Geometry from a differentiable viewpoint
著者
書誌事項
Geometry from a differentiable viewpoint
Cambridge University Press, 1994
- : pbk
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注記
Includes bibliography (p. 297-302), and symbol, name and subject indexes
内容説明・目次
内容説明
Differential geometry has developed in many directions since its beginnings with Euler and Gauss. This often poses a problem for undergraduates: which direction should be followed? What do these ideas have to do with geometry? This book is designed to make differential geometry an approachable subject for advanced undergraduates. The text serves as both an introduction to the classical differential geometry of curves and surfaces and as a history of the non-Euclidean plane. The book begins with the theorems of non-Euclidean geometry, then introduces the methods of differential geometry and develops them towards the goal of constructing models of the hyperbolic plane. Interesting diversions are offered, such as Huygens' pendulum clock and mathematical cartography; however, the focus of the book is on the models of non-Euclidean geometry and the modern ideas of abstract surfaces and manifolds. Although the main use of this text is as an advanced undergraduate course book, the historical aspect of the text should appeal to most mathematicians.
目次
- Prelude and themes
- Part I. Synthetic Methods and Results: 1. Spherical geometry
- 2. Euclid
- 3. The theory of parallels
- 4. Non-Euclidean geometry I
- 5. Non-Euclidean geometry II
- Part II. Development: Differential Geometry: 6. Curves
- 7. Curves in space
- 8. Surfaces
- 9. Curvature for surfaces
- 10. Metric equivalence of surfaces
- 11. Geodesics
- 12. The Gauss-Bonnet theorem
- 13. Constant curvature surfaces
- Part III. Recapitulation
- 14. Abstract surfaces
- 15. Modelling non-Euclidean geometry
- 16. Coda: generalizations.
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