  Let $M$ and $N$ be compact smooth oriented Riemannian $n$-manifolds without
boundary embedded in $\mathbb{R}^{n+1}$. Several problems about minimal
distortion bending and morphing of $M$ to $N$ are posed. Cost functionals that
measure distortion due to stretching or bending produced by a diffeomorphism
$h:M \to N$ are defined, and new results on the existence of minima of these
cost functionals are presented. In addition, the definition of a morph between
two manifolds $M$ and $N$ is given, and the theory of minimal distortion
morphing of compact manifolds is reviewed.
