Almost-complex structure

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

$$ I ^ {2} = - \mathop{\rm id} , $$

i.e. a field of complex structures in the tangent spaces $ T _ {p} M $, $ p \in M $. An almost-complex structure $ I $ determines a decomposition $ T ^ {\mathbf C } M = V _ {+} + V _ {-} $ of the complexification $ T ^ {\mathbf C } M $ of the tangent bundle in a direct sum of two complex mutually-conjugate subbundles $ V _ {+} $ and $ V _ {-} $ consisting of eigen vectors of the affinor $ I $( extended by linearity on $ T ^ {\mathbf C } M $) with eigen values $ i $ and $ -i $, respectively. Conversely, a decomposition of $ T ^ {\mathbf C } M $ in a direct sum of mutually-conjugate vector subbundles $ S , \overline{S}\; $ defines an almost-complex structure on $ M $ for which $ V _ {+} = S $.

An almost-complex structure $ I $ is called integrable if it is induced by a complex structure on $ M $, i.e. if there exists an atlas of admissible charts of the manifold $ M $ in which the field $ I $ has constant coordinates $ I _ {k} ^ {j} $. A necessary and sufficient condition for the integrability of an almost-complex structure is that the subbundle $ V _ {+} $ 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 $ V _ {+} $ to be involutive is equivalent to the vanishing of the vector-valued $ 2 $- form $ N ( I , I ) $ associated with $ I $ and given by the formula

$$ N ( I , I ) ( X , Y ) = [ I X , I Y ] - I [ X , I Y ] - I [ I , X Y ] - [ X , Y ] , $$

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

$$ N ( I , I ) = [ I , [ I , d ] ] + d , $$

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

From the point of view of the theory of $ G $- structures an almost-complex structure is a $ \mathop{\rm GL} ( m , \mathbf C ) $- structure, where $ m = (1 / 2 ) \mathop{\rm dim} M $, and the torsion tensor $ N ( I , I ) $ is the tensor defined by the first structure function of this structure. A $ \mathop{\rm GL} ( m , \mathbf C ) $- 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 $ G $ 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 $ G $ acts transitively on the manifold $ M $, then the almost-complex structure $ I $ is uniquely defined by its value $ I _ {p} $ at a fixed point $ p \in M $. This represents an invariant complex structure in the tangent space $ T _ {p} M $ 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 $ G / H $ of a Lie group $ G $ by the subgroup $ H $ consisting of fixed points of an automorphism of even order of $ G $ has an invariant almost-complex structure. An example is the $ 6 $- dimensional sphere $ S ^ {6} $, considered as the homogeneous space $ G _ {2} / \mathop{\rm SU} (3) $; none of the invariant almost-complex structures on $ S ^ {6} $ 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