On the local character of the extension of traces for Sobolev mappings

We prove that a mapping $u\text{\hspace{0.17em}⁣}:{\mathcal{M}}^{\mathrm{\prime }}\to \mathcal{N}$, where ${\mathcal{M}}^{\mathrm{\prime }}$ and $\mathcal{N}$ are compact Riemannian manifolds, is the trace of a Sobolev mapping $U\text{\hspace{0.17em}⁣}:{\mathcal{M}}^{\mathrm{\prime }}\times [0,1)\to \mathcal{N}$ if and only if it is on some open covering of ${\mathcal{M}}^{\mathrm{\prime }}$. In the global case where $\mathcal{M}$ is a compact Riemannian manifold with boundary, this implies that the analytical obstructions to the extension of a mapping $u\text{\hspace{0.17em}⁣}:\mathrm{\partial }\mathcal{M}\to \mathcal{N}$ to some Sobolev mapping $U\text{\hspace{0.17em}⁣}:\mathcal{M}\to \mathcal{N}$ are purely local.