Introduction to Hyperbolic Geometry

1. Front Matter

   Pages i-xii

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2. Introduction

   • Arlan Ramsay, Robert D. Richtmyer

   Pages 1-8

3. Axioms for Plane Geometry

   • Arlan Ramsay, Robert D. Richtmyer

   Pages 9-29

4. Some Neutral Theorems of Plane Geometry

   • Arlan Ramsay, Robert D. Richtmyer

   Pages 30-68

5. Qualitative Description of the Hyperbolic Plane

   • Arlan Ramsay, Robert D. Richtmyer

   Pages 69-127

6. ℍ3 and Euclidean Approximations in ℍ2

   • Arlan Ramsay, Robert D. Richtmyer

   Pages 128-148

7. Differential Geometry of Surfaces

   • Arlan Ramsay, Robert D. Richtmyer

   Pages 149-189

8. Quantitative Considerations

   • Arlan Ramsay, Robert D. Richtmyer

   Pages 190-201

9. Consistency and Categoricalness of the Hyperbolic Axioms; The Classical Models

   • Arlan Ramsay, Robert D. Richtmyer

   Pages 202-217

10. Matrix Representation of the Isometry Group

    • Arlan Ramsay, Robert D. Richtmyer

    Pages 218-231

11. Differential and Hyperbolic Geometry in More Dimensions

    • Arlan Ramsay, Robert D. Richtmyer

    Pages 232-241

12. Connections with the Lorentz Group of Special Relativity

    • Arlan Ramsay, Robert D. Richtmyer

    Pages 242-253

13. Constructions by Straightedge and Compass in the Hyperbolic Plane

    • Arlan Ramsay, Robert D. Richtmyer

    Pages 254-282

14. Back Matter

    Pages 283-289

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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.

• derivation
• differential equation
• differential geometry
• differential geometry of surfaces
• hyperbolic geometry

"The book is well laid out with no shortage of diagrams and with each chapter prefaced with its own useful introduction...Also well written, it makes pleasurable reading." Proceedings of the Edinburgh Mathematical Society

• Department of Mathematics, University of Colorado, Boulder, USA

  Arlan Ramsay, Robert D. Richtmyer

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