I want to show that if $F : \mathbb{R}^n \rightarrow \mathbb{R}^{n-k}$ is a $C^1$ function and $rank(DF) = n-k$ then $M:=F^{-1}(\{0\})$ defines a manifold.
My idea: Without loss of generality I assumed that $F: \mathbb{R}^k \times \mathbb{R}^{n-k} \rightarrow \mathbb{R}^{n-k}$ such that $D_2F$ is invertible. Hence, by the implicit function theorem there is a $C^1$ function $f$ such that $F(x,f(x))=0$ on an open neighbourhood $U(x_0)$ for some $x_0$ in $\mathbb{R}^k.$
My problem is: I now want to define the parametrisation as $g(x):=F(x,f(x))$ on $U(x_0)$ but I don't see how this is possible cause $Dg(x_0) = DF(x_0,f(x_0)) (Id,Df(x_0))^T $ is not necessarily injective, I guess. The first matrix has maximal rank (is surjective) and the second one has maximal rank (is surjective), but this does not give me injectivity of $Dg$. In particular, I am not sure, if this is helpful at all.
What I am looking for are a few hints how I can define a chart of a parametrization now.
Your parametrization should just be the map $g\colon U(x_0) \to \Bbb R^n$ given by $g(x)= (x,f(x))$. :)