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A tensor field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011930/a0119301.png" /> of linear transformations of the tangent spaces on a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011930/a0119302.png" /> satisfying the condition
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011930/a0119303.png" /></td> </tr></table>
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i.e. a field of complex structures in the tangent spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011930/a0119304.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011930/a0119305.png" />. An almost-complex structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011930/a0119306.png" /> determines a decomposition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011930/a0119307.png" /> of the complexification <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011930/a0119308.png" /> of the tangent bundle in a direct sum of two complex mutually-conjugate subbundles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011930/a0119309.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011930/a01193010.png" /> consisting of eigen vectors of the affinor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011930/a01193011.png" /> (extended by linearity on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011930/a01193012.png" />) with eigen values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011930/a01193013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011930/a01193014.png" />, respectively. Conversely, a decomposition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011930/a01193015.png" /> in a direct sum of mutually-conjugate vector subbundles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011930/a01193016.png" /> defines an almost-complex structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011930/a01193017.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011930/a01193018.png" />.
+
A tensor field $  I $
 +
of linear transformations of the tangent spaces on a manifold  $  M $
 +
satisfying the condition
  
An almost-complex structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011930/a01193019.png" /> is called integrable if it is induced by a complex structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011930/a01193020.png" />, i.e. if there exists an atlas of admissible charts of the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011930/a01193021.png" /> in which the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011930/a01193022.png" /> has constant coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011930/a01193023.png" />. A necessary and sufficient condition for the integrability of an almost-complex structure is that the subbundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011930/a01193024.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011930/a01193025.png" /> to be involutive is equivalent to the vanishing of the vector-valued <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011930/a01193026.png" />-form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011930/a01193027.png" /> associated with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011930/a01193028.png" /> and given by the formula
+
$$
 +
I  ^ {2}  = - \mathop{\rm id} ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011930/a01193029.png" /></td> </tr></table>
+
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 $.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011930/a01193030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011930/a01193031.png" /> vector fields. This form is called the torsion tensor, or the Nijenhuis tensor, of the almost-complex structure. The torsion tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011930/a01193032.png" /> can be considered as first-order differentiation on the algebra of differential forms on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011930/a01193033.png" /> of the form
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011930/a01193034.png" /></td> </tr></table>
+
$$
 +
N ( I , I ) ( X , Y )  = [ I X , I Y ] - I
 +
[ X , I Y ] - I [ I , X Y ] - [ X , Y ] ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011930/a01193035.png" /> is the exterior differential and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011930/a01193036.png" /> is considered as a differentiation of order zero.
+
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
  
From the point of view of the theory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011930/a01193037.png" />-structures an almost-complex structure is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011930/a01193038.png" />-structure, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011930/a01193039.png" />, and the torsion tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011930/a01193040.png" /> is the tensor defined by the first structure function of this structure. A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011930/a01193041.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011930/a01193042.png" /> 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.
+
$$
 +
N ( I , I )  = [ I , [ I , d ] ] + d ,
 +
$$
  
If the automorphism group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011930/a01193043.png" /> acts transitively on the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011930/a01193044.png" />, then the almost-complex structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011930/a01193045.png" /> is uniquely defined by its value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011930/a01193046.png" /> at a fixed point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011930/a01193047.png" />. This represents an invariant complex structure in the tangent space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011930/a01193048.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011930/a01193049.png" /> of a Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011930/a01193050.png" /> by the subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011930/a01193051.png" /> consisting of fixed points of an automorphism of even order of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011930/a01193052.png" /> has an invariant almost-complex structure. An example is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011930/a01193053.png" />-dimensional sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011930/a01193054.png" />, considered as the homogeneous space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011930/a01193055.png" />; none of the invariant almost-complex structures on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011930/a01193056.png" /> is integrable.
+
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.
 
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.
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====References====
 
====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>
 
<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====
 
====Comments====

Latest revision as of 16:10, 1 April 2020


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
How to Cite This Entry:
Almost-complex structure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Almost-complex_structure&oldid=17647
This article was adapted from an original article by D.V. Alekseevskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article