I'm looking for a proof of that the only spheres with almost complex structure are $S^2$ and $S^6$. I've googled "almost complex structure sphere", but all I get is comments saying that "this fact is well-known".


Are there good write-ups on this topic? Thanks in advance.



References on almost complex structures on spheres
I think this "well known fact" was proved first by Borel and Serre,


Borel, A., Serre, J. P.: Groupes de Lie et puissances réduites de Steenrod. Amer. J. Math.75, 409–448 (1953)


For a more detailed timeline, see [Differential Geometry: Geometry in mathematical physics and related topics](http://books.google.de/books?id=1s-UePH5KfAC&pg=PA100&lpg=PA100&dq=almost+complex+structure+on+spheres+S%5E2+S%5E6&source=bl&ots=HZDRuFRPB6&sig=T_jsFRQlEIhPPBTvfGHOZp5426I&hl=de&ei=9qn3S9ShO6KemwPUgenHAg&sa=X&oi=book_result&ct=result&resnum=6&ved=0CC8Q6AEwBTgU#v=onepage&q&f=false) by Greene and Yau (p.100).


Or as BCnrd suggested [Almost complex and complex structures](http://books.google.de/books?id=-yz03cskUOgC&pg=PA233&lpg=PA233&dq=almost+complex+structure+on+spheres+S%5E2+S%5E6&source=bl&ots=KxGtjBRHCP&sig=AANscUt-KM5KILqIaPMJ1zRId94&hl=de&ei=9qn3S9ShO6KemwPUgenHAg&sa=X&oi=book_result&ct=result&resnum=3&ved=0CCEQ6AEwAjgU#v=onepage&q&f=false) by C. C. Hsiung (Chapter VI)


