View source for Almost-complex structure

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

====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 [[#References|[a1]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Newlander,   L. Nirenberg,   "Complex analytic coordinates in almost complex manifolds"  ''Ann. of Math.'' , '''65'''  (1957)  pp. 391–404</TD></TR></table>