  In this announcement we consider the following problem. Let $n,m\geq 1$,
$U\subset\mathbb R^n$ open. In this paper we provide a sharp solution to the
following Whitney distortion extension problems: (a) Let $\phi:U\to \mathbb
R^n$ be a $C^m$ map. If $E\subset U$ is compact (with some geometry) and the
restriction of $\phi$ to $E$ is an almost isometry with small distortion, how
to decide when there exists a $C^m(\mathbb R^n)$ one-to-one and onto almost
isometry $\Phi:\mathbb R^n\to \mathbb R^n$ with small distortion which agrees
with $\phi$ in a neighborhood of $E$ and a Euclidean motion $A:\mathbb R^n\to
\mathbb R^n$ away from $E$. (b) Let $\phi:U\to \mathbb R^n$ be $C^{\infty}$
map. If $E\subset U$ is compact (with some geometry) and the restriction of
$\phi$ to $E$ is an almost isometry with small distortion, how to decide when
there exists a $C^{\infty}(\mathbb R^n)$ one-to-one and onto almost isometry
$\Phi:\mathbb R^n\to \mathbb R^n$ with small distortion which agrees with
$\phi$ in a neighborhood of $E$ and a Euclidean motion $A:\mathbb R^n\to
\mathbb R^n$ away from $E$. Our results complement those of [14,15,20] where
there, $E$ is a finite set. In this case, the problem above is also a problem
of interpolation and alignment of data in $\mathbb R^n$. The material in this
paper appears in the memoir [14].
