# Any real analytic Frobenius theorem used in the proof of integrable almost complex manifolds to arise from complex manifolds?

The reference book is S.S.Chern's Complex Manifolds Without Potential Theory.

It's used in integrability condition for an almost complex structure to arise from a complex structure. It's claimed that if the almost complex structure is locally defined by the forms $\theta^k$ of type $(1,0)$ which are linearly independent over the ring of complex-valued smooth functions, then the condition is $d\theta^k\equiv0\mod (\theta^1,\dotsc,\theta^n)$. Here $(\theta^1,\dotsc,\theta^n)$ is the ideal generated by $\theta^1,\dotsc,\theta^n$ in the exterior algebra.

Under the hypothesis that the almost complex structure is real analytic (i.e. the field of endomorphisms $J_x\colon T_xM\to T_xM$ such that $J_x^2=-1_x$, then in some local coordinates, $J_x$ is real-analytic), then it follows from Frobenius's theorem that there exist complex local coordinates $z^k$ such that the form is of type $(1,0)$ are linear combinations of $dz^k$. It's found in Chern's book, pp 17.

I don't know which version of Frobenius's theorem is used. The cotangent bundle, and therefore the exterior algebra of differential forms is complexified (i.e., $\otimes_\mathbb{R}\mathbb C$). The version I know can be found in many canonical textbooks, say Warner's GTM94, or Chern's Lectures on Differential Geometry. It's used for smooth distributions. I wonder the real-analytic version used in the preceding illustration, the idea of the proof of the theorem (it's complexified, therefore the original proof cannot apply directly, I think), and especially where the real-analyticity is really used.

Any help? Thanks!