Geodesic
Surjective isometries of metric geometries
Canadian mathematical bulletin, Tome 64 (2021) no. 4, pp. 828-839

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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.
DOI : 10.4153/S0008439520000867
Mots-clés : Constant curvature geometries, isometries 
Beardon, A. F.; Minda, D. Surjective isometries of metric geometries. Canadian mathematical bulletin, Tome 64 (2021) no. 4, pp. 828-839. doi: 10.4153/S0008439520000867
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