Foundations of Hyperbolic Manifolds

John G. Ratcliffe (auth.)9780387331973, 0387331972

This book is an exposition of the theoretical foundations of hyperbolic manifolds. It is intended to be used both as a textbook and as a reference. The reader is assumed to have a basic knowledge of algebra and topology at the first year graduate level of an American university. The book is divided into three parts. The first part is concerned with hyperbolic geometry and discrete groups. The second part is devoted to the theory of hyperbolic manifolds. The third part integrates the first two parts in a development of the theory of hyperbolic orbifolds.

The second edition contains hundreds of changes and corrections, and new additions include: A more thorough discussion of polytopes; Discussion of Simplex Reflection groups has been expanded to give a complete classification of the Gram matrices of spherical, Euclidean and hyperbolic n-simplices; A new section on the volume of a simplex, in which a derivation of Schlafli’s differential formula is presented; A new section with a proof of the n-dimensional Gauss-Bonnet theorem.

The exercises have been thoroughly reworked, pruned, and upgraded, and over 100 new exercises have been added. The author has also prepared a solutions manual which is available to professors who choose to adopt this text for their course.

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Table of contents :
Front Matter….Pages i-xi
Euclidean Geometry….Pages 1-35
Spherical Geometry….Pages 36-55
Hyperbolic Geometry….Pages 56-104
Inversive Geometry….Pages 105-147
Isometries of Hyperbolic Space….Pages 148-191
Geometry of Discrete Groups….Pages 192-262
Classical Discrete Groups….Pages 263-329
Geometric Manifolds….Pages 330-370
Geometric Surfaces….Pages 371-430
Hyperbolic 3-Manifolds….Pages 431-502
Hyperbolic n -Manifolds….Pages 503-572
Geometrically Finite n -Manifolds….Pages 573-651
Geometric Orbifolds….Pages 652-714
Back Matter….Pages 715-750