Titel
Discontinuous Normals in Non-Euclidean Geometries and Two-Dimensional Gravity
Giampiero Esposito
Dipartimento di Fisica “Ettore Pancini” and INFN Sezione di Napoli, Complesso Universitario di Monte S. Angelo
Abstract
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 𝒫, which is the Euclidean closure of the hyperbolic plane ℍ, bounded by n hyperbolic geodesic segments. The polygon 𝒫 is built by considering the unique geodesic that connects the 𝑛+2 vertices 𝑧˜,𝑧0,𝑧1,…,𝑧𝑛−1,𝑧𝑛. 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.
Stichwort
geometric measure theorynon-euclidean geometriestwo-dimensional quantum gravity
Objekt-Typ
Sprache
Englisch [eng]
Persistent identifier
Erschienen in
Titel
Symmetry
Band
14
Ausgabe
10
ISSN
2073-8994
Erscheinungsdatum
2022
Publication
MDPI AG
Erscheinungsdatum
2022
Zugänglichkeit
Rechte
Rechteangabe
© 2022 by the authors
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