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We study Möbius and quasimöbius mappings in spaces with a semimetric meeting the Ptolemy inequality. We construct a bimetrization of a Ptolemeic space which makes it possible to introduce a Möbius-invariant metric (angular distance) in the complement to each nonsingleton. This metric coincides with the hyperbolic metric in the canonical cases. We introduce the notion of generalized angle in a Ptolemeic space with vertices a pair of sets, determine its magnitude in terms of the angular distance and study distortion of generalized angles under quasimöbius embeddings. As an application to noninjective mappings, we consider the behavior of the generalized angle under projections and obtain an estimate for the inverse distortion of generalized angles under quasimeromorphic mappings (mappings with bounded distortion).

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Under study are the sets in R n (NED sets) each of which does not affect the conformal capacity of any condenser with connected plates disjoint from this set. These sets are removable singularities of quasiconformal mappings, which explains our interest in them. For compact sets on a hyperplane we obtain a geometric criterion of the NED property; we point out a simple sufficient condition for an NED set in terms of the connected attainability of its points from its complement in the hyperplane. For compact sets on a hypersphere we obtain a criterion for an NED set in terms of the reduced module at a pair of points in its complement. We establish that a compact set on a hypersphere S, removable for the capacity in at least one spherical ring concentric with S and containing S, is an NED set.Keywords: module of a family of curves, NED set, quasiconformal mapping, removable singularity, capacity of a condenser, reduced generalized module, capacity defect, attainable boundary point By a removable set (removable singularity) for a given class F of mappings we understand some compact set E such that for every open neighborhood U ⊃ E and every mapping f ∈ F with domain U \ E there exists a mapping of the same class F on U coinciding with f on U \ E. The problem of describing removable sets occurs in almost all branches of classical mathematics: general topology, analysis, function theory, and so on. In 1950 Ahlfors and Beurling described [1] the removable singularities on the plane for the class AD of analytic functions with bounded Dirichlet integral (the so-called zero sets); in particular, they established that the removable sets for analytic functions of class AD coincide with the sets preserving the modules of families of curves (NED sets). In 1956 Pesin obtained [2] the same result for quasiconformal plane mappings. In 1962 Väisälä described [3] the main properties of NED sets in R n , while their removability for quasiconformal space mappings is established independently in the articles by Aseev and Sychëv [4] as well as Vodop yanov and Gol dshteȋn [5]. The question whether each removable set for quasiconformal mappings in R n with n > 2 is an NED set still remains open. Moreover, the answer to this question is unknown in the particular case of a compact set on a hyperplane in R n . In 1969 Kopylov and Pesin gave [6,7] examples of removable sets of this type in R 3 ; for a broader class of compact sets in R n with zero (n − 1)-dimensional Hausdorff measure (which are NED sets [3]) Väisälä proved the removability for mappings with bounded distortion [8] (multivalent quasiconformal mappings); in yet more general a situation, for sets with zero projections, Miklyukov obtained in [9] the same property also in 1969. In 1974 the author gave in [10] an example of an NED set on a hyperplane in R n of nonzero (n − 1)-dimensional Hausdorff measure. Hedberg considered in [11, § 7, pp. 199-200] the question of the removability of a compact set E on a hyperplane for the p-capacity of condensers of a particular form (the opp...