# Foundations of Hyperbolic Manifolds

Part of the Graduate Texts in Mathematics book series (GTM, volume 149)

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Part of the Graduate Texts in Mathematics book series (GTM, volume 149)

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 book is divided into three parts. The first part is concerned with hyperbolic geometry and discrete groups. The main results are the characterization of hyperbolic reflection groups and Euclidean crystallographic groups. The second part is devoted to the theory of hyperbolic manifolds. The main results are Mostow’s rigidity theorem and the determination of the global geometry of hyperbolic manifolds of finite volume. The third part integrates the first two parts in a development of the theory of hyperbolic orbifolds. The main result is Poincare«s fundamental polyhedron theorem.

The exposition if at the level of a second year graduate student with particular emphasis placed on readability and completeness of argument. After reading this book, the reader will have the necessary background to study the current research on hyperbolic manifolds.

The second edition is a thorough revision of the first edition that embodies hundreds of changes, corrections, and additions, including over sixty new lemmas, theorems, and corollaries. The new main results are Schl\¬afli’s differential formula and the $n$-dimensional Gauss-Bonnet theorem.

John G. Ratcliffe is a Professor of Mathematics at Vanderbilt University.

Dimension Grad Isometrie Volume derivation hyperbolic manifolds manifold polytope topology

- DOI https://doi.org/10.1007/978-0-387-47322-2
- Copyright Information Springer Science+Business Media, LLC 2006
- Publisher Name Springer, New York, NY
- eBook Packages Mathematics and Statistics
- Print ISBN 978-0-387-33197-3
- Online ISBN 978-0-387-47322-2
- Series Print ISSN 0072-5285
- Buy this book on publisher's site