If $\gamma$ and $\eta$ are two smooth curves in a smooth manifold $M$, is it possible to find a diffeomorphism of $M$ such that $f \circ \gamma = \eta$? What if one removes the assumption of smoothness, working with topological manifolds (or even plain topological spaces) and homeomorphisms?


(Later edit: Argghh! I apologize, I forgot to specify: both curves do not have self-intersections and are homotopic.)



Diffeomorphism with prescribed behaviour
If your definition of "curve" is embeddings of say $[0,1]$ in a manifold, then the result follows in dimensions $n\geq 2$ by the isotopy extension theorem, together with the observation that the unit tangent bundle of a connected manifold is connected. In dimension $n=1$ the diffeomorphism would not always be isotopic to the identity, as the corresponding unit tangent bundle is not connected. The proof is primarily the *isotopy extension theorem* whose proof is an ODEs result and can be found in Hirsch's textbook on differential topology. 


Some more details: there is precisely one isotopy-class of smoooth embedding of $[0,1]$ into a manifold if and only if the unit tangent bundle of that manifold is connected. This follows from the isotopy extension theorem in a standard way. For manifolds of dimension $n \geq 2$ their unit tangent bundles are connected if and only if the manifold is connected. For $n=1$ the unit tangent bundle is never connected.


