  Given bounded domains $\Omega_1$ and $\Omega_2$ in $\mathds{R}^N$ and an
isometry $T$ from $W^{1,p}(\Omega_1)$ to $W^{1,p}(\Omega_2)$, we give
sufficient conditions ensuring that $T$ corresponds to a rigid motion of the
space, i.e., $Tu = \pm (u \circ \xi)$ for an isometry $\xi$, and that the
domains are congruent. More general versions of the involved results are
obtained along the way.
