A New method to the study of the behavior of the geodesics on hyperbolic manifolds. Applications

Abstract

The presented work is a research in the field of the geometry of two-dimensional hyperbolic (equipped with a metric of constant negative curvature) manifolds. In this survey article we gather resent results of the global behavior of geodesics on hyperbolic manifolds, giving special attention to the two-dimensional case. This paper discribe new method (is developed a new method of colour multilaterals) for solving this problem - an algorithm (the construction of a practical approach) that allows determine the behavior of this geodesic on hyperbolic manifolds. With the help of this technique, the question of the qualitative behavior of geodesics in general on hyperbolic 2-manifolds is solved. Applications and future direction are discussed. JEL: MSC 53C60,30 F60,53 C22