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.
KeywordsModulus Space Closed Geodesic Geometric Surface Klein Bottle Hyperbolic Surface
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