Let $M$ be a $C^\infty$ manifold of dimension $m$. A $C^r$ foliations of dimension $n$ of $M$ is a $C^r$ atlas $\mathcal{F}$ of $M$ which is maximal (not needed) with the following properties:
a) If $(U,\phi)\in\mathcal{F}$ then $\phi(U)=U_1\times U_2\subset \mathbb{R}^n\times \mathbb{R}^{m-n}$ respectively;
b) If $(U,\phi)$ and $(V,\psi)\in\mathcal{F}$ are such that $U\cap V\neq\emptyset$ then the change of coordinates map $\psi\circ \phi^{-1}:\phi(U\cap V)\rightarrow \psi(U\cap V)$ is of the form $\psi\circ\phi{-1}(x,y)=(h_1(x,y),h_2(y))$.
Now, suppose we have $f:M^m\rightarrow N^n$ a $C^r$ submersion. I can use the local form of submersions to, given $p\in M$ and $q=f(p)\in N$ such that $p\in U$, $q\in V$, obtain local charts $\phi$ and $\psi$ with $\phi(U)=U_1\times U_2$ as above, and $\psi(V)=V_2$ with $U_2\subset V_2$, such that $\psi\circ f \phi^{-1}:U_1\times U_2\rightarrow U_2$ coincides with the projection $\pi(x,y)=y$.
The question is: how can I show that the local charts ($U,\phi$) define a $C^r$ foliated manifold structure on $M$? I need to show that they have the form as pointed on the beginning, but I don't know how to do this. The leaves are the connected components of the level sets $f^{-1}(c)$, $c\in N$.
Let $f: \, M^m \rightarrow N^n$ be a submersion $\Rightarrow \, \forall p \in M, q=f(p) \in N, \exists (U,\varphi)$ is a local chart of $p, (V, \psi)$ is a local chart of $q$ such that $\varphi (U) = U_1 \times U_2 \subset \mathbb{R}^{m-n} \times \mathbb{R}^n, \psi(V) = V_2 \supset U_2$.
$$ \psi \circ f \circ \varphi^{-1} :\, U_1 \times U_2 \rightarrow V_2 \, \mbox{is of the form} \, (x,y) \rightarrow y$$
Indeed, $f$ in submersion at $p \Rightarrow \exists \, (\tilde{U} , \tilde{\varphi})$ is a local chart of $p$, $(\tilde{V}, \tilde{\psi})$ is a local chart of $f(p)$ such that $\psi(f(p) =0, f(U) \subset V$, and the transition mapping has following form:
$$ \tilde{\psi} \circ f \circ \tilde{\varphi}^{-1} :\, \tilde{\varphi}(\tilde{U}) \rightarrow \tilde{\psi}(\tilde{V}) \, \mbox{is of the form} \, (*,y) \rightarrow y$$
In $\tilde{\varphi}(\widetilde{U})$ we consider the neighbor $U_1 \times U_2$,so we got a local chart $(U=\tilde{\varphi}^{-1} (U_1 \times U_2), \tilde{\varphi}\big|_{\tilde{\varphi}^{-1} (U_1 \times U_2)})$.In the other hand, $ \tilde{\psi} \circ f \circ \tilde{\varphi}^{-1} $ has form as $pr_1 \Rightarrow U_2 \subset V_2= \tilde{\psi}(\tilde{V})$. We will prove that charts $(U,\varphi)$ with $p$ ranges over $M (\forall \, p \in M)$ form a folation codimension n in $M \,(m \geqslant n )$ .
We consider a point p $\in M$
We need to show that $\varphi_2 \circ \varphi_1^{-1}: \, \varphi_1(U_1) \rightarrow \varphi_2 (U_2) $ has form of an foliation chang of coordinate map codimension $n$, that is $$ \varphi_2 \circ \varphi_1^{-1} (x,y) = (h(x,y), g(y))$$
It's means that : $ pr_2 \circ \varphi_2 \circ \varphi_1^{-1} : \mathbb{R}^n \rightarrow \mathbb{R}^n$ is smooth. We have \begin{eqnarray*} p{r_2} &=& {\psi _2} \circ f \circ \varphi _2^{ - 1}\\ \Rightarrow p{r_2} \circ {\varphi _2} \circ \varphi _1^{ - 1} &=& {\psi _2} \circ f \circ \varphi _2^{ - 1} \circ {\varphi _2} \circ \varphi _1^{ - 1}\\ &=& {\psi _2} \circ \psi _1^{ - 1} \circ {\psi _1} \circ f \circ \varphi _1^{ - 1}\\ \Rightarrow p{r_2} \circ {\varphi _2} \circ \varphi _1^{ - 1}(x,y) &=& {\psi _2} \circ \psi _1^{ - 1} \circ ({\psi _1} \circ f \circ \varphi _1^{ - 1}(x,y))\\ &=& {\psi _2} \circ \psi _1^{ - 1}(y):= g(y) \end{eqnarray*}
So $(U,\varphi)$ is a chart of a foliaiton atlas of codimension n. Plaque $(U, \varphi)$ is $\varphi^{-1}(\alpha), \alpha \in \varphi(U)$. This plaque is on the leaf $f^{-1}(c)$ với $c=\psi^{-1}(pr_2(\alpha))$.