  A morph between two Riemannian $n$-manifolds is an isotopy between them
together with the set of all intermediate manifolds equipped with Riemannian
metrics. We propose measures of the distortion produced by some classes of
morphs and diffeomorphisms between two isotopic Riemannian $n$-manifolds and,
with respect to these classes, prove the existence of minimal distortion morphs
and diffeomorphisms. In particular, we consider the class of time-dependent
vector fields (on an open subset $\Omega$ of $ \R^{n+1}$ in which the manifolds
are embedded) that generate morphs between two manifolds $M$ and $N$ via an
evolution equation, define the bending and the morphing distortion energies for
these morphs, and prove the existence of minimizers of the corresponding
functionals in the set of time-dependent vector fields that generate morphs
between $M$ and $N$ and are $L^2$ functions from $[0,1]$ to the Sobolev space
$W^{k,2}_0(\Omega,\R^{n+1})$.
