathbb) and \mathbb^\times in \operatorname(4n, \R) is their mutual center \R^\times (the group of scalar matrices with nonzero real coefficients), we have the isomorphism :\operatorname(n, \mathbb)\cdot\mathbb^\times \cong (\operatorname(n, \mathbb)\times\mathbb^\times)/\R^\times.


Almost quaternionic structure

An almost quaternionic structure on a smooth manifold M is just a \operatorname(n, \mathbb)\cdot\mathbb^\times-structure on M. Equivalently, it can be defined as a subbundle H of the
endomorphism bundle In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every po ...
\operatorname(TM) such that each fiber H_x is isomorphic (as a real algebra) to the
quaternion algebra In mathematics, a quaternion algebra over a field ''F'' is a central simple algebra ''A'' over ''F''See Milies & Sehgal, An introduction to group rings, exercise 17, chapter 2. that has dimension 4 over ''F''. Every quaternion algebra becomes a ma ...
\mathbb. The subbundle 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 H naturally admits a bundle metric coming from the quaternionic algebra structure, and, with this metric, H splits into an orthogonal direct sum of vector bundles H = L\oplus E where L is the trivial line bundle through the identity operator, and E is a rank-3 vector bundle corresponding to the purely imaginary quaternions. Neither the bundles H or E are necessarily trivial. The unit sphere bundle Z = S(E) inside E corresponds to the pure unit imaginary quaternions. These are endomorphisms of the tangent spaces that square to −1. The bundle Z is called the twistor space of the manifold M, and its properties are described in more detail below. Local sections of Z are (locally defined)
almost complex structure In mathematics, an almost complex manifold is a smooth manifold equipped with a smooth linear complex structure on each tangent space. Every complex manifold is an almost complex manifold, but there are almost complex manifolds that are not complex ...
s. There exists a neighborhood U of every point x in an almost quaternionic manifold M with an entire
2-sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ce ...
of almost complex structures defined on U. One can always find I, J, K\in\Gamma(Z, _U) such that :I^2 = J^2 = K^2 = IJK = -1 Note, however, that none of these operators may be extendable to all of M. That is, the bundle Z may admit no ''global'' sections (e.g. this is the case with quaternionic projective space \mathbb^n). 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 M is an almost quaternionic structure Q which admits a torsion-free
affine connection In differential geometry, an affine connection is a geometric object on a smooth manifold which ''connects'' nearby tangent spaces, so it permits tangent vector fields to be differentiated as if they were functions on the manifold with values i ...
\nabla preserving 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 M together with a quaternionic structure on M.


Special cases and additional structures


Hypercomplex manifolds

A hypercomplex manifold is a quaternionic manifold with a torsion-free \operatorname(n, \mathbb)-structure. The reduction of the structure group to \operatorname(n, \mathbb) is possible if and only if the almost quaternionic structure bundle H\subset \operatorname(TM) is trivial (i.e. isomorphic to M\times\mathbb). An almost hypercomplex structure corresponds to a global frame of H, or, equivalently, triple of almost complex structures I, J, and K such that :I^2 = J^2 = K^2 = IJK = -1 A hypercomplex structure is an almost hypercomplex structure such that each of I, J, and K are integrable.


Quaternionic Kähler manifolds

A quaternionic Kähler manifold is a quaternionic manifold with a torsion-free \operatorname(n)\cdot\operatorname(1)-structure.


Hyperkähler manifolds

A
hyperkähler manifold In differential geometry, a hyperkähler manifold is a Riemannian manifold (M, g) endowed with three integrable almost complex structures I, J, K that are Kähler with respect to the Riemannian metric g and satisfy the quaternionic relations I^2 ...
is a quaternionic manifold with a torsion-free \operatorname(n)-structure. A hyperkähler manifold is simultaneously a hypercomplex manifold and a quaternionic Kähler manifold.


Twistor space

Given a quaternionic n-manifold M, the unit 2-sphere subbundle Z=S(E) corresponding to the pure unit imaginary quaternions (or almost complex structures) is called the twistor space of M. It turns out that, when n\ge 2, there exists a natural complex structure on Z such that the fibers of the projection Z\to M are isomorphic to \mathbb^1. When n=1, the space Z admits a natural
almost complex structure In mathematics, an almost complex manifold is a smooth manifold equipped with a smooth linear complex structure on each tangent space. Every complex manifold is an almost complex manifold, but there are almost complex manifolds that are not complex ...
, but this structure is integrable only if the manifold is self-dual. It turns out that the quaternionic geometry on M can be reconstructed entirely from holomorphic data on 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 \mathbb^n.


References

* * {{Manifolds Differential geometry Manifolds Quaternions Structures on manifolds