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!
