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Fb2 Foundations of Hyperbolic Manifolds (Graduate Texts in Mathematics) ePub

by John G. Ratcliffe

Category: Mathematics
Subcategory: Science books
Author: John G. Ratcliffe
ISBN: 354094348X
ISBN13: 978-3540943488
Language: English
Publisher: Springer-Verlag Berlin and Heidelberg GmbH & Co. K (October 1, 1994)
Pages: 758
Fb2 eBook: 1647 kb
ePub eBook: 1449 kb
Digital formats: mobi lrf mbr doc

This book is an exposition of the theoretical foundations of hyperbolic manifolds. The reader is assumed to have a basic knowledge of algebra and topology at the first year graduate level of an American university.

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 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.

Graduate Texts in Mathematics. Foundations of Hyperbolic Manifolds. This book is an exposition of the theoretical foundations of hyperbolic manifolds. The main results are the characterization of hyperbolic reflection groups and Euclidean crystallographic groups.

Foundations of Hyperbolic Manifolds book Foundations of Hyperbolic Manifolds. Graduate Texts in Mathematics by. John G. Ratcliffe.

Foundations of Hyperbolic Manifolds book. This heavily class-tested book is an exposition of the theoretical foundations of hyperbolic manifolds. It is a both a textbook and a reference. Ratcliffe is a Professor of Mathematics at Vanderbilt University. Series: Graduate Texts in Mathematics (Book 149). Hardcover: 782 pages.

Graduate Texts in Mathematics (continued after index). Ratcliffe Department of Mathematics Stevenson Center 1326 Vanderbilt University Nashville, Tennessee 37240 john. Ribet Department of Mathematics Department of Mathematics San Francisco State University University of California, Berkeley San Francisco, CA 94132 Berkeley, CA 94720-3840 USA USA axler. edu Mathematics Subject Classification (2000)

Foundations of Hyperbolic Manifolds Graduate Texts in Mathematics.

Foundations of Hyperbolic Manifolds Graduate Texts in Mathematics.

Graduate Texts in Mathematics (GTM) (ISSN 0072-5285) is a series of graduate-level textbooks in mathematics published by Springer-Verlag. The GTM series is easily identified by a white band at the top of the book.

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This book is an exposition of the theoretical foundations of hyperbolic manifolds Foundations of Hyperbolic Manifolds Graduate Texts in Mathematics (Том 149). Издание: иллюстрированное. Particular emphasis has been placed on readability and completeness of ar gument. The treatment of the material is for the most part elementary and self-contained. The reader is assumed to have a basic knowledge of algebra and topology at the first-year graduate level of an American university. Foundations of Hyperbolic Manifolds Graduate Texts in Mathematics (Том 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 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, Chapters 1-7, is concerned with hyperbolic geometry and discrete groups. The second part, Chapters 8-12, is devoted to the theory of hyperbolic manifolds. The third part, Chapter 13, integrates the first two parts in a development of the theory of hyperbolic orbifolds. There are over 500 exercises in this book and more than 180 illustrations.
Comments to eBook Foundations of Hyperbolic Manifolds (Graduate Texts in Mathematics)
Hawk Flying
I'm impressed with how comprehensive this book is. It's a fantastic reference, and in particular I found its treatment of hyperbolic structures on knot complements very illuminating. Another reviewer mentioned that the prerequisites can vary greatly from chapter to chapter, which is definitely true, but the clarity of the presentation is pretty solid throughout which helps keep the more technical bits manageable.
Gindian
I bought this after reading the table of contents at Amazon online.
I was looking for some manifold gluing as working examples.
I'm sure three phd's are going to show up telling me I don't know Jack,
so here is a link to my own work:
[...]
which of course Amazon removed!
This book is too general and doesn't deliver specifics.
I've made a list of specific things that are lacking in this text:
1) Alexander Polynomials
2) Cartan groups
3) Knot Complement
4) D-branes
5) Discriminant
6) Fano
7) Fibers
8) Haken
9) Knot
10) Mandelbrot set
11) Minimal polynomial
12) Seifert surface
13) Seifert Matrix
14) String
15) Thurston
16) Turing
17 Weeks
18) Wang tile
A link to a better book:
Topology of 3-Manifolds and Related Topics (Dover Books on Mathematics)
I might be able to get something out of this book after several years
of rereading ( Fort's book has taken at least a year),
but since I paid good money for the book
I expected something besides n dimensional generalizations
and some new manifold type designated B^n ( I think that may
be related to Braid groups as opposed to B_n Cartan groups or Brouwer groups).
I'm trying to learn about Manifold gluing in hyberbolic knot-Link manifolds
which is in modern terms Kirby calculus ( not mentioned in
this index). So for me as a self-learning tool,
this book fails on first reading.
At second reading the mention of the Gieseking manifold is useful
but not descriptive enough.
Jieylau
Love the book
Mezilabar
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.
Survivors
The book is nothing if not comprehensive, and if you work in the field, it is a useful reference to have close at hand. However, I would not recommend it to a student, since there is a good chance a student would be bored to death by the time he slogged his way through this. Thurston's notes (NOT his book), available for free from MSRI are available for free from MSRI, and are vastly superior as an introduction. Various survey papers by Vinberg (including the one in the Russian Encyclopedia of Mathematics: Geometry: Volume 2: Spaces of Constant Curvature (v. 2) are very lucid, and bring out the beauty of the subject much better than the book under review.
Sermak Light
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.
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