Abstract
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).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Gehring F. W. and Palka B., “Quasiconformally homogeneous domains,” J. Anal. Math., 30, No.5, 172–199 (1976).
Gehring F. W. and Osgood B. G., “Uniform domains and the quasi-hyperbolic metric,” J. Anal. Math., 45, 50–74 (1979).
Ferrand J., “A characterization of quasiconformal mappings by the behavior of a function of three points,” in: Proc. 13th Rolf Nevanlinna Colloquium (Joensuu, 1987), Springer-Verlag, Berlin; Heidelberg; New York; London; Paris; Tokyo, 1988, pp. 110–123. (Lecture Notes in Math.; 1351.)
Seittenranta P., “Möbius-invariant metrics,” Math. Proc. Cambridge Phil. Soc., 125, 511–533 (1999).
Agard S. and Gehring F. W., “Angles and quasiconformal mappings,” Proc. London Math. Soc. (3), 14A, 1–21 (1965).
Agard S. B., “Angles and quasiconformal mappings in space,” J. Anal. Math., 22, 177–200 (1969).
Taari O., “Characterisierung der Quasikonformität mit Hilfe der Winkelverzerrung,” Ann. Acad. Sci. Fenn. A1, 362, 1–11 (1965).
Aseev V. V., “Möbius-invariant metrics of space domains,” in: Abstracts: The Fourth Siberian Congress on Applied and Industrial Mathematics Dedicated to the Memory of M. A. Lavrent’ev. Part 1, Inst. Mat. (Novosibirsk), Novosibirsk, 2000, pp. 148–149.
Aseev V. V. and Tetenov A. V., “Angular characteristics of the pairs of sets and a problem of quasisymmetric glueing,” in: Proceedings of the Lobachevski\(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\imath}\) Center in Mathematics: Function Theory, Its Applications, and Related Problems. Materials of the 6th Kazan’ International Summer School-Conference [in Russian], Izdat. Kazan’ Mat. Obshch., Kazan’, 2003, 19, pp. 16–17.
Blumenthal M. L., Theory and Applications of Distance Geometry, Clarendon Press, Oxford (1953).
Bourbaki N., General Topology. Use of Real Numbers in General Topology. Function Spaces. Summary of Results [Russian translation], Nauka, Moscow (1968).
Berdon A. F., The Geometry of Discrete Groups [Russian translation], Nauka, Moscow (1986).
Wermer J., Potential Theory, Springer-Verlag, Berlin; Heidelberg; New York; London; Paris; Tokyo (1974). (Lecture Notes in Math.; 408.)
Vuorinen M., Conformal Geometry and Quasiregular Mappings, Springer-Verlag, Berlin; Heidelberg; New York; London; Paris; Tokyo (1988). (Lecture Notes in Math., 1319.)
Berger M., Geometry. Vol. 1 [Russian translation], Mir, Moscow (1984).
Jiang Yueping, “On Clifford cross-ratio and its application,” J. Hunan Univ. (in China), 20, No.2, 21–25 (1993).
Aseev V. V., “Quasisymmetric embeddings and mappings with bounded modulus distortion,” submitted to VINITI on November 6, 1984, No. 7190-84.
Väisälä J., “Quasimöbius maps,” J. Anal. Math., 44, 218–234 (1984/85).
Aseev V. V., “On metrization with distortion coefficients of a space of domains,” in: Group and Metric Properties of Mappings [in Russian], Novosibirsk State University, Novosibirsk, 1995, 1, pp. 97–105.
Martio O., Rickman S., and Väisälä J., “Distortion and singularities of quasiregular mappings,” Ann. Acad. Sci. Fenn. Ser. A 1 Math., 465, 1–13 (1970).
Reshetnyak Yu. G., Space Mappings with Bounded Distortion [in Russian], Nauka, Novosibirsk (1982).
Martio O., Rickman S., and Väisälä J., “Definitions for quasiregular mappings,” Ann. Acad. Sci. Fenn. Ser. A 1 Math., 448, 1–40 (1969).
Author information
Authors and Affiliations
Additional information
Original Russian Text Copyright © 2005 Aseev V. V., Sychëv A. V., and Tetenov A. V.
Translated from Sibirski \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\imath}\) Matematicheski \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\imath}\) Zhurnal, Vol. 46, No. 2, pp. 243–263, March–April, 2005.
Rights and permissions
About this article
Cite this article
Aseev, V.V., Sychëv, A.V. & Tetenov, A.V. Möbius-invariant metrics and generalized angles in Ptolemeic spaces. Sib Math J 46, 189–204 (2005). https://doi.org/10.1007/s11202-005-0020-3
Received:
Issue Date:
DOI: https://doi.org/10.1007/s11202-005-0020-3