Abstract
In this paper we study the connection between $\eta$-quasisymmetric homomorphisms and $K$-quasi\-con\-for\-mal mappings on $n$-dimensional smooth connected Riemannian manifolds. The main result of our research is the Theorem 3.1. For its proof we use a partition of unity method, which subordinate to the locally finite atlas of the manifold. Several results on the boundary behavior of $\eta$-quasisymmetric homomorphisms between two arbitrary domains, QED (uniform) domains and domains with weakly flat boundaries and compact closures on the Riemannian manifolds are also obtained in view of the above relations. The obtained results can be applied to Finsler manifolds with the addition of some conditions, which will take into account the specific of the initial manifold.
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