In
differential geometry
Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...
, a quaternionic manifold is a
quaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. The algebra of quater ...
ic analog of a
complex manifold
In differential geometry and complex geometry, a complex manifold is a manifold with a ''complex structure'', that is an atlas (topology), atlas of chart (topology), charts to the open unit disc in the complex coordinate space \mathbb^n, such th ...
. 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 function
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex de ...
s 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 may ...
of real dimension 4''n'' equipped with a torsion-free
-structure. More naïve, but straightforward, definitions lead to a dearth of examples, and exclude spaces like
quaternionic projective space
In mathematics, quaternionic projective space is an extension of the ideas of real projective space and complex projective space, to the case where coordinates lie in the ring of quaternions \mathbb. Quaternionic projective space of dimension ''n'' ...
which should clearly be considered as quaternionic manifolds.
Early history
Marcel Berger's 1955 paper 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 and Kraines who have independently proven that any such manifold admits a parallel 4-form
.The long-awaited analog of strong Lefschetz theorem was published
in 1982 :
Definitions
The enhanced quaternionic general linear group
If we regard the
quaternionic vector space as a
right -module, we can identify the algebra of right
-linear maps with the algebra of
quaternionic matrices acting on
''from the left''. The invertible right
-linear maps then form a subgroup
of
. We can enhance this group with the group
of nonzero quaternions acting by scalar multiplication on
''from the right''. Since this scalar multiplication is
-linear (but ''not''
-linear) we have another embedding of
into
. The group
is then defined as the product of these subgroups in
. Since the intersection of the subgroups
and
in
is their mutual center
(the group of scalar matrices with nonzero real coefficients), we have the isomorphism
:
Almost quaternionic structure
An almost quaternionic structure on a smooth manifold
is just a
-structure on
. Equivalently, it can be defined as a
subbundle of the
endomorphism bundle such that each fiber
is isomorphic (as a
real algebra) to the
quaternion algebra
In mathematics, a quaternion algebra over a field (mathematics), 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 (vector space), dimension 4 ove ...
. The subbundle
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
naturally admits a
bundle metric In differential geometry, the notion of a metric tensor can be extended to an arbitrary vector bundle, and to some principal fiber bundles. This metric is often called a bundle metric, or fibre metric.
Definition
If ''M'' is a topological manifold ...
coming from the quaternionic algebra structure, and, with this metric,
splits into an orthogonal
direct sum
The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently but analogously for different kinds of structures. As an example, the direct sum of two abelian groups A and B is anothe ...
of vector bundles
where
is the trivial line bundle through the identity operator, and
is a rank-3 vector bundle corresponding to the purely imaginary quaternions. Neither the bundles
or
are necessarily trivial.
The
unit sphere bundle
inside
corresponds to the pure unit imaginary quaternions. These are endomorphisms of the tangent spaces that square to −1. The bundle
is called the twistor space of the manifold
, and its properties are described in more detail below.
Local sections of
are (locally defined)
almost complex structures. There exists a neighborhood
of every point
in an almost quaternionic manifold
with an entire
2-sphere
A sphere (from Greek , ) is a surface analogous to the circle, a curve. In solid geometry, 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 ''center' ...
of almost complex structures defined on
. One can always find
such that
:
Note, however, that none of these operators may be extendable to all of
. That is, the bundle
may admit no ''global'' sections (e.g. this is the case with
quaternionic projective space
In mathematics, quaternionic projective space is an extension of the ideas of real projective space and complex projective space, to the case where coordinates lie in the ring of quaternions \mathbb. Quaternionic projective space of dimension ''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
is an almost quaternionic structure
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 ...
preserving
. Such a connection is never unique, and is not considered to be part of the quaternionic structure. A quaternionic manifold is a smooth manifold
together with a quaternionic structure on
.
Special cases and additional structures
Hypercomplex manifolds
A
hypercomplex manifold is a quaternionic manifold with a torsion-free
-structure. The reduction of the structure group to
is possible if and only if the almost quaternionic structure bundle
is trivial (i.e. isomorphic to
). An almost hypercomplex structure corresponds to a global frame of
, or, equivalently, triple of almost complex structures
, and
such that
:
A hypercomplex structure is an almost hypercomplex structure such that each of
, and
are integrable.
Quaternionic Kähler manifolds
A
quaternionic Kähler manifold
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. The algebra of quaternions ...
is a quaternionic manifold with a torsion-free
-structure.
Hyperkähler manifolds
A
hyperkähler manifold is a quaternionic manifold with a torsion-free
-structure. A hyperkähler manifold is simultaneously a hypercomplex manifold and a quaternionic Kähler manifold.
Twistor space
Given a quaternionic
-manifold
, the unit 2-sphere subbundle
corresponding to the pure unit imaginary quaternions (or almost complex structures) is called the twistor space of
. It turns out that, when
, there exists a natural
complex structure on
such that the fibers of the projection
are isomorphic to
. When
, the space
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
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
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...
. Unfortunately, the twistor space of a quaternionic manifold can be quite complicated, even for simple spaces like
.
References
*
*
{{Manifolds
Differential geometry
Manifolds
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...
Structures on manifolds