  We give a quantitative characterization of traces on the boundary of Sobolev
maps in $\dot{W}^{1,p}(\mathcal M, \mathcal N)$, where $\mathcal{M}$ and
$\mathcal{N}$ are compact Riemannian manifolds, $\partial \mathcal{M} \neq
\emptyset$: the Borel-measurable maps $u\colon \partial \mathcal M \to
\mathcal{N}$ that are the trace of a map $U\in \dot{W}^{1,p}(\mathcal M,
\mathcal{N})$ are characterized as the maps for which there exists an extension
energy density $w \colon \partial \mathcal{M} \to [0,\infty]$ that controls the
Sobolev energy of extensions from $\lfloor p - 1 \rfloor$-dimensional subsets
of $\partial \mathcal{M}$ to $\lfloor p\rfloor$-dimensional subsets of
$\mathcal{M}$.
