Canonical submersion theorem

Let $f : M \to N$ be a submersion at $p \in M$ , then $m = \dim M \geq n = \dim N$ and there exist charts $(φ_{1}, U_{1}, V_{1})$ around $p$ and $(ψ_{1}, X_{1}, Y_{1})$ around $f (p)$ such that

ψ_{1} \circ f \circ φ_{1}^{- 1} = {π |}_{V_{1}} .

Canonical submersion theorem

REMARK: As can be deduced from the following proof, a submersion gives to $M$ the structure of a product around $p$ since we can construct a diffeomorphism

t_{p} : p \in U_{2} ⟶ f (U_{2}) \times I_{p}^{m - n}

being $I_{p} \subseteq R$ an interval depending on $p$ .
Proof of the Canonical Submersion Theorem. Take a chart ${φ, U, V}$ near $p$ and a chart ${ψ, X, Y}$ near $f (p)$ so that $f (U) \subset X$ . Since $f$ is a submersion,

d (ψ \circ f \circ φ^{- 1})_{φ (p)} = d ψ_{q} \circ d f_{p} \circ d φ_{φ (p)}^{- 1} : T_{φ (p)} V = R^{m} \to T_{ψ (q)} Y = R^{n}

is surjective. Denote $F = ψ \circ f \circ φ^{- 1}$ . Then the Jacobian matrix $(\frac{\partial F^{i}}{\partial x^{j}})$ is an $n \times m$ matrix of rank $n$ at $φ (p)$ . By reordering the coordinates if necessary, we may assume the sub-matrix

(\frac{\partial F^{i}}{\partial x^{j}}), 1 \leq i \leq n, 1 \leq j \leq n

is nonsingular at $φ (p)$ . Note that this re-ordering procedure can be done by modifying $(ψ, X, Y)$ to another chart $(ψ_{1}, X_{1}, Y_{1})$ , and thus we really have $F = ψ_{1} \circ f \circ φ^{- 1}$ . Define

G : V \to R^{m}, (x^{1}, \dots, x^{m}) \mapsto (F^{1}, \dots, F^{n}, x^{n + 1}, \dots, x^{m}) .

Then obviously $d G_{φ (p)}$ is nonsingular. By the inverse function theorem, there is a neighborhood $V_{0}$ of $φ (p)$ so that $G$ is a diffeomorphism from $V_{0}$ to $G (V_{0})$ . Let $H$ be the inverse of $G$ on $G (V_{0})$ . Note that $F = π \circ G$ . Let $U_{1} = φ^{- 1} (V_{0})$ , $V_{1} = G (V_{0})$ , and $φ_{1} = G \circ φ$ . Then $(φ_{1}, U_{1}, V_{1})$ is a chart near $p$ , and

ψ_{1} \circ f \circ φ_{1}^{- 1} = ψ_{1} \circ f \circ (φ^{- 1} \circ H) = F \circ H = π \circ G \circ H = π .

\begin{matrix} ◻ \end{matrix}

In some sense this is "dual" to the canonical immersion theorem.

The proof use the inverse function theorem.