Almost-complex structure

A tensor field of linear transformations of the tangent spaces on a manifold satisfying the condition

i.e. a field of complex structures in the tangent spaces , . An almost-complex structure determines a decomposition of the complexification of the tangent bundle in a direct sum of two complex mutually-conjugate subbundles and consisting of eigen vectors of the affinor (extended by linearity on ) with eigen values and , respectively. Conversely, a decomposition of in a direct sum of mutually-conjugate vector subbundles defines an almost-complex structure on for which .

An almost-complex structure is called integrable if it is induced by a complex structure on , i.e. if there exists an atlas of admissible charts of the manifold in which the field has constant coordinates . A necessary and sufficient condition for the integrability of an almost-complex structure is that the subbundle is involutive, i.e. that the space of its sections is closed with respect to commutation of (complex) vector fields. The condition for the subbundle to be involutive is equivalent to the vanishing of the vector-valued -form associated with and given by the formula

where and vector fields. This form is called the torsion tensor, or the Nijenhuis tensor, of the almost-complex structure. The torsion tensor can be considered as first-order differentiation on the algebra of differential forms on of the form

where is the exterior differential and is considered as a differentiation of order zero.

From the point of view of the theory of -structures an almost-complex structure is a -structure, where , and the torsion tensor is the tensor defined by the first structure function of this structure. A -structure is a structure of elliptic type, therefore the Lie algebra of infinitesimal automorphisms of an almost-complex structure satisfies a second-order system of elliptic differential equations [1]. In particular, the Lie algebra of infinitesimal automorphisms of an almost-complex structure on a compact manifold is finite-dimensional, and the group of all automorphisms of a compact manifold with an almost-complex structure is a Lie group. For non-compact manifolds this statement is, in general, not true.

If the automorphism group acts transitively on the manifold , then the almost-complex structure is uniquely defined by its value at a fixed point . This represents an invariant complex structure in the tangent space relative to the isotropic representation (see Invariant object on a homogeneous space). Methods of the theory of Lie groups allow one to construct a wide class of homogeneous spaces having an invariant almost-complex structure (both integrable and non-integrable) and to classify invariant almost-complex structures under different assumptions (see [2]). For instance, any quotient space of a Lie group by the subgroup consisting of fixed points of an automorphism of even order of has an invariant almost-complex structure. An example is the -dimensional sphere , considered as the homogeneous space ; none of the invariant almost-complex structures on is integrable.

The existence of an almost-complex structure on a manifold imposes certain restrictions on its topology — it must be of even dimension, oriented, and in the compact case all its odd-dimensional Stiefel–Whitney classes must vanish. Among the spheres only the spheres of dimensions 2 and 6 admit an almost-complex structure.

References

[1]	S. Kobayashi, "Transformation groups in differential geometry" , Springer (1972)
[2]	B.P. Komrakov, "Structure on manifolds and homogeneous spaces" , Minsk (1978) (In Russian)
[3]	A. Lichnerowicz, "Global theory of connections and holonomy groups" , Noordhoff (1976) (Translated from French)
[4]	R.O. Wells jr., "Differential analysis on complex manifolds" , Springer (1980)
[5]	L. Hörmander, "An introduction to complex analysis in several variables" , North-Holland (1973)

Comments

The theorem that an almost-complex structure is integrable, i.e. comes from a complex structure, if and only if its Nijenhuis tensor vanishes, is due to A. Newlander and L. Nirenberg [a1].

References

[a1]	A. Newlander, L. Nirenberg, "Complex analytic coordinates in almost complex manifolds" Ann. of Math. , 65 (1957) pp. 391–404