I have the following problem which I don't know if it's true:
Let $\varphi: I\rightarrow \mathbb{R}^n$ and $\psi: J\rightarrow \mathbb{R}^n$ be parameterizations of a curve $\Gamma$ such that $\varphi '(t), \psi '(s)$ are unitary vectors for every $t\in I, s\in J$. Then there exist a smooth map $\alpha: I\rightarrow J$ surjective such that $\alpha '(t)\neq 0$ for every $t\in I$ and $\psi = \varphi \circ \alpha$.
If such map exist, I proved that this map must be $\alpha(t) = t_0 +t$ or $\alpha(t) = t_0 -t$. But, I don't see how this map works...
Any idea?..