This is from Bredon's Topology and Geometry, pp 83~83. He's definitions of immersion and submersion are the following:
Now, in 7.3 Theorem, he explains that (using the implicit function theorem) if $\theta:M^{m}\to N^{n}$ is a smooth map such that the differential $\theta_{*}:T_{p}(M^{m})\to T_{\theta(p)}(N^{n})$ is onto, then in some small neighborhood and an appropriate local coordinates, $\theta$ looks just like the usual projection map from $\mathbb{R}^m$ to $\mathbb{R}^n$. I think I understand the meaning of this theorem.
But right after the proof, he says, without any explanation:
I just don't get it. Why is this an immeadiate corollay of theorem 7.3? More precisely, I think that:
- To say that $\theta^{-1}(y)$ is an embedded submanifold, we must first make sure what functional structure $\theta^{-1}(y)$ is given and check that such an functional structure makes it a manifold.
- After that, we must show that the inclusion map $\iota:\theta^{-1}(y)\to M^{m}$ is an immersion, one-one, and so on. (the condition (4) of 5.7)
Are these points trivially established?? If so, why? Please enlighten me.
Edit: Bredon's defines a regular value to be any point of $N^{n}$ which is not critical value. A critival value is the image in $N^{n}$ of a critical point.
Let's prove $\theta^{-1}(y)$ is an embedded submanifold. We are considering it with the subspace topology.
First, let's get the local charts. For every point $p \in \theta^{-1}(y)$, take local charts $(\phi,U)$ and $(\psi,V)$ around $p$ and $f(p)$ such that the following diagram commutes.
$\require{AMScd}$ \begin{CD} U @>\theta>> V\\ @V \phi V V @VV \psi V\\ \mathbb{R}^m @>>\pi> \mathbb{R}^n, \end{CD} where $\psi(\theta(p))=0$ and $\phi(p)=0$.
It follows that $\theta^{-1}(y) \cap U=\phi^{-1}(\mathbb{R}^{m-n}) \cap U$. We take $(\phi|_{\theta^{-1}(y) \cap U}^{\mathbb{R}^{m-n}}, \theta^{-1}(y) \cap U)$ as a local chart around $p$, and call it $\widetilde{\phi}$. Those are open sets in the subspace topology, so the charts are valid. It is then clear that the change of charts will be smooth, since they are given by the following commutative diagram.
$\require{AMScd}$ \begin{CD} \mathbb{R}^{m-n} @>\widetilde{\phi} \circ \widetilde{\psi}^{1}>> \mathbb{R}^{m-n}\\ @V iVV @A\pi ' AA \\ \mathbb{R}^m @>>\phi \circ \psi^{-1}> \mathbb{R}^m. \end{CD}
The inclusion being an homeomorphism with the image is merely topological (and almost tautological), since it is equivalent to saying that the identity $(\theta^{-1}(y),\tau_{inc}) \to (\theta^{-1}(y),\tau_{inc})$ is a homeomorphism. Being one-one is also clear. Being smooth and $\phi_*$ being a monomorphism follows readily from the local charts, since the inclusion translates to local charts as an inclusion from $\mathbb{R}^{m-n} \hookrightarrow \mathbb{R}^m$, which is smooth and has injective derivative at all points (the derivative being itself).