Let $W \subset \Bbb R^n$ open and $F:W \to \Bbb R$, $F$ is $C^{\infty}$ and $a\lt b \in F(W)$ so that $L=F^{-1}([a,b])$ is compact and $\nabla F(p) \ne 0$ for every $p \in L$
By taking the field $X = \frac {\nabla F}{|\nabla F|^2} $ then the flow ${φ^X}_{b-a}$ gives us a diffeomormphism between $F^{-1}(a)$ and $F^{-1}(b)$.
My question is if we can generalise this for $\Bbb R^k$ instead of $\Bbb R$?
No.
For example, think of a regular curve (so $\nabla \gamma \neq 0$) $\gamma : (-\epsilon, \epsilon) \to \mathbb R^2$ which has a self intersection at $a=(0,0)\in \mathbb R^2$. Then $\gamma^{-1}(a)$ has more than one points while $\gamma^{-1}(b)$ has only one point in general.