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This self-contained 2007 textbook presents an exposition of the correctly-acknowledged classical two-dimensional geometries, harking back to Euclidean, spherical, hyperbolic, and the regionally Euclidean torus, and introduces the important concepts of Euler numbers for topological triangulations, and Riemannian metrics. The cautious dialogue of these classical examples provides school college students with an introduction to the additional primary precept of curved areas developed later inside the book, as represented by embedded surfaces in Euclidean three-space, and their generalization to abstract surfaces outfitted with Riemannian metrics. Themes working all by way of embrace these of geodesic curves, polygonal approximations to triangulations, Gaussian curvature, and the hyperlink to topology provided by the Gauss-Bonnet theorem. Fairly a couple of diagrams help ship the essential factor elements to life and helpful examples and exercises are included to help understanding. All by means of the emphasis is positioned on particular proofs, making this textual content material excellent for any scholar with a main background in analysis and algebra.
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