# $C^1$ manifold with complex structure

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?

• 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. Nov 7, 2023 at 3:21
• @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. Nov 7, 2023 at 3:29
• Nov 7, 2023 at 3:30