Abstract
The lines of Euclidean and hyperbolic geometries are characterized by Benz (Monatsh Math 141:1–10, 2004) as metric lines in the sense of Blumenthal and Menger (Studies in Geometry. San Francisco: Freeman, 1970). In this paper, we extend the notion of metric lines to metric hyperplanes and characterize the hyperplanes of Euclidean geometries as metric hyperplanes. In addition to this we give a new proof that there do not exist metric hyperplanes in hyperbolic geometry and this result implies that corresponding higher dimensional Euclidean and hyperbolic spaces are not isometric. Moreover, as in hyperbolic geometry, there do not exist metric hyperplanes in elliptic and spherical geometries.
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Demirel, O. A new proof of the nonexistence of isometries between higher dimensional Euclidean and hyperbolic spaces. Aequat. Math. 89, 1449–1459 (2015). https://doi.org/10.1007/s00010-014-0316-0
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DOI: https://doi.org/10.1007/s00010-014-0316-0