THE BEHAVIOR OF GEODESICS IN TWO-DIMENSIONAL HYPERBOLIC MANIFOLDS

The main purpose of this work is the given a new constructive method for solving the problem of the behavior of geodesic on hyperbolic surfaces of genus g, k punctures and with n geodesic boundary components. At first:1) we obtain a complete classification of all possible geodesic curves on the simplest hyperbolic 2-manifolds (hyperbolic horn; hyperbolic cylinder; parabolic horn (cusp), hyperbolic pants); 2) on surface of genus 2; Finally: 3) on compact closed hyperbolic surface without boundarie (general case); 4) on hyperbolic surface of genus g and with n geodesic boundary components; 5) on hyperbolic 1-punctured torus; on generalized hyperbolic pants; in general case: for any punctured hyperbolic surface of genus g and k punctures; 6) in the most general case: or in any hyperbolic surface of genus g, k punctures and with n boundary curves.