I may be conflating a number of different concepts, but I am confused about measurements using the first fundamental form.

Here is what I think I know:

- The first fundamental form can be used to compute distance on a parametric surface
- The coefficients of the first fundamental form can be re-parameterized through substitution, as with the relation given in Sec. 1.3 here.

Here is what confuses me:

- Re-parameterizing the sphere through a map projection (e.g. gnomonic projection, mercator projection, etc.) results in different distances on the mapped surface. That is, $ds_{map}^2 \ne ds_{sphere}^2$. I can walk through this derivation and it makes logical sense.
- Yet, "arclength is invariant to coordinate system transformations," at least according to the proof on the Metric Tensor Wikipedia page (screenshot pasted below) and a handful of other Google search results on the topic. But isn't a spherical projection a coordinate transformation $M : (\lambda, \phi) \rightarrow (x, y)$?

I believe that this text clears it up a bit (bottom of page 8):

A mapping of a portion of a manifold M to a portion of a manifold N is called isometric, if the length of any curve on N is the same as the length of its pre-image on M.

In other words, arclength is preserved by isometric transformations. Spherical map projections are provably not isometric, so, sure, that's why $ds_{map}^2 \ne ds_{sphere}^2$. I think I follow this.

Why I'm confused is because I thought I followed the logic of proof on the Wikipedia page, and I don't understand why it wouldn't generalize to non-isometric mappings. All the same derivatives can be calculated through a spherical projection, for example.

So does arclength hold or not?

[**Note:** when answering this question, feel free to use the example of spherical projections, because it's something I think I understand fairly well. Also, I'm not a mathematician, so formal mathematical language, while likely the most accurate way to answer this question, is probably going to be lost on me.]