I've heard a lot of arguments about (smooth) curves $\gamma$ on a (smooth) manifold $M$, like the following one:
If the curve $\gamma$ has a certain property $P$, and for every reparameterization $\delta$ of $\gamma$ we can show that the curve $\delta$ also has property $P$, then the property $P$ is true independent of the parametrization of the curve.
Is this reasoning always valid? A crucial ingredient that seems to be used is the fact/assumption that if any two smooth curves $\gamma:I\to M$ and $\delta:J\to M$ have the same image in $M$ then they must be reparameterizations of one another, i.e., $\delta = \gamma \circ \phi$, where $\phi:J\to I$ is a diffeomorphism.
Is this indeed a true statement? I tried to prove it myself but did not get very far. It seems to me that we need at least some extra requirements, e.g. that the curves are immersions, or that they are injective, etc. Or maybe the statement is only true locally?
Thanks!