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 $\mathbb C$?

  • 1
    $\begingroup$ Hi this absolutely isn’t my field but... I think one can show the vanishing of the Nijenhuis tensor easily: N_J(X, X) = 0 and N_J(X, JX) = 0 by inspection. Let me know if I’ve said something dumb (true with high probability)! $\endgroup$ – alpoge May 18 '19 at 1:05
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    $\begingroup$ If you can find a vector field $X$ on a neighborhood U such that $[X,JX]=0$ on $U$, then that neighborhood is isomorphic to $\mathbb{C}$. This is because the Lie bracket is the obstruction to $X,JX$ "integrating" to a coordinate grid $\endgroup$ – Daniel Barter May 18 '19 at 1:28
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    $\begingroup$ See mathoverflow.net/questions/17565/… $\endgroup$ – Raju May 18 '19 at 8:02

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):enter image description here

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).


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