On admissible changes of variables for Sobolev functions on (sub)Riemannian manifolds (Q334432)

The article under review is concerned with the following change of variables question: Which mappings \(\phi\) between metric spaces \(X\) and \(Y\) induce isomorphisms between the Sobolev spaces \(W^{1,p}(X)\) and \(W^{1,p}(Y)\) through composition? While this question has been settled in the case that \(X\) and \(Y\) are domains of some \(\mathbb{R}^n\), the author of this article presents a characterization in the case that \(X\) and \(Y\) are domains of some Riemannian \(n\)-manifold. The main results are Theorem 1 and Theorem 2. In the case \(p=n\), a mapping \(\phi\) has the desired property if and only if \(\phi\) coincides with a quasiconformal homeomorphism almost everywhere. In the case \(p\neq n\), \(\phi\) has the desired property if and only if \(\phi\) coincides with a quasi-isometry (i.e., a homeomorphism that resembles bi-Lipschitz on large scale) almost everywhere.