TITLE: Almost complex manifold of dimension 2... locally isomorphic to ℂ?
QUESTION [6 upvotes]: I know that this is supposed to be standard, but I don't know how to search for it... hence the question:

Let $J$ be an almost complex structure on $M:=\mathbb R^2$, i.e., a $C^\infty$ section of $\mathrm{End}(TM)$ which squares to $-\mathrm{id}_{TM}$.
How does one prove that $(M,J)$ is integrable, i.e. how does one prove that it is locally isomorphic to $\mathbb C$?

REPLY [9 votes]: Eckmann-Frölicher say so right after they introduce the (“Nijenhuis”) torsion tensor in (1951, p. 2284), bottom: “dans $\mathrm E_2$, $t^j_{kl}$ est toujours $=0$.”
And this was known to imply integrability in the plane, as Newlander-Nirenberg start by recalling with proof in (1957, p. 394):
On p. 392 they also write

For $n=1$ the compatibility conditions are vacuous and the problem becomes that of introducing isothermal coordinates with respect to the Riemannian metric $ds^2=|dz + ad\bar z|^2$

— a problem whose solution Lichtenstein (1919, §24e, p. 264) traces back to Gauss (1825); I would add Lagrange’s Sur la construction des Cartes Géographiques (1781, pp. 167-169).