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- Title
Discontinuous Normals in Non-Euclidean Geometries and Two-Dimensional Gravity.
- Authors
Battista, Emmanuele; Esposito, Giampiero
- 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 ˜ , z 0 , z 1 , ... , z n − 1 , z n . 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.
- Subjects
NON-Euclidean geometry; GEOMETRIC measure theory; GEODESIC equation; QUANTUM gravity; VECTOR fields; HYPERBOLIC geometry
- Publication
Symmetry (20738994), 2022, Vol 14, Issue 10, pN.PAG
- ISSN
2073-8994
- Publication type
Article
- DOI
10.3390/sym14101979