The Newlander-Nirenberg theorem states that any Integrable Almost Complex manifold is a complex manifold. I am looking for natural examples of complex structures that are not integrable.
Nonintegrable almost complex structures
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geometry
differential-geometry
complex-geometry
almost-complex
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0I changed the title by adding "almost", because *nonintegrable complex structure* is an oxymoron. Also, I added some relevant tags. – 2011-06-23
1 Answers
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The sphere $S^6$ naturally sits inside of the imaginary octonians $\operatorname{Im}\mathbb{O}$. At the point $p\in S^6$, multiplication by $p$ on $ T_p S^6 = p^\bot \subseteq \operatorname{Im}\mathbb{O}$ defines an almost complex structure.
This almost complex structure is not integrable, due to the nonassociativity of the octonians.
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0@Michael: It's probably also worth adding that my only interaction with (almost) comple$x$ manifolds has been through a few cursory google searches - I've probably spent a grand total of a couple of days of my life thinking about them in any kind of detail. Sorry to be useless! – 2015-01-21