= Solution
A <submersion> is a <smooth map between manifolds> $F:X^n\to Y^m$ for which
$$
D_pF:T_pX\longrightarrow T_{F(p)}Y
$$
is surjective at every $p\in X$.
The local submersion theorem says that around every $p\in X$ there are coordinates $(x^1,\ldots,x^n)$ centered at $p$ and $(y^1,\ldots,y^m)$ centered at $F(p)$ in which
$$
F(x^1,\ldots,x^n)=(x^1,\ldots,x^m).
$$
To prove it, surjectivity lets us choose $m$ source coordinates such that $dF^1,\ldots,dF^m$ are independent. Complete them by source coordinates $x^{m+1},\ldots,x^n$ and define
$$
G=(F^1,\ldots,F^m,x^{m+1},\ldots,x^n).
$$
The derivative of $G$ is invertible at $p$, so the <inverse function theorem> makes $G$ a local coordinate system; in these coordinates $F$ is the displayed projection.
For $q\in Y$, the fiber is locally given by
$$
x^1=q^1,\ldots,x^m=q^m.
$$
These are slice coordinates, so $F^{-1}(q)$ is an <embedded submanifold> of dimension $n-m$.
Back to article page
