Hyperbolic Manifolds and the Compact Two-dimensional Case

In this chapter we are going to introduce the notion of hyperbolic manifold (i.e. a manifold modeled on hyperbolic space) via the introduction of a much more general class of manifolds. We shall prove the first essential properties of such manifolds (namely, the fact that if a hyperbolic manifold is complete then it can be obtained as a quotient of hyperbolic space). Afterwards we shall consider the special case of compact surfaces and we shall give a complete classification of the hyperbolic structures on a surface of fixed genus (that is we shall give a parametrization of the so-called Teichmüller space).