Transition functions of foliation charts

Question

Let $M$ be a smooth $n$-manifold (without boundary), and let $E \subset TM$ be an integrable (smooth) distribution on $M$ of codimension $q$. From the Frobenius theorem, we know that this distribution induces a foliation on $M$ in the sense that for every $p \in M$, there is a "foliation" chart $(U, \varphi)$ around $p$ with the property that $\varphi(U)$ is an open cube in $\mathbb{R}^n$ with $$ E|_U = \ker (d(\pi_2 \circ \varphi)), $$ where $\pi_2 : \mathbb{R}^n \to \mathbb{R}^q$ is the projection on the last $q$ coordinates.

I have seen in some classical books about foliation theory that the above implies (and in fact is equivalent to) that $M$ can be covered by a collection of charts $(U_i, \varphi_i)$, with $\varphi_i : U_i \to \mathbb{R}^{n - q} \times \mathbb{R}^q$ so that the transition functions $$\varphi_{ij} := \varphi_i \circ \varphi_j^{-1}$$ are of the form $$\varphi_{ij}(x, y) = (g_{ij}(x, y), h_{ij}(y)). $$

However, it is not clear to me at all why the foliation charts for an integrable distribution must have the transition functions of the above particular form. The way I understand the foliation charts is that they are particular charts depending on the structure of the distribution, and I do not see any connection of that with the corresponding transition functions.

I feel I am missing something very simple here.

Answer 1

Roughly speaking, the form of the transition cocycle $\varphi_{ij}$ encodes not only the local smooth/topological structure of the ambient manifold $M$, but also the local smooth/topological structure of each single leaf of the foliation as well as how/how well these leaves fit together.

Each leaf is a horizontal line segment in foliation charts (as the foliation is codimension $q$), and the fact that the second component of the transition cocycle is constant in the first coordinate means that once a height $y^\ast$ is fixed (i.e. once the local piece of a specific leaf is fixed), then the cocycle $\varphi_{ij}$ takes the horizontal segment at the speficied height to some other horizontal segment at some other height, namely at $h_{ij}(y^\ast)$. Note that one can, in an invariant manner, consider a splitting $\mathbb{R}^{n-q}\times\mathbb{R}^{q}$, and likewise one can consider fixed heights in local coordinates, but one can not consider heights numerically in an invariant fashion.

Thus $\varphi_{ij}=(g_{ij},h_{ij})$ determines uniquely the smooth manifold structure on $M$ with the local model having a specified splitting; (for appropriately chosen heights) the first component $g_{ij}$ determines uniquely the smooth manifold structures of leaves of the foliation; and the second component $h_{ij}$ determines uniquely how different leaves of the foliation locally fit together.

I should also note that the transition cocycle formalism is somewhat more versatile than the formalism using differential forms, as it allows for finer descriptions of the objects (e.g. in (hyperbolic) dynamics often the foliations are continuous, but each leaf is $C^1$; consequently one can and does consider bundles that are tangential to the leaves, although integrating these bundles to obtain foliations require more work than referring to the Clebsch-Deahna-Frobenius theorem).

Answer 2

This is more of an extended comment, answering your questions which arise from @AlpUzmna's answer.

As @AlpUzman explains in the second paragraph of their answer, the transition functions must map leaves to leaves; since a leaf is determined by the last $q$ coordinates, this implies that the last $q$ coordinates of transition function cannot depend on the first $n - q$ coordinates.

I believe this can be formalized as in this simple case (with a straightforward generalization to arbitrary dimensions): say we have a smooth map $f = (f_1, f_2) : \mathbb{R}^2 \to \mathbb{R}^2$; assume that $f$ maps the lines of the form $\mathbb{R} \times \{c\}$ into lines of the form $\mathbb{R} \times \{c'\}$. Then, $\partial_1 f_2 = 0$.

And this should be immediate: fix $(x, y) \in \mathbb{R}^2$. By assumption, $$f_2(x + t, y) \text{ is constant as a function of } t \in \mathbb{R}. $$ Then, by definition, $$\partial_1f_2(x, y) = \lim_{h \to 0} \frac{f_2(x + h, y) - f_2(x, y)}{h} = \left. \frac{d}{dt} \right|_{t = 0} f_2(x + t, y) = 0. $$