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On the Circle Preserving Property of Möbius Transformations

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Mathematics Without Boundaries

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Abstract

This paper is mainly concerned with the study of circle-preserving property of Möbius transformations acting on \(\widehat{\mathbf{R}}^{n}=\mathbf{R} ^{n}\cup \left\{\infty \right\}\). The circle-preserving property is the most known invariant characteristic property of Möbius transformations. Obviously, a Möbius transformation acting on \(\widehat{\mathbf{R}}^{n}\) is circle-preserving. Recently, for the converse statement, some interesting and nice results have been obtained. Here, we investigate these studies. We consider the relationships between Möbius transformations and sphere-preserving maps in \(\widehat{\mathbf{R}} ^{n}\) since the studies about the circle-preserving property of maps in \(\widehat{\mathbf{R}}^{n}\) are related to the study of sphere-preserving maps. For the case n = 2, we also consider the problem whether or not the circle-preserving property is an invariant characteristic property of Möbius transformations for the circles corresponding to any norm function \(\left\Vert.\right\Vert\) on \(\mathbf{C}\).

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Acknowledgments

The author would like to thank Prof. Th. M. Rassias for his encouragement to prepare this survey.

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Authors and Affiliations

  1. Faculty of Science and Literature, Department of Mathematics, Balıkesir University, 10145, Balıkesir, Turkey

    Nihal Yılmaz Özgür

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  1. Nihal Yılmaz Özgür
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Correspondence to Nihal Yılmaz Özgür .

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Editors and Affiliations

  1. Department of Mathematics, National Technical University of Athens, Athens, Greece

    Themistocles M. Rassias

  2. Dept. Industrial & Systems Engineering, University of Florida, Gainesville, Florida, USA

    Panos M. Pardalos

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Özgür, N. (2014). On the Circle Preserving Property of Möbius Transformations. In: Rassias, T., Pardalos, P. (eds) Mathematics Without Boundaries. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-1106-6_17

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