Foundations of Hyperbolic Manifolds

  • John G. Ratcliffe

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

Table of contents

  1. Front Matter
    Pages i-xii
  2. Pages 1-34
  3. Pages 35-53
  4. Pages 54-99
  5. Pages 100-143
  6. Pages 334-374
  7. Pages 375-434
  8. Pages 435-507
  9. Pages 508-599
  10. Pages 681-744
  11. Back Matter
    Pages 745-779

About this book


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

Authors and affiliations

  • John G. Ratcliffe
    • 1
  1. 1.Department of Mathematics, Stevenson Center 1326Vanderbilt UniversityNashville

Bibliographic information