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Let $M$ be a manifold. A complex structure on $M$ is an endomorphism $J \in \text{End}(TM)$ such that $J^2 = -\text{id}$ together with the vanishing of the Nijenhuis tensor. If $J$ is real-analytic, the Frobenius theorem implies that the data $(M,J)$ defines a complex manifold, i.e., there exist local holomorphic coordinates on $M$. If $J$ is smooth, then $(M,J)$ is a complex manifold by the Newlander-Nirenberg theorem. If $J$ is only assumed to be $C^1$, does the vanishing of the Nijenhuis tensor imply that $(M,J)$ is a complex manifold?

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  • $\begingroup$ I don't understand the "we may assume" phrase. If M is odd-dimensional, or if M fails to satisfy certain restrictive conditions (the 4-sphere is maybe the simplest example), then we cannot assume M is endowed with a complex structure (which I am taking to mean what differential geometers usually call an "almost complex" structure), since in those cases it has none. $\endgroup$ Nov 7, 2023 at 3:21
  • $\begingroup$ @Daniel Asimov. Thanks, I should detele the "may". I meant to say that $M$ satisfy certain restrictive conditions, i.e. $M$ is endowed with an $C^1$-alomost complex structure whose Nijenhuis tensor vanishes. $\endgroup$ Nov 7, 2023 at 3:29
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    $\begingroup$ Check arxiv.org/pdf/0710.2310.pdf $\endgroup$ Nov 7, 2023 at 3:30

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