Can every path connected $1$-surface in $\mathbb R^{n+1}$ be globally parametrized?

Question

Let $f : \mathbb R^{n+1} \to \mathbb R^n$ be a smooth function such that $S:=f^{-1}(\{0\})$ is a path connected $1$-surface

( i.e. rank $Df(a)=n , \forall a\in S$ ) .

Then is it true that there exist a smooth function $g: I \to \mathbb R^{n+1}$ such that $g(I)=S$ and rank $Dg(a)=1 , \forall a \in I$ , where $I$ is an open connected subset of real line ?

Answer 1

Well, any one-dimensional connected manifold is diffeomorphic to either $\mathbb{R}$ or $S^1$. If $S$ is diffeomorphic to $\mathbb{R}$, you can choose $g$ to be a diffeomorphism and if $S$ is diffeomorphic to $S^1$, choose a diffeomorphism $h \colon S^1 \rightarrow S$ and compose it with the cover map $\pi \colon \mathbb{R} \rightarrow S^1$ to get your $g = h \circ \pi$. It won't be a diffeomorphism but $g(\mathbb{R}) = S$ and $g$ will be an immersion (so $dg$ will be of constant rank one).