Is There Any Metric Line Which Can Be Represented By A Single Fixed Point?

Year 2019, Volume: 4 Issue: 1, 54 - 62, 27.06.2019

Oğuzhan Demirel

https://doi.org/10.33484/sinopfbd.469669

Abstract

In this paper, determining metric lines in Levenberg plane, a special metric line example which can be written by using a single fixed point is presented. Moreover, we prove nonexistence of periodic lines in Levenberg plane and give a problem whether there exist a distance space in which periodic lines written by using a single fixed point?

Keywords

Metric spaces, functional equations of metric and periodic lines and their solutions, Levenberg plane

References

• [1] Benz, W., Metric and periodic lines in real inner product space geometries. Monatsh. Math.141 (2004), no. 1, 1–10.
• [2] Benz, W., Classical Geometries in Modern Contexts, Birkhauser, 2007.
• [3] Blumenthal LM, Menger K (1970) Studies in Geometry. San Francisco: Freeman.
• [4] Demirel, O., Seyrantepe E.S, Sönmez, N., Metric and periodic lines in the Poincare ball modelof hyperbolic geometry, Bull. Iran. Math. Soc. 38 (2012), no. 3, 805–815.
• [5] Demirel, O., Seyrantepe E.S, The cogyrolines of Möbius gyrovector spaces are metric but notperiodic, Aequationes Math. 85 (2013), no. 1-2, 185–200.
• [6] Gelisgen, O., On the Congruence of Triangles in Levenberg Plane, Dumlupinar ¨ Universitesi ¨Fen Bilimleri Enstitüsü 14, 2007.
• [7] Höfer, R., Metric lines in Lorentz-Minkowski geometry, Aequationes Math. 71 (2006), no. 1-2,162–173.
• [8] Höfer, R., Metric and Periodic lines in de Sitter’s World, J.Geom. 90 (2008), 66–82.