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

We give a description of metric properties of measurable mappings of domains on Riemannian manifolds inducing isomorphisms of Sobolev spaces by the composition rule. We prove that any such mapping can be redefined on a set of measure zero to be quasi-isometric, when the exponent of summability is different from the dimension of a Riemannian manifold or to coincide with a quasi-conformal mapping otherwise. © 2016, Pleiades Publishing, Ltd.