Ebook: Introduction to Hyperbolic Geometry

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.

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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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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.
Content:
Front Matter....Pages i-xii
Introduction....Pages 1-8
Axioms for Plane Geometry....Pages 9-29
Some Neutral Theorems of Plane Geometry....Pages 30-68
Qualitative Description of the Hyperbolic Plane....Pages 69-127
?3 and Euclidean Approximations in ?2 ....Pages 128-148
Differential Geometry of Surfaces....Pages 149-189
Quantitative Considerations....Pages 190-201
Consistency and Categoricalness of the Hyperbolic Axioms; The Classical Models....Pages 202-217
Matrix Representation of the Isometry Group....Pages 218-231
Differential and Hyperbolic Geometry in More Dimensions....Pages 232-241
Connections with the Lorentz Group of Special Relativity....Pages 242-253
Constructions by Straightedge and Compass in the Hyperbolic Plane....Pages 254-282
Back Matter....Pages 283-289

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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.
Content:
Front Matter....Pages i-xii
Introduction....Pages 1-8
Axioms for Plane Geometry....Pages 9-29
Some Neutral Theorems of Plane Geometry....Pages 30-68
Qualitative Description of the Hyperbolic Plane....Pages 69-127
?3 and Euclidean Approximations in ?2 ....Pages 128-148
Differential Geometry of Surfaces....Pages 149-189
Quantitative Considerations....Pages 190-201
Consistency and Categoricalness of the Hyperbolic Axioms; The Classical Models....Pages 202-217
Matrix Representation of the Isometry Group....Pages 218-231
Differential and Hyperbolic Geometry in More Dimensions....Pages 232-241
Connections with the Lorentz Group of Special Relativity....Pages 242-253
Constructions by Straightedge and Compass in the Hyperbolic Plane....Pages 254-282
Back Matter....Pages 283-289
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