Manifolds with a unique embedding
We show that if $X, Y$ are smooth, compact $k$-dimensional submanifolds of $\mathbb R^n$ and $2k+2\le n$, then each diffeomorphism $\phi: X\to Y$ can be extended to a diffeomorphism $\Phi: \mathbb R^n\to \mathbb R^n$ which is tame (to be defined in this paper). Moreover, if $X, Y$ are real analytic manifolds and the mapping $\phi$ is analytic, then we can choose $\Phi$ to be also analytic.
We extend this result to some interesting categories of closed (not necessarily compact) subsets of $\mathbb R^n$, namely, to the category of Nash submanifolds (with Nash, real-analytic and smooth morphisms) and to the category of closed semi-algebraic subsets of $\mathbb R^n$ (with morphisms being semi-algebraic continuous mappings). In each case we assume that $X, Y$ are $k$-dimensional and $\phi$ is an isomorphism, and under the same dimension restriction $2k+2\le n$ we assert that there exists an extension $\Phi :\mathbb R^n\to\mathbb R^n$ which is an isomorphism and it is tame.The same is true in the category of smooth algebraic subvarieties of $\mathbb C^n$, with morphisms being holomorphic mappings and with morphisms being polynomial mappings.