If I have two regular surfaces $S_1, S_2$, with first fundamental forms expressed by two matrices $M_1, M_2$ and an isometry $\phi$ between them, does this mean that the length of a curve $\beta$ on $S_1$ measured according to $M_1$ is the same as that of $\phi\circ\beta$ measured according to $M_2$?
Or the only lengths preserved are the ones of the geodesics?
(in the comments I've posted a little longer explanation of what I'm trying to do)
Best Answer
Well, turns out I had made a little mistake in effectively computing $M_2$. Indeed, the answer is YES, as the theory states. (I don't know if to delete the question, since it may be useful to someone, one day)