# 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)

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
How to Cite This Entry:
Almost-complex structure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Almost-complex_structure&oldid=45082
This article was adapted from an original article by D.V. Alekseevskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article