Editing Almost complex manifold (section)

==Examples==
For every integer n, the flat space '''R'''<sup>2''n''</sup> admits an almost complex structure. An example for such an almost complex structure is (1 ≤ ''j'', ''k'' ≤ 2''n''): <math>J_{jk} = -i\delta_{j,k-1} </math> for odd ''j'', <math>J_{jk} = i\delta_{j,k+1} </math> for even ''j''.

The only [[sphere]]s which admit almost complex structures are '''S'''<sup>2</sup> and '''S'''<sup>6</sup> ({{harvtxt|Borel|Serre|1953}}). In particular, '''S'''<sup>4</sup> cannot be given an almost complex 
structure (Ehresmann and Hopf). In the case of '''S'''<sup>2</sup>, the almost complex structure comes from an honest complex structure on the [[Riemann sphere]]. The 6-sphere, '''S'''<sup>6</sup>, when considered as the set of unit norm imaginary [[octonion]]s, inherits an almost complex structure from the octonion multiplication; the question of whether it has a [[#Integrable almost complex structures|complex structure]] is known as the ''Hopf problem,'' after [[Heinz Hopf]].<ref>{{cite journal|last1=Agricola |first1=Ilka |authorlink1=Ilka Agricola |first2=Giovanni |last2=Bazzoni |first3=Oliver |last3=Goertsches |first4=Panagiotis |last4=Konstantis |first5=Sönke |last5=Rollenske |title=On the history of the Hopf problem |arxiv=1708.01068 |journal=[[Differential Geometry and Its Applications]] |year=2018 |volume=57 |pages=1–9|doi=10.1016/j.difgeo.2017.10.014 |s2cid=119297359 }}</ref>