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

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.