论文标题
非欧几里得几何形状和二维重力中的不连续正态
Discontinuous normals in non-Euclidean geometries and two-dimensional gravity
论文作者
论文摘要
本文构建了两个在非欧亚人空间中的广义正常的详细示例，即双曲线和椭圆形几何形状。在双曲机平面中，我们定义了N侧双曲多边形P，即双曲线平面H的欧几里得闭合，由N双曲线测量片段界定。多边形P是通过考虑连接n+2个顶点（Tilde Z），Z0，Z1，...，Z（n-1），Z（n）的独特的大地测量物来构建的。将顶点连接的大地测量学是以真实轴为中心的欧几里得半圆。评估了与测量连接两个连续顶点的媒介正常，结果是不连续的。在椭圆形几何形状的框架内，我们求解了地球方程并构建一个大地三角形。同样在这种情况下，我们获得了不连续的正常矢量场。最后，概述了对二维欧几里得量子重力的可能应用。
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.
