On the Role of Riemannian Metrics in Conformal and Quasiconformal Geometry

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.