Rank Theorem

Statements

Theorem. Let $A_0\subset {\mathbb{R}}^n$ and $B_0\subset {\mathbb{R}}^m$ be open sets and, for a given $r\in {\mathbb{N}}_{>0}$, let $F\in {\mathcal{C}}^r(A_0,B_0)$ be a vector-valued function of rank $k$ on $A_0$. For every $a\in A_0$ and $b=F(a)$, there exist open sets $U\subset {\mathbb{R}}^n$, $V\subset {\mathbb{R}}^m$, $A\subset A_0$ and $B\subset B_0$ with $a\in A$ and $b\in B$, and there exist diffeomorphisms $G\in {\mathcal{C}}^r(A,U)$ and $G\in {\mathcal{C}}^r(B,V)$ such that the image of the composition

H\circ F\circ G^{-1}:&U&\to&V\\ &(x^1,\ldots,x^n)&\mapsto&(x^1,\ldots,x^k,0,\ldots,0) \end{array}

is contained in $V$.

Remarks

Remark. The rank theorem tells us that a mapping of constant rank $k$ behaves locally like a projection of ${\mathbb{R}}^n={\mathbb{R}}^k\times {\mathbb{R}}^{n-k}$ to ${\mathbb{R}}^k$, i.e., $F\circ G^{-1}:{\mathbb{R}}^n\to {\mathbb{R}}^k$ followed by a injection of ${\mathbb{R}}^k$ onto ${\mathbb{R}}^k\times \{0\}\subset {\mathbb{R}}^k\times {\mathbb{R}}^{m-k}={\mathbb{R}}^m$, i.e., $H\circ F\circ G^{-1}:{\mathbb{R}}^n\to {\mathbb{R}}^k\times \{0\}$.

References

@Boo75 Theorem 7.1