TY - JOUR

T1 - Discontinuous Normals in Non-Euclidean Geometries and Two-Dimensional Gravity

AU - Battista, Emmanuele

AU - Esposito, Giampiero

N1 - Funding Information:
E.B. is grateful to Nicola Fusco for invaluable discussions regarding many relevant topics of geometric measure theory. The authors thank Marco Abate for the enlightening correspondence. This work is supported by the Austrian Science Fund (FWF) grant P32086. G.E. dedicates this research to Margherita.
Publisher Copyright:
© 2022 by the authors.

PY - 2022/10

Y1 - 2022/10

N2 - 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.

AB - 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.

KW - geometric measure theory

KW - non-euclidean geometries

KW - two-dimensional quantum gravity

UR - http://www.scopus.com/inward/record.url?scp=85140928765&partnerID=8YFLogxK

U2 - 10.3390/sym14101979

DO - 10.3390/sym14101979

M3 - Article

AN - SCOPUS:85140928765

VL - 14

JO - Symmetry

JF - Symmetry

SN - 2073-8994

IS - 10

M1 - 1979

ER -