Abstract
In this chapter, we study the geometry of geometric surfaces. The chapter begins with a review of the topology of compact surfaces. In Section 9.2, a geometric method for constructing spherical, Euclidean, and hyperbolic surfaces is given. The fundamental relationship between the Euler characteristic of a closed geometric surface and its area is derived in Section 9.3. In Section 9.4, the set of similarity equivalence classes of Euclidean or hyperbolic structures on a closed surface is shown to have a natural topology. The geometry of closed geometric surfaces is studied in Sections 9.5 and 9.6. The chapter ends with a study of the geometry of complete hyperbolic surfaces of finite area.
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© 2006 Springer Science+Business Media, LLC
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(2006). Geometric Surfaces. In: Foundations of Hyperbolic Manifolds. Graduate Texts in Mathematics, vol 149. Springer, New York, NY. https://doi.org/10.1007/978-0-387-47322-2_9
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DOI: https://doi.org/10.1007/978-0-387-47322-2_9
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-33197-3
Online ISBN: 978-0-387-47322-2
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