## Introduction to Hyperbolic GeometryThis 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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### Contents

Introduction | 1 |

Axioms for Plane Geometry | 9 |

Some Neutral Theorems of Plane Geometry | 30 |

Qualitative Description of the Hyperbolic Plane | 69 |

H3 and Euclidean Approximations in H2 | 130 |

Differential Geometry of Surfaces | 153 |

Quantitative Considerations | 194 |

Consistency and Categoricalness of the Hyperbolic Axioms The Classical Models | 206 |

Matrix Representation of the Isometry Group | 222 |

Differential and Hyperbolic Geometry in More Dimensions | 236 |

Connections with the Lorentz Group of Special Relativity | 246 |

Constructions by Straightedge and Compass in the Hyperbolic Plane | 258 |

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### Common terms and phrases

according angle angle of parallelism angular defect assume asymptotic axioms bounded called Chapter circle common complete congruent consider consistent constant constructed contains continuous coordinates corresponding curve defined Definition denote derivative described determined differential direction discussed distance equal equation Euclidean example Exercise fact field fixed follows formulas function geodesic geometry given gives half-plane hence horocycle hyperbolic plane ideal point increasing infinity interior intersection invariant isometry Lemma length less limit Lorentz mapping matrix metric Note obtained opposite origin parallel perpendicular polar coordinates polygon positive proof proved quantities radius rational reflection region represent result rotation satisfies segment shown side sinh space special relativity surface tangent tangent vector tanh tensor Theorem transformation translation triangle unique unit vector vertices zero

### References to this book

Beyond the Einstein Addition Law and Its Gyroscopic Thomas Precession: The ... A.A. Ungar No preview available - 2001 |