Discontinuous Normals in Non-Euclidean Geometries and Two-Dimensional Gravity

Autor(en)

Emmanuele Battista, Giampiero Esposito

Abstrakt

This paper builds two detailed examples of generalized normal in non-Euclidean spaces, i.e., the hyperbolic and elliptic geometries. In the hyperbolic plane we define a n-sided hyperbolic polygon (Formula presented.), which is the Euclidean closure of the hyperbolic plane (Formula presented.), bounded by n hyperbolic geodesic segments. The polygon (Formula presented.) is built by considering the unique geodesic that connects the (Formula presented.) vertices (Formula presented.). The geodesics that link the vertices are Euclidean semicircles centred on the real axis. The vector normal to the geodesic linking two consecutive vertices is evaluated and turns out to be discontinuous. Within the framework of elliptic geometry, we solve the geodesic equation and construct a geodesic triangle. Additionally in this case, we obtain a discontinuous normal vector field. Last, the possible application to two-dimensional Euclidean quantum gravity is outlined.

Organisation(en)

Mathematische Physik

Externe Organisation(en)

Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Napoli, Università degli studi di Napoli Federico II

Journal

Symmetry

Band

14

Anzahl der Seiten

18

ISSN

2073-8994

DOI

https://doi.org/10.3390/sym14101979

Publikationsdatum

10-2022

Peer-reviewed

Ja

ÖFOS 2012

103028 Relativitätstheorie, 103019 Mathematische Physik

Schlagwörter

ASJC Scopus Sachgebiete

Computer Science (miscellaneous), Chemistry (miscellaneous), Allgemeine Mathematik, Physics and Astronomy (miscellaneous)

Link zum Portal

https://ucrisportal.univie.ac.at/de/publications/ecd2f7bb-e958-47b9-b35f-2e4ca1b97411