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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 studythe 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.
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 studythe 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.
From the reviews of the second edition:
"Designed to be useful as both textbook and a reference, this book renders a real service to the mathematical community by putting together the tools and prerequisites needed to enter the territory of Thurston's formidable theory of hyperbolic 3-mainfolds ... . Every chapter is followed by historical notes, with attributions to the relevant literature, both of the originators of the idea present in the chapter and of modern presentation thereof. The bibliography contains 463 entries." (Victor V. Pambuccian, Zentralblatt MATH, Vol. 1106 (8), 2007)
"Designed to be useful as both textbook and a reference, this book renders a real service to the mathematical community by putting together the tools and prerequisites needed to enter the territory of Thurston's formidable theory of hyperbolic 3-mainfolds ... . Every chapter is followed by historical notes, with attributions to the relevant literature, both of the originators of the idea present in the chapter and of modern presentation thereof. The bibliography contains 463 entries." (Victor V. Pambuccian, Zentralblatt MATH, Vol. 1106 (8), 2007)
"The author provides a book that will serve both as a reference to experts in the area for many years to come, and potentially as a textbook to introduce this area to the more sophisticated student...This book has a tremendous amount of depth. In addition to the careful and complete exposition, each chapter ends with a fascinating section containing historical notes, putting many of the ideas into context. This volume will play an important role in the continuing development of this fascinating field." -- Colin Adams, Mathematical Reviews
"A detailed and extensive study of geometric manifolds, esp. of hyperbolic ones, is preceded by an expose of foundations of non-Euclidean spaces, of their models and of related groups of transformations." -- A. Szybiak, Zentralblatt
"A detailed and extensive study of geometric manifolds, esp. of hyperbolic ones, is preceded by an expose of foundations of non-Euclidean spaces, of their models and of related groups of transformations." -- A. Szybiak, Zentralblatt