Curves, Surfaces, and Hyperbolic Geometry
Curves, Surfaces, and Hyperbolic Geometry
This chapter explains the basics of working with simple closed curves, focusing on the case of the closed surface Sɡ of genus g. When g is greater than or equal to 2, hyperbolic geometry enters as a useful tool since each homotopy class of simple closed curves has a unique geodesic representative. The chapter begins by recalling some basic results about surfaces and hyperbolic geometry, with particular emphasis on the boundary of the hyperbolic plane and hyperbolic surfaces. It then considers simple closed curves in a surface S, along with geodesics and intersection numbers. It also discusses the bigon criterion, homotopy versus isotopy for simple closed curves, and arcs. Finally, it describes the change of coordinates principle and three facts about homeomorphisms.
Keywords: simple closed curve, hyperbolic geometry, homotopy, hyperbolic plane, hyperbolic surface, intersection number, bigon criterion, isotopy, coordinates principle, homeomorphism
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