Does "integrability condition" have a specific meaning or is it used in a casual way?

Question

The following is an excerpt from Marco Gualtieri's thesis

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?

Answer 1

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

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.