Abstract
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).
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© 1992 Springer-Verlag Berlin Heidelberg
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Benedetti, R., Petronio, C. (1992). Hyperbolic Manifolds and the Compact Two-dimensional Case. In: Lectures on Hyperbolic Geometry. Universitext. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-58158-8_2
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DOI: https://doi.org/10.1007/978-3-642-58158-8_2
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-55534-6
Online ISBN: 978-3-642-58158-8
eBook Packages: Springer Book Archive