I was dealing with a question that for me seems vague, surely because I'm starting in differential geometry. Well:
One vector field of the torus $T$ is obtained by parametrization of the meridian by arc lenght e defining $w(p)$ as the velocity of meridian by $p$. Prove that the field is diferenciable.
Well, I am confused because maybe there's more than one parametrization... I could not understand how to approach can run in general.
Many thanks!
For any Riemannian manifold $M$, and for any smooth connected oriented 1-dimensional manifold $G \subset M$, the arc length parameterization of $G$ is unique up to translation of the parameter.
What this means is that if $\gamma : (a,b) \to G$ and $\rho : (c,d) \to G$ are orientation preserving arc length parameters of subsegments of $G$, and if $p = \gamma(s) = \rho(t)$ for some $s \in (a,b)$, $t \in (c,d)$, and if $r>0$ is taken so small that $(s-r,s+r) \subset (a,b)$, and $(t-r,t+r) \subset (c,d)$, then for each $u \in (s-r,s+r)$ we have $$\gamma(u) = \rho(u-s+t) $$ And from this it is pretty obvious, taking $u=s$, that the tangent vectors $\gamma'(s)$ and $\rho'(t)$ at the point $p$ are equal.