caa a22 4500 605508224 CHVBK 20210128100635.0 cr unu---uuuuu 210128e20151201xx s 000 0 eng 10.1007/s00010-014-0316-0 doi (NATIONALLICENCE)springer-10.1007/s00010-014-0316-0 Demirel Oğuzhan Department of Mathematics, Faculty of Arts and Sciences, ANS Campus, Afyon Kocatepe University, 03200, Afyonkarahisar, Turkey aut A new proof of the nonexistence of isometries between higher dimensional Euclidean and hyperbolic spaces [Elektronische Daten] [Oğuzhan Demirel] 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. Springer Basel, 2014 Metric spaces nationallicence functional equations of metric and their solutions nationallicence hyperbolic geometry nationallicence gyrogroups nationallicence Aequationes mathematicae Springer International Publishing 89/6(2015-12-01), 1449-1459 0001-9054 89:6<1449 2015 89 10 https://doi.org/10.1007/s00010-014-0316-0 text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s00010-014-0316-0 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Demirel Oğuzhan Department of Mathematics, Faculty of Arts and Sciences, ANS Campus, Afyon Kocatepe University, 03200, Afyonkarahisar, Turkey aut NATIONALLICENCE 773 0- Aequationes mathematicae Springer International Publishing 89/6(2015-12-01), 1449-1459 0001-9054 89:6<1449 2015 89 10