Discontinuous normals in non-Euclidean geometries and two-dimensional gravity

TL;DR

This paper constructs examples of discontinuous normal vectors in hyperbolic and elliptic geometries and discusses potential applications to two-dimensional Euclidean quantum gravity.

Contribution

It provides detailed constructions of generalized normal vectors in non-Euclidean geometries, highlighting their discontinuities and potential relevance to quantum gravity models.

Findings

Discontinuous normals in hyperbolic polygons

Discontinuous normals in elliptic geodesic triangles

Potential applications to 2D Euclidean quantum gravity

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 (tilde z),z0,z1,...,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. Also in this case, we obtain a discontinuous normal vector field. Last, the possible application to two-dimensional Euclidean quantum gravity is outlined.