I have found two different definitions of equivalence for curves. The first states that two curves are equivalent if they share the range (which is fine: it is an equivalence relationship). Another, finer, requires them to have a function $g: I\rightarrow J$, where $I$ and $J$ are the sets on which the two curves are defined such that $$ C_1(x)=C_2(g(x)) $$ where $g$ must be $C^1$, subjective, and such that $g'(x)\neq0$.
Now, come the questions.
- Does sharing the range imply that there exists a continuous monotonic, subjective, increasing function such that $C_1(x)=C_2(g(x))$?
- Can we furnish an example of two curves that are equivalent according to the first and not to the second definition?
- If the answer to the second is yes, is it possible that the line integrals of the two functions that only satisfy the first condition differ? (more broadly, does the first vs the second kind of equivalence actually imply differences in the fundamental theorems associated with equivalence?)
Thank you.