Hyperbolic Surfaces and Fuchsian groups: Summary

Abstract

Abstract The real :1-dimensional Euclidean space JR3 is given a sense of reality by being considered as a model of the physical world and most of its elementary properties are well known. Mathematically, the hyperbolic space, which is one type of non-Euclidean geometry, has the same right to claim its own existence as the Euclidean space, but few of its properties have been established as common knowledge. In the present volume, we give a glimpse into the charms of hyperbolic manifolds, which are generalizations of the hyperbolic space in the sense that they can be regarded as the hyperbolic space in a “neighborhood” of any point on it.