In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, an almost complex manifold is 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 ...
equipped with a smooth
linear complex structure
In mathematics, a complex structure on a real vector space V is an automorphism of V that squares to the minus identity, - \text_V . Such a structure on V allows one to define multiplication by complex scalars in a canonical fashion so as to re ...
on each
tangent space
In mathematics, the tangent space of a manifold is a generalization of to curves in two-dimensional space and to surfaces in three-dimensional space in higher dimensions. In the context of physics the tangent space to a manifold at a point can be ...
. Every
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 ...
is an almost complex manifold, but there are almost complex manifolds that are not complex manifolds. Almost complex structures have important applications in
symplectic geometry
Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form. Symplectic geometry has its origins in the ...
.
The concept is due to
Charles Ehresmann
Charles Ehresmann (19 April 1905 – 22 September 1979) was a German-born French mathematician who worked in differential topology and category theory.
He was an early member of the Bourbaki group, and is known for his work on the differentia ...
and
Heinz Hopf
Heinz Hopf (19 November 1894 – 3 June 1971) was a German mathematician who worked on the fields of dynamical systems, topology and geometry.
Early life and education
Hopf was born in Gräbschen, German Empire (now , part of Wrocław, Poland) ...
in the 1940s.
Formal definition
Let ''M'' be a smooth manifold. An almost complex structure ''J'' on ''M'' is a linear complex structure (that is, a
linear map
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that p ...
which squares to −1) on each tangent space of the manifold, which varies smoothly on the manifold. In other words, we have a
smooth tensor field
In mathematics and physics, a tensor field is a function assigning a tensor to each point of a region of a mathematical space (typically a Euclidean space or manifold) or of the physical space. Tensor fields are used in differential geometry, ...
''J'' of
degree such that
when regarded as a
vector 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 eve ...
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
on the
tangent bundle
A tangent bundle is the collection of all of the tangent spaces for all points on a manifold, structured in a way that it forms a new manifold itself. Formally, in differential geometry, the tangent bundle of a differentiable manifold M is ...
. A manifold equipped with an almost complex structure is called an almost complex manifold.
If ''M'' admits an almost complex structure, it must be even-dimensional. This can be seen as follows. Suppose ''M'' is ''n''-dimensional, and let be an almost complex structure. If then . But if ''M'' is a real manifold, then is a real number – thus ''n'' must be even if ''M'' has an almost complex structure. One can show that it must be
orientable
In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "anticlockwise". A space is o ...
as well.
An easy exercise in
linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrix (mathemat ...
shows that any even dimensional vector space admits a linear complex structure. Therefore, an even dimensional manifold always admits a -rank tensor ''pointwise'' (which is just a linear transformation on each tangent space) such that at each point ''p''. Only when this local tensor can be patched together to be defined globally does the pointwise linear complex structure yield an almost complex structure, which is then uniquely determined. The possibility of this patching, and therefore existence of an almost complex structure on a manifold ''M'' is equivalent to a
reduction of the structure group
In differential geometry, a ''G''-structure on an ''n''-manifold ''M'', for a given structure group ''G'', is a principal ''G''- subbundle of the tangent frame bundle F''M'' (or GL(''M'')) of ''M''.
The notion of ''G''-structures includes vario ...
of the tangent bundle from to . The existence question is then a purely
algebraic topological one and is fairly well understood.
Examples
For every integer n, the flat space R
2''n'' admits an almost complex structure. An example for such an almost complex structure is (1 ≤ ''j'', ''k'' ≤ 2''n''):
for odd ''j'',
for even ''j''.
The only
sphere
A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
s which admit almost complex structures are S
2 and S
6 (). In particular, S
4 cannot be given an almost complex
structure (Ehresmann and Hopf). In the case of S
2, the almost complex structure comes from an honest complex structure on the
Riemann sphere
In mathematics, the Riemann sphere, named after Bernhard Riemann,
is a Mathematical model, model of the extended complex plane (also called the closed complex plane): the complex plane plus one point at infinity. This extended plane represents ...
. The 6-sphere, S
6, when considered as the set of unit norm imaginary
octonion
In mathematics, the octonions are a normed division algebra over the real numbers, a kind of Hypercomplex number, hypercomplex Number#Classification, number system. The octonions are usually represented by the capital letter O, using boldface or ...
s, inherits an almost complex structure from the octonion multiplication; the question of whether it has a
complex structure is known as the ''Hopf problem,'' after
Heinz Hopf
Heinz Hopf (19 November 1894 – 3 June 1971) was a German mathematician who worked on the fields of dynamical systems, topology and geometry.
Early life and education
Hopf was born in Gräbschen, German Empire (now , part of Wrocław, Poland) ...
.
Differential topology of almost complex manifolds
Just as a complex structure on a vector space ''V'' allows a decomposition of ''V''
C into ''V''
+ and ''V''
− (the
eigenspace
In linear algebra, an eigenvector ( ) or characteristic vector is a Vector (mathematics and physics), vector that has its direction (geometry), direction unchanged (or reversed) by a given linear map, linear transformation. More precisely, an e ...
s of ''J'' corresponding to +''i'' and −''i'', respectively), so an almost complex structure on ''M'' allows a decomposition of the complexified tangent bundle ''TM''
C (which is the vector bundle of complexified tangent spaces at each point) into ''TM''
+ and ''TM''
−. A section of ''TM''
+ is called a
vector field
In vector calculus and physics, a vector field is an assignment of a vector to each point in a space, most commonly Euclidean space \mathbb^n. A vector field on a plane can be visualized as a collection of arrows with given magnitudes and dire ...
of type (1, 0), while a section of ''TM''
− is a vector field of type (0, 1). Thus ''J'' corresponds to multiplication by
''i'' on the (1, 0)-vector fields of the complexified tangent bundle, and multiplication by −''i'' on the (0, 1)-vector fields.
Just as we build
differential form
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications ...
s out of
exterior power
In mathematics, the exterior algebra or Grassmann algebra of a vector space V is an associative algebra that contains V, which has a product, called exterior product or wedge product and denoted with \wedge, such that v\wedge v=0 for every vector ...
s of the
cotangent bundle
In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold. It may be described also as the dual bundle to the tangent bundle. This m ...
, we can build exterior powers of the complexified cotangent bundle (which is canonically isomorphic to the bundle of dual spaces of the complexified tangent bundle). The almost complex structure induces the decomposition of each space of ''r''-forms
:
In other words, each Ω
''r''(''M'')
C admits a decomposition into a sum of Ω
(''p'', ''q'')(''M''), with ''r'' = ''p'' + ''q''.
As with any
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 ...
, there is a canonical projection π
''p'',''q'' from Ω
''r''(''M'')
C to Ω
(''p'',''q''). We also have the
exterior derivative
On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The re ...
''d'' which maps Ω
''r''(''M'')
C to Ω
''r''+1(''M'')
C. Thus we may use the almost complex structure to refine the action of the exterior derivative to the forms of definite type
:
:
so that
is a map which increases the holomorphic part of the type by one (takes forms of type (''p'', ''q'') to forms of type (''p''+1, ''q'')), and
is a map which increases the antiholomorphic part of the type by one. These operators are called the
Dolbeault operators.
Since the sum of all the projections must be the
identity map
Graph of the identity function on the real numbers
In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unc ...
, we note that the exterior derivative can be written
:
Integrable almost complex structures
Every
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 ...
is itself an almost complex manifold. In local holomorphic coordinates
one can define the maps
:
(just like a counterclockwise rotation of π/2) or
:
One easily checks that this map defines an almost complex structure. Thus any complex structure on a manifold yields an almost complex structure, which is said to be 'induced' by the complex structure, and the complex structure is said to be 'compatible with' the almost complex structure.
The converse question, whether the almost complex structure implies the existence of a complex structure is much less trivial, and not true in general. On an arbitrary almost complex manifold one can always find coordinates for which the almost complex structure takes the above canonical form at any given point ''p''. In general, however, it is not possible to find coordinates so that ''J'' takes the canonical form on an entire
neighborhood
A neighbourhood (Commonwealth English) or neighborhood (American English) is a geographically localized community within a larger town, city, suburb or rural area, sometimes consisting of a single street and the buildings lining it. Neigh ...
of ''p''. Such coordinates, if they exist, are called 'local holomorphic coordinates for J'. If ''M'' admits local holomorphic coordinates for ''J'' around every point then these patch together to form a
holomorphic
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 deri ...
atlas
An atlas is a collection of maps; it is typically a bundle of world map, maps of Earth or of a continent or region of Earth. Advances in astronomy have also resulted in atlases of the celestial sphere or of other planets.
Atlases have traditio ...
for ''M'' giving it a complex structure, which moreover induces ''J''. ''J'' is then said to be '
integrable'. If ''J'' is induced by a complex structure, then it is induced by a unique complex structure.
Given any linear map ''A'' on each tangent space of ''M''; i.e., ''A'' is a tensor field of rank (1, 1), then the Nijenhuis tensor is a tensor field of rank (1,2) given by
:
or, for the usual case of an almost complex structure ''A=J'' such that
,
:
The individual expressions on the right depend on the choice of the smooth vector fields ''X'' and ''Y'', but the left side actually depends only on the pointwise values of ''X'' and ''Y'', which is why ''N''
''A'' is a tensor. This is also clear from the component formula
:
In terms of the
Frölicher–Nijenhuis bracket, which generalizes the Lie bracket of vector fields, the Nijenhuis tensor ''N
A'' is just one-half of
'A'', ''A''
The Newlander–Nirenberg theorem states that an almost complex structure ''J'' is integrable if and only if ''N
J'' = 0. The compatible complex structure is unique, as discussed above. Since the existence of an integrable almost complex structure is equivalent to the existence of a complex structure, this is sometimes taken as the definition of a complex structure.
There are several other criteria which are equivalent to the vanishing of the Nijenhuis tensor, and which therefore furnish methods for checking the integrability of an almost complex structure (and in fact each of these can be found in the literature):
*The Lie bracket of any two (1, 0)-vector fields is again of type (1, 0)
*
*
Any of these conditions implies the existence of a unique compatible complex structure.
The existence of an almost complex structure is a topological question and is relatively easy to answer, as discussed above. The existence of an integrable almost complex structure, on the other hand, is a much more difficult analytic question. For example, it is still not known whether S
6 admits an integrable almost complex structure, despite a long history of ultimately unverified claims. Smoothness issues are important. For
real-analytic
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...
''J'', the Newlander–Nirenberg theorem follows from the
Frobenius theorem; for ''C''
∞ (and less smooth) ''J'', analysis is required (with more difficult techniques as the regularity hypothesis weakens).
Compatible triples
Suppose ''M'' is equipped with a
symplectic form
In mathematics, a symplectic vector space is a vector space V over a field F (for example the real numbers \mathbb) equipped with a symplectic bilinear form.
A symplectic bilinear form is a mapping \omega : V \times V \to F that is
; Bilinear: ...
''ω'', a
Riemannian metric
In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...
''g'', and an almost complex structure ''J''. Since ''ω'' and ''g'' are
nondegenerate, each induces a bundle isomorphism ''TM → T*M'', where the first map, denoted ''φ''
''ω'', is given by the
interior product
In mathematics, the interior product (also known as interior derivative, interior multiplication, inner multiplication, inner derivative, insertion operator, contraction, or inner derivation) is a degree −1 (anti)derivation on the exterio ...
''φ''
''ω''(''u'') = ''i''
''u''''ω'' = ''ω''(''u'', •) and the other, denoted ''φ''
''g'', is given by the analogous operation for ''g''. With this understood, the three structures (''g'', ''ω'', ''J'') form a compatible triple when each structure can be specified by the two others as follows:
*''g''(''u'', ''v'') = ''ω''(''u'', ''Jv'')
*ω(''u'', ''v'') = ''g''(''Ju'', ''v'')
*''J''(''u'') = (''φ''
''g'')
−1(''φ''
''ω''(''u'')).
In each of these equations, the two structures on the right hand side are called compatible when the corresponding construction yields a structure of the type specified. For example, ''ω'' and ''J'' are compatible if and only if ''ω''(•, ''J''•) is a Riemannian metric. The bundle on ''M'' whose sections are the almost complex structures compatible to ''ω'' has contractible fibres: the complex structures on the tangent fibres compatible with the restriction to the symplectic forms.
Using elementary properties of the symplectic form ''ω'', one can show that a compatible almost complex structure ''J'' is an
almost Kähler structure for the Riemannian metric ''ω''(''u'', ''Jv''). Also, if ''J'' is integrable, then (''M'', ''ω'', ''J'') is a
Kähler manifold
In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. The concept was first studied by Jan Arnol ...
.
These triples are related to the
2 out of 3 property of the unitary group.
Generalized almost complex structure
Nigel Hitchin introduced the notion of a
generalized almost complex structure on the manifold ''M'', which was elaborated in the doctoral dissertations of his students
Marco Gualtieri and
Gil Cavalcanti. An ordinary almost complex structure is a choice of a half-dimensional
subspace of each fiber of the complexified
tangent bundle
A tangent bundle is the collection of all of the tangent spaces for all points on a manifold, structured in a way that it forms a new manifold itself. Formally, in differential geometry, the tangent bundle of a differentiable manifold M is ...
''TM''. A generalized almost complex structure is a choice of a half-dimensional
isotropic
In physics and geometry, isotropy () is uniformity in all orientations. Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence '' anisotropy''. ''Anisotropy'' is also ...
subspace of each fiber of the
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 the complexified tangent and
cotangent bundle
In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold. It may be described also as the dual bundle to the tangent bundle. This m ...
s. In both cases one demands that the direct sum of the
subbundle and its
complex conjugate
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, if a and b are real numbers, then the complex conjugate of a + bi is a - ...
yield the original bundle.
An almost complex structure integrates to a complex structure if the half-dimensional subspace is closed under the
Lie bracket
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identit ...
. A generalized almost complex structure integrates to a
generalized complex structure
In the field of mathematics known as differential geometry, a generalized complex structure is a property of a differential manifold that includes as special cases a complex structure and a symplectic structure. Generalized complex structures w ...
if the subspace is closed under the
Courant bracket. If furthermore this half-dimensional space is the annihilator of a nowhere vanishing
pure spinor then ''M'' is a
generalized Calabi–Yau manifold.
See also
*
*
*
*
*
*
*
References
*
* Information on compatible triples, Kähler and Hermitian manifolds, etc.
* Short section which introduces standard basic material.
*
*
{{Manifolds
Smooth manifolds