Notes on quasiregular maps between Riemannian manifolds

TL;DR

This paper reviews how core properties of quasiregular maps in Euclidean spaces extend to Riemannian manifolds, emphasizing Sobolev space definitions and key theorems transfer.

Contribution

It systematically demonstrates the equivalence of different Sobolev space definitions and transfers fundamental quasiregular map results from Euclidean spaces to manifolds.

Findings

Equivalence of Sobolev space definitions on manifolds

Transfer of Reshetnyak's theorem to manifolds

Pull-backs preserve Sobolev differential forms

Abstract

These notes provide an exposition on obtaining the well-known standard results of quasiregular maps on Riemannian manifolds, given the corresponding theory in the Euclidean setting. We recall several different approaches to first-order Sobolev spaces between Riemannian manifolds, and show that they result in equivalent definitions of quasiregular maps. We explain how e.g. Reshetnyak's theorem, degree and local index theory, and the quasiregular change of variables formula are transferred into the manifold setting from Euclidean spaces. Finally, we conclude with a proof of the basic fact that pull-backs with quasiregular maps preserve Sobolev differential forms of the conformal exponent