Geometry from a Differentiable Viewpoint

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Cambridge University Press, 1994 - Mathematics - 308 pages
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This book offers a new treatment of the topic, one which is designed to make differential geometry an approachable subject for advanced undergraduates. Professor McCleary considers the historical development of non-Euclidean geometry, placing differential geometry in the context of geometry students will be familiar with from high school. The text serves as both an introduction to the classical differential geometry of curves and surfaces and as a history of a particular surface, the non-Euclidean or hyperbolic plane. The main theorems of non-Euclidean geometry are presented along with their historical development. The author then introduces the methods of differential geometry and develops them toward the goal of constructing models of the hyperbolic plane. While interesting diversions are offered, such as Huygen's pendulum clock and mathematical cartography, the book thoroughly treats the models of non-Euclidean geometry and the modern ideas of abstract surfaces and manifolds.
  

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Contents

Spherical geometry
3
NonEuclidean geometry I
34
Curves
63
Curves in space
80
Surfaces
95
8 Map projections
116
Metric equivalence of surfaces
145
Geodesics
157
Constantcurvature surfaces
186
Abstract surfaces
201
Modeling the nonEuclidean plane
217
Where from here?
242
On the hypotheses which lie at the foundations
269
Notes on selected exercises
279
Bibliography 297
301
Copyright

The GaussBonnet theorem
171

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About the author (1994)

John McCleary is Professor of Mathematics at Vassar College on the Elizabeth Stillman Williams Chair. His research interests lie at the boundary between geometry and topology, especially where algebraic topology plays a role. His papers on topology have appeared in Inventiones Mathematicae, the American Journal of Mathematics and other journals, and he has written expository papers that have appeared in American Mathematical Monthly. He is also interested in the history of mathematics, especially the history of geometry in the nineteenth century and of topology in the twentieth century. He is the author of A User's Guide to Spectral Sequences and A First Course in Topology: Continuity and Dimension and he has edited proceedings in topology and in history, as well as a volume of the collected works of John Milnor. He has been a visitor to the mathematics institutes in Goettingen, Strasbourg and Cambridge, and to MSRI in Berkeley.

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