Let $M,N$ smooth manifolds and $F:M \to N$ a smooth submersion. Show that the connected components of nonempty level sets form a foliation of $M$.

My idea is to use the Global Fröbenius theorem for the distribution $D$, where $D_{p}=ker (dF)_{p}$, I proved that $D$ is involutive distribution and for each $p \in M$, $F^{-1}(F(p))$ is an integral manifold for $M$ passing by $p$. My question is:

If I prove that the connected components of $F^{-1}(F(p))$ are integral manifolds for $M$, then is each one a maximal integral manifold passing by $p$ and the global Frobenius theorem assures that connected components are a foliation for $M$?


It's not necessary to use the Frobenius theorem. You can simply apply the implicit function theorem.

Since $F : M \to N$ is a smooth submersion, the implicit function theorem gives you a collection of 4-tuples $\{(U_i,V_i,\alpha_i,\beta_i)\}$ such that $\{U_i\}$ is an open cover of $M$, $\{V_i\}$ is an open cover of $N$, and $\alpha_i,\beta_i$ are homeomorphisms fitting into a commutative diagram as follows: $\require{AMScd}$ \begin{CD} U_i @>F \, \mid \, U_i >> V_i\\ @V \alpha_i V V @VV \beta_i V\\ (0,1)^m @>>p> (0,1)^n \end{CD} where $\alpha_i,\beta_i$ are homeomorphisms and $p$ is projection. Furthermore, you can directly check the overlap condition which shows that this gives a foliation atlas for $M$, where the local leaves (known as "plaques") have the form $\alpha_i^{-1} \circ p^{-1}(x)$ for $x \in (0,1)^n$. So each plaque is homeomorphic to $(0,1)^{m-n}$, and the folication atlas tells you that the plaques fit together in local product structures given by the commutative diagram.


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