GEOMETRY OF GEODESICS ON HYPERBOLIC MANIFOLDS

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. We introduce a new method (a way) to describe the global behavior of geodesics on hyperbolic manifolds of dimension two. We use this construction (method of colour multilaterals ) to investigate typical behavior of geodesics on a arbitrary hyperbolic surfaces of signature . Applications and future direction are discussed. For this purpose with the help of proposed practical approach at first:1) we obtain a complete classification of all possible geodesics on the simplest hyperbolic 2-manifolds (hyperbolic horn; hyperbolic cylinder; parabolic horn (cusp)); 2) describe the behavior of geodesics on the following cases: a) on a genus two hyperbolic surface (double-glued from two pair of pants); b) we investigate the typical behavior of geodesic on a compact closed hyperbolic surface without boundary (general case); c) on a hyperbolic surface of genus g and with n boundary components; d) on a hyperbolic 1- punctured torus; e) on a generalized hyperbolic pants; f) on a hyperbolic thrice-punctured sphere; in general case: g) for any (oriented) punctured hyperbolic surface M of genus g and k punctures; in the most general case: h) behavior of geodesic on any hyperbolic surface of signature (with genus g n boundary components and k cusps).