Introduction to hyperbolic geometry
著者
書誌事項
Introduction to hyperbolic geometry
（Universitext）
Springer-Verlag, c1995
- : us
- : gw
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注記
Includes bibliographical references and index
内容説明・目次
- 巻冊次
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: us ISBN 9780387943398
内容説明
This book is an introduction to hyperbolic and differential geometry that provides material in the early chapters that can serve as a textbook for a standard upper division course on hyperbolic geometry. For that material, the students need to be familiar with calculus and linear algebra and willing to accept one advanced theorem from analysis without proof. The book goes well beyond the standard course in later chapters, and there is enough material for an honors course, or for supplementary reading. Indeed, parts of the book have been used for both kinds of courses. Even some of what is in the early chapters would surely not be nec essary for a standard course. For example, detailed proofs are given of the Jordan Curve Theorem for Polygons and of the decomposability of poly gons into triangles, These proofs are included for the sake of completeness, but the results themselves are so believable that most students should skip the proofs on a first reading. The axioms used are modern in character and more "user friendly" than the traditional ones. The familiar real number system is used as an in gredient rather than appearing as a result of the axioms. However, it should not be thought that the geometric treatment is in terms of models: this is an axiomatic approach that is just more convenient than the traditional ones.
目次
- Preface
- Introduction
- 1. Axioms for Plane Geometry
- 2. Some Neutral Theorems of Plane Geometry
- 3. Qualitative Description of the Hyperbolic Plane
- 4. H3 and Euclidean Approximations in H2
- 5. Differential Geometry of Surface
- 6. Quantitative Considerations
- 7. Consistency and Categoricalness of the Hyperbolic Axioms- the Classical Models
- 8. Matrix Representation of the Isometry Group
- 9. Differential and Hyperbolic Geometry in More Dimensions
- 10. Connections with the Lorentz Group of Special Relativity
- 11. Constructions by Straightedge and Compass in the Hyperbolic Plane
- Index
- 巻冊次
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: gw ISBN 9783540943396
内容説明
This text for advanced undergraduates emphasizes the logical connections of the subject. The derivations of formulas from the axioms do not make use of models of the hyperbolic plane until the axioms are shown to be categorical; the differential geometry of surfaces is developed far enough to establish its connections to the hyperbolic plane; and the axioms and proofs use the properties of the real number system to avoid the tedium of a completely synthetic approach. The development includes properties of the isometry group of the hyperbolic plane, tilings, and applications to special relativity. Elementary techniques from complex analysis, matrix theory, and group theory are used, and some mathematical sophistication on the part of students is thus required, but a formal course in these topics is not a prerequisite.
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