Consider $1 < n\in \mathbb {N} $, let $\gamma_1,\gamma_2 : [0,1] \to \mathbb{R}^{n}$ be smooth paths such that $$\gamma_1 (0) = \gamma_2(0) \neq \gamma_1(1) = \gamma_2(1) $$ and $\gamma_1, \gamma_2$ are injetive functions.
Question: Does there exist a diffeomorphism $\varphi: \mathbb{R}^n \to \mathbb{R}^n$ such that $\varphi (\gamma_1 ([0,1])) = \gamma_2 ([0,1]).$
Does anyone know if this result is true?
This seems true but I do not know how to prove it, can anyone help me?
Counterexample in $\mathbb R^2$: Define $$\gamma_1 (t)=\begin{cases} (e^{1/(2t-1)},0)&t\in [0,1/2)\\0& t=1/2\\(0,e^{1/(1-2t)})&t\in (1/2,1]\end{cases}$$
This is a $C^\infty$ injective mapping that that traces out the line segment $[(1/e,0),(0,0)]$ followed by the segment $[(0,0),(0,1/e)].$
Note that $\gamma_1$ makes a right angle turn at $(0,0).$ There is no diffeomorphism that can turn $\gamma_1$ into the usual straight line segment $\gamma_2$ from $(1/e,0)$ to $(0,1/e).$