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I am trying to figure out, whether the IFT can be generalized to curves. Let's say I have a function $G(x,u)$ mapping $\mathbb{R}^{n+m}\rightarrow \mathbb{R}^n$ with invertible jacobian $\frac{\partial G}{\partial x}$. However, not only do we assume $G(x_0,u_0) = 0$, but $G(x(t),u(t)) = 0$ for a compact curve $t\mapsto (x(t),u(t))$ in $\mathbb{R}^{n+m}$.

Needless to say, for each point on the curve there is an associated neighborhood $V(t)$ where the IFT applies. I guess the main question is: Can we make it so that the neighborhoods with corresponding graphs glue? Or does this need additional assumptions?

Any help is appreciated!

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  • $\begingroup$ I don't see a question here. What does your curve have to do with the function $G$? It looks like you are assuming the curve belongs to a level-set. But what do you mean about $V$ and "the IFT"? $\endgroup$
    – Ryan Budney
    Commented Dec 6, 2022 at 16:26
  • $\begingroup$ With "the implicit function theorem" I mean the standard theorem for $\mathcal{C}^1$ functions $\mathbb{R}^{n+m}\rightarrow \mathbb{R}^n$. The curve a priori is any smooth curve satisfying $G(x(t), u(t)) = 0$ such that the corresponding Jacobian is invertible along the curve. $V(t)$ are the open neighborhoods the classic implicit function theorem provides at every point of the curve. $\endgroup$
    – Matthias Himmelmann
    Commented Dec 7, 2022 at 9:18
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    $\begingroup$ What do you mean by gluing neighbourhoods? Could you turn this into a precise question, i.e. is there an actual theorem you are looking to prove? $\endgroup$
    – Ryan Budney
    Commented Dec 7, 2022 at 16:52

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