On the Whitney Extension-Interpolation-Alignment problem for almost isometries with small distortion in ${\mathbb{R}}^D$

In this paper, we study the following problem. Let $D\ge 2$, $S\subset \mathbb R^D$befiniteandlet$\phi:S\to \mathbb R^D$with$\phi$ a small distortion on $S$. We solve the Whitney extension-interpolation-alignment problem of how to understand when $\varphi$ can be extended to a function $\mathrm{\Phi }:{\mathbb{R}}^D\to {\mathbb{R}}^D$ which is a smooth small distortion on ${\mathbb{R}}^D$. The work in this paper appears in the research memoir [14]