Foundations of Hyperbolic Manifolds (Graduate Texts in Mathematics) | 
 enlarge | Author: John Ratcliffe
Publisher: Springer
Category: Book
List Price: $69.95
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Rating:  3 reviews
Sales Rank: 271547
Media: Hardcover
Edition: 2nd
Pages: 783
Number Of Items: 1
Shipping Weight (lbs): 2.8
Dimensions (in): 9.2 x 6 x 1.6
ISBN: 0387331972
Dewey Decimal Number: 516
EAN: 9780387331973
Publication Date: August 23, 2006
Availability: Usually ships in 1-2 business days
Condition: THIS ITEM IS UNUSED AND IN GOOD CONDITION. IT MAY HAVE SLIGHT SHELFWEAR BUT OTHERWISE IT IS FINE.
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Product Description
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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An excellent overview for mathematicians and physicists August 26, 2001
Dr. Lee D. Carlson (Saint Louis, Missouri USA)
13 out of 13 found this review helpful
The advent of non-Euclidean geometry resulted in many different areas of mathematics, some being specifically related to geometry, others being more general, such as proof theory and model theory. This book is an excellent overview of a particular branch of non-Euclidean geometry called hyperbolic geometry. There are good exercises in the book, and the author gives a detailed history of the subjects after the end of each chapter. After a brief review of Euclidean geometry in chapter 1, emphasizing the metric properties of Euclidean space, orthogonal transformations, and isometries, the author discusses spherical geometry in chapter 2. Spherical and hyperbolic geometries are dual to each other, in the sense that in spherical geometry, a line through a point outside a given line is never parallel to the given line; but in hyperbolic geometry there are infinitely many such lines. Also, the sum of the angles of a spherical triangle is always greater than 180 degrees ; but in hyperbolic geometry less than 180 degrees. Hyperbolic geometry is of crucial importance in physics, particularly in the theory of relativity, and the author begins a discussion of this kind of geometry in chapter 3. Hyperbolic n-space is viewed more as dual to elliptic geometry in the sense that it is modeled as a unit sphere of imaginary radius with only the positive sheet of this (disconnected) set retained. The author outlines in detail the important properties of hyperbolic geometry along with its trigonometry. This is followed in the next chapters by a model of hyperbolic n-space as a conformal ball and an upper half-space, and a consideration of the isometries of hyperbolic space. The Mobius transformations are given detailed treatment. The famous classical discrete groups are introduced, along with the crystallographic groups. The discussion gets more abstract in some parts here, for the author introduces some algebraic notions such as valuation rings, in order to prove Selberg's lemma. The author finally lays the groundwork for a theory of hyperbolic manifolds in chapter 8, by first introducing geometric spaces. These are defined by four axioms, which are generalizations of Euclid's first four axioms, and two of these axioms imply that any geometric manifold is an n-manifold. The discussion is specialized in the next chapter to geometric surfaces, where the famous Gauss-Bonnet theorem, relating the area of a surface to its Euler characteristic, is proved for spherical, Euclidean, or hyperbolic surfaces. The author studies the collection of similarity equivalence classes of complete structures for a geometric surface, namely the moduli space of such structures. Physicists, particularly string theorists, will appreciate the resulting discussion on Teichmuller space and the Dehn-Nielsen theorem. Considerations of a nature more familiar to geometric topologists follows in the next chapter, where it is shown how to explicitly construct hyperbolic 3-manifolds. Dehn surgery is employed to study the complement of the figure 8 knot. The discussion is very interesting, for it employs explicit detailed constructions that would take many hours to dig out of the literature. The general case of n-dimensional hyperbolic manifolds is the subject of chapter 11, with the constructions in chapter 10 generalized to deal with high dimensions. The author considers also the two closed, orientable, hyperbolic manifolds of the same homotopy type have the same volume by using the Gromov invariant, a quantity defined in terms of the singular homology on the manifold. The reader will get a taste of the Haar measure in the proof of the result, and later an overview of measure homology. The later is very interesting, as it brings in techniques from differential topology and the de Rham complex, and it also defines a notion of a "straightening" and smearing of a singular complex. Mostow rigidity, which says that for any two closed, connected, orientable, hyperbolic n-manifolds, with n greater than 2, a homotopy between these will also be an isometry, is also proven here. The next chapter is more involved than the rest, and deals with the case of geometrically finite n-manifolds. Dealing with cusps and "sharp corners" from the actions of discrete groups is given detailed and rigorous discussion here. The discussion leads naturally to a treatment of orbifolds in the next chapter. These objects have been extremely important in string theories in high energy physics, and the author does an excellent job of detailing their properties.
Best on the market October 26, 2008
Fadi E.
This is a wonderful book on both hyperbolic geometry **and** spherical geometry--non-Euclidean geometry in general. It's more comprehensive than all of the others. The prerequisites for this book vary greatly from chapter to chapter. If you want to read, and understand, all of the material right away, the prerequisites are somewhat steep. I would study smooth and riemannian manifolds first (I heavily recommend John Lee's two books). I would also get some basic algebraic topology (Hatcher's is a classic). If you have these, it's smooth sailing ahead.
A Comprehensive Approach May 6, 2007
Karan Mohan Puri
0 out of 1 found this review helpful
The book is excellent as a reference book and approaches hyperbolic geometry from the Lorentzian viewpoint (which seems to be different from other authors). A great book to have for graduate students studying hyperbolic geometry.
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