On the Role of Riemannian Metrics in Conformal and Quasiconformal Geometry

TL;DR

This paper introduces a new coordinate-invariant definition of quasiregular and quasiconformal mappings on Riemannian manifolds, generalizing classical Euclidean concepts and establishing foundational properties and applications.

Contribution

It proposes a natural, invariant definition of quasiregular and quasiconformal maps on Riemannian manifolds, extending classical Euclidean theory.

Findings

New invariant definition of quasiregular and quasiconformal mappings

Basic properties and convergence theorems established

Countable quasiconformal groups admit invariant conformal structures

Abstract

This article is the introductory part of authors PhD thesis. The article presents a new coordinate invariant definition of quasiregular and quasiconformal mappings on Riemannian manifolds that generalizes the definition of quasiregular mappings on $\R^n$. The new definition arises naturally from the inner product structures of Riemannian manifolds. The basic properties of the mappings satisfying the new definition and a natural convergence theorem for these mappings are given. These results are applied in a subsequent paper, arXiv:1209.1285. In the current article, an application, likewise demonstrating the usability of the new definition, is given. It is proven that any countable quasiconformal group on a general Riemannian manifolds admits an invariant conformal structure. This result generalizes a classical result by Pekka Tukia in the countable case.