The following is an excerpt from Marco Gualtieri's [thesis](https://arxiv.org/abs/math/0401221)



> 
> A central theme of this thesis is that classical geometrical
> structures which appear, at first glance, to be completely different
> in nature, may actually be special cases of a more general unifying
> structure. Of course, there is wide scope for such generalization, as
> we may consider structures defined by sections of any number of
> natural bundles present on manifolds. What must direct us in deciding
> which tensor structures to study is the presence of natural
> integrability conditions.
> 
> 
> 



> 
> Good examples of such conditions include the
> closure of a symplectic form, the Einstein or special holonomy
> constraint on a Riemannian metric, the vanishing of the Nijenhuis
> tensor of a complex structure, and the Jacobi identity for a Poisson
> bivector, among many others.
> 
> 
> 


I think by closure of symplectic form, it means $d\omega=0$.


The only notion of integrability I know is integrability of a subbundle of $TM$; that is, for every point of $M$, there is an integrable manifold corresponding to the distribution.


Frobenius theorem says $L\subseteq TM$ is integrable if and only if $[X,Y]\in L$ for every $X,Y\in L$.


But, I do not fully understand what is "integrable" in a differential form being closed, or a bivector to satisfy Jacobi identity or for Nijenhuis tensor to be zero.


I would understand if we call it "a smoothly varying condition" or something similar, but, why would some one want to refer them as integrability conditions?



Does "integrability condition" have a specific meaning or is it used in a casual way?
It turns out there is a well defined notion of integrability of a G-structure on a manifold.


Thanks to the user Thomas Rot who has given the reference [Linear $G$-structures by examples](https://www.few.vu.nl/%7Epasquott/course16.pdf)


Definition $2.1$ is that of $G$-structure on a manifold M.


It defines the notion of integrability of a $G$-structure in Definition $2.4$.


1. A ($p$-dimensional) distribution $\mathcal{F}$ on a ($n$-dimensional) manifold $M$ can be seen as a $GL(p,n-p)$ structure on the manifold $M$. We can talk about integrability of this $G$-structure. It says (Theorem $2.25$) $\mathcal{F}$ is involutive if and only if $\mathcal{F}$ is an integrable G-structure. This goes by the name Frobenius theorem.
2. An almost complex structure J on a manifold $M$ (of dimension $2k$) can be seen as a $GL\_k(\mathbb{C})$-structure on the manifold $M$. We can talk about integrability of this $G$-structure. It says that (Theorem $2.31$) Nijenhuis tensor of $J$ vanishes if and only if $J$ is an integrable $G$-structure. This goes by the name Newlander-Nirenberg theorem.
3. A non degenerate $2$-form $\omega$ on a manifold $M$ (of dimension $2k$) can be seen as a $Sp\_k(\mathbb{R})$-structure on the manifold $M$. We can talk about integrability of this $G$-structure. It says (Theorem $2.41$) $\omega$ is closed if and only if $\omega$ is an integrable $G$-structure. This goes by the name of Darboux theorem.


