Curved Spaces From Classical Geometries to Elementary Differential Geometry
- Textbook
Description
This self-contained 2007 textbook presents an exposition of the well-known classical two-dimensional geometries, such as Euclidean, spherical, hyperbolic, and the locally Euclidean torus, and introduces the basic concepts of Euler numbers for topological triangulations, and Riemannian metrics. The careful discussion of these classical examples provides students with an introduction to the more general theory of curved spaces developed later in the book, as represented by embedded surfaces in Euclidean 3-space, and their generalization to abstract surfaces equipped with Riemannian metrics.…
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Key features
- A concrete approach to the theory, with emphasis on self-contained explicit proofs; uses the classical geometries to motivate the basic ideas of elementary differential geometry
- Provides a link between basic undergraduate courses on Analysis and Algebra, and more advanced theoretical courses in geometry
- Rigorous treatment of the classical geometries, via analytical ideas, with exercises at the end of each chapter, reinforcing the material in the text
- A novel approach to defining curvature on abstract surfaces, and to proving the topological invariance of the Euler number
- Coverage of a wide range of topics, starting with very elementary material and concluding with rather more advanced mathematical ideas
- Certain geometrical themes, such as geodesics, curvature, and the Gauss-Bonnet theorem, running throughout the book, provide a unifying philosophy
About the book
- DOI https://doi.org/10.1017/9781139507677
- Subjects Geometry and Topology,Mathematical Physics,Mathematics,Physics and Astronomy
- Format: Hardback
- Publication date: 21 January 2008
- ISBN: 9780521886291
- Dimensions (mm): 247 x 174 mm
- Weight: 0.52kg
- Contains: 79 b/w illus. 105 exercises
- Page extent: 198 pages
- Availability: Available
- Format: Paperback
- Publication date: 14 January 2008
- ISBN: 9780521713900
- Dimensions (mm): 247 x 174 mm
- Weight: 0.33kg
- Contains: 79 b/w illus. 105 exercises
- Page extent: 198 pages
- Availability: Available
- Format: Digital
- Publication date: 14 September 2019
- ISBN: 9781139507677
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