I apologies in advance if this question is so trivial, I had read some textbook on Riemannian geometry and I dont have some things clear. This is the thing: I saw some questions about the arc length showing that it can't be a differential form in the ambient space and I would like to clarify some points. So these are my questions:
Let a regular curve $C$ in $\mathbb{R}^n$, then in view of the comments of this question using recursively the interior multiplication over the canonical volume form in $\mathbb{R}^n$ I can get an induced volume form $\omega$ in $C$, so
What exactly mean the integral of this induced volume form?
There is a relation with the arc length $ds$, maybe something like $\|\omega \|_2=ds$?
Thank you in advance.