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Geometry of Surfaces
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John Stillwell
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Springer-Verlag
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228
ISBN
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0387977430
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This text intends to provide the student with the knowledge of a geometry of greater scope than the classical geometry taught today, which is no longer an adequate basis for mathematics or physics, both of which are becoming increasingly geometric. The geometry of surfaces is an ideal starting point for students learning geometry for the following reasons; first, the extensions offer the simplest possible introduction to fundamentals of modern geometry: curvature, group actions and covering spaces. Second, the prerequisites are modest and standard and include only a little linear algebra, calculus, basic group theory and basic topology. Third and most important, the theory of surfaces of constant curvature has maximal connectivity with the rest of mathematics. They realize all the topological types of compact two-dimensional manifolds, and historically, they are the source of the main concepts of complex analysis, differential geometry, topology, and combinatorial group theory, as well as such hot topics as fractal geometry and string theory. The formal coverage is extended by exercises and informal discussions throughout the text.
Preface
1. The Euclidean Plane
2. Euclidean Surfaces
3. The Sphere
4. The Hyperbolic Plane
5. Hyperbolic Surfaces
6. Pathe and Geodesics
7. Planar and Spherical Tessellations
8. Tessellations of Compact Surfaces
Referencs
Index
"The three basic geometries of constant curvature are the Euclidean (zero curvature), spherical (positive curvature) and hyperbolic (negative curvature). These may be studied through their isometries (chapters 1, 3, 4, respectively). This is pretty. Other than these three "planes" one may obtain surfaces that are locally isometric to them by taking quotients by certain groups of isometries. That's easy in the Euclidean case (chapter 2), trivial in the spherical case, and hard in the hyperbolic case (chapter 5), which needs to be complemented by a whole chapter of topology (chapter 6). These groups of isometries have "fundamental regions", i.e. polygons that tessellate the plane in which they live. Not all tessellations are obtained in this way, however, so one is lead to study tesellations in general, corresponding to more general groups of isometries (chapters 7-8). The presentation is well motivated within its own aesthetic framework, but some discomfort results from the feeling that this clever approach is a virtuoso post-construction (some may say that this is consistent with "Klein's spirit" (cf. p. viii)). Historical background and related topics are treated in informal discussion sections at the end of each chapter. The main theme here is the deep connections with complex function theory."
"An excellent exposition of the three geometries of surfaces. I highly recommend this book to advanced undergraduates and beginning graduates in mathematics."
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