Editing Almost complex manifold (section)

== Formal definition ==
Let ''M'' be a smooth manifold. An '''almost complex structure''' ''J'' on ''M'' is a linear complex structure (that is, a [[linear map]] which squares to −1) on each tangent space of the manifold, which varies smoothly on the manifold. In other words, we have a [[smooth function|smooth]] [[tensor field]] ''J'' of [[Tensor#Tensor degree|degree]] {{nowrap|(1, 1)}} such that <math>J^2=-1</math> when regarded as a [[vector bundle]] [[isomorphism]] <math>J\colon TM\to TM</math> on the [[tangent bundle]]. A manifold equipped with an almost complex structure is called an '''almost complex manifold'''.

If ''M'' admits an almost complex structure, it must be even-dimensional. This can be seen as follows. Suppose ''M'' is ''n''-dimensional, and let {{nowrap|''J'' : ''TM'' → ''TM''}} be an almost complex structure. If {{nowrap|1=''J''{{i sup|2}} = −1}} then {{nowrap|1=(det ''J'')<sup>2</sup> = (−1){{sup|''n''}}}}. But if ''M'' is a real manifold, then {{nowrap|det ''J''}} is a real number – thus ''n'' must be even if ''M'' has an almost complex structure. One can show that it must be [[orientable manifold|orientable]] as well.

An easy exercise in [[linear algebra]] shows that any even dimensional vector space admits a linear complex structure. Therefore, an even dimensional manifold always admits a {{nowrap|(1, 1)}}-rank tensor ''pointwise'' (which is just a linear transformation on each tangent space) such that {{nowrap|1=''J''{{sub|''p''}}{{sup|2}} = −1}} at each point ''p''. Only when this local tensor can be patched together to be defined globally does the pointwise linear complex structure yield an almost complex structure, which is then uniquely determined. The possibility of this patching, and therefore existence of an almost complex structure on a manifold ''M'' is equivalent to a [[reduction of the structure group]] of the tangent bundle from {{nowrap|GL(2''n'', '''R''')}} to {{nowrap|GL(''n'', '''C''')}}. The existence question is then a purely [[algebraic topology|algebraic topological]] one and is fairly well understood.