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Surjective isometries of metric geometries

Part of: Riemann surfaces Classical differential geometry

Published online by Cambridge University Press:  28 October 2020

A. F. Beardon  and
D. Minda
A. F. Beardon
Affiliation:
Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, CambridgeCB3 0WB, UKe-mail:afb@dpmms.cam.ac.uk
D. Minda*
Affiliation:
Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH 45221-0025, USA
*
e-mail:minda@ucmail.uc.edu
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Abstract

Many authors define an isometry of a metric space to be a distance-preserving map of the space onto itself. In this note, we discuss spaces for which surjectivity is a consequence of the distance-preserving property rather than an initial assumption. These spaces include, for example, the three classical (Euclidean, spherical, and hyperbolic) geometries of constant curvature that are usually discussed independently of each other. In this partly expository paper, we explore basic ideas about the isometries of a metric space, and apply these to various familiar metric geometries.

Keywords

Constant curvature geometriesisometries

MSC classification

Primary: 53A30: Conformal differential geometry
Secondary: 53A35: Non-Euclidean differential geometry 30F99: None of the above, but in this section
Type
Article
Information
Canadian Mathematical Bulletin , Volume 64 , Issue 4 , December 2021 , pp. 828 - 839
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Copyright
© Canadian Mathematical Society 2020

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