Solution
= Solution
Under the <Euclidean isometry> $\widetilde{\mathbf r}=\mathbf a+R\mathbf r$, where $R^TR=I$,
$$
\partial_u\widetilde{\mathbf r}=R\partial_u\mathbf r.
$$
Therefore
$$
\widetilde g
=(R\mathbf r_u)\mathbin{\cdot}(R\mathbf r_u)
=\mathbf r_u\mathbin{\cdot}\mathbf r_u=g.
$$
Since the integration parameter is unchanged,
$$
\widetilde S(u)=\int_0^u\sqrt{\widetilde g(u')}\,du'
=S(u).
$$
Both the <induced metric> and <arc length> are invariant.