Three-Dimensional Geometry and Topology

Book Description

This book develops some of the extraordinary richness, beauty, and power of geometry in two and three dimensions, and the strong connection of geometry with topology. Hyperbolic geometry is the star. A strong effort has been made to convey not just denatured formal reasoning (definitions, theorems, and proofs), but a living feeling for the subject. There are many figures, examples, and exercises of varying difficulty.

This book was the origin of a grand scheme developed by Thurston that is now coming to fruition. In the 1920s and 1930s the mathematics of two-dimensional spaces was formalized. It was Thurston's goal to do the same for three-dimensional spaces. To do this, he had to establish the strong connection of geometry to topology--the study of qualitative questions about geometrical structures. The author created a new set of concepts, and the expression "Thurston-type geometry" has become a commonplace.

Three-Dimensional Geometry and Topology had its origins in the form of notes for a graduate course the author taught at Princeton University between 1978 and 1980. Thurston shared his notes, duplicating and sending them to whoever requested them. Eventually, the mailing list grew to more than one thousand names. The book is the culmination of two decades of research and has become the most important and influential text in the field. Its content also provided the methods needed to solve one of mathematics' oldest unsolved problems--the Poincaré Conjecture.

Thurston received the Fields Medal, the mathematical equivalent of the Nobel Prize, in 1982 for the depth and originality of his contributions to mathematics. In 1979 he was awarded the Alan T. Waterman Award, which recognizes an outstanding young researcher in any field of science or engineering supported by the National Science Foundation.

Customer Reviews:

A refreshing style of writing.......2001-06-21

Stanislaw Ulam once compared learning mathematics to learning a language, in that some people learn mathematics by "grammar" while other learn it by ear. Thurston's book is a bit like learning by ear.

fun and geometric-intuition-minded.......1998-12-23

A must for anyone entering the field of three-dimensional topology and geometry. Most of it is about hyperbolic geometry, which is the biggest area of research in 3-d geometry and topology nowdays.

Most of it is readable to undergraduates. Its target audience, though, is beginning graduate students in mathematics. If not already familiar with hyperbolic geometry, you might want to get an introduction to the subject first. Once with this background, though, you will discover there is another level of understanding of hyperbolic space you never realized was possible. One imagines Thurston able to skateboard around hyperbolic space with the kind of geometric understanding he conveys here.

Book Description

Complex hyperbolic geometry is a particularly rich area of study, enhanced by the confluence of several areas of research including Riemannian geometry, complex analysis, symplectic and contact geometry, Lie group theory, and harmonic analysis. The boundary of complex hyperbolic geometry, known as spherical CR or Heisenberg geometry, is equally rich, and although there exist accounts of analysis in such spaces there is currently no account of their geometry. This book redresses the balance and provides an overview of the geometry of both the complex hyperbolic space and its boundary. Motivated by applications of the theory to geometric structures, moduli spaces and discrete groups, it is designed to provide an introduction to this fascinating and important area and invite further research and development.

Experiencing Geometry: In Euclidean, Spherical and Hyperbolic Spaces (2nd Edition)

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First-principle arguments: a great intro to theoretical geo.
Geometry Teacher Likes It!!

Experiencing Geometry: In Euclidean, Spherical and Hyperbolic Spaces (2nd Edition)
David W. Henderson
Manufacturer: Pearson Education
ProductGroup: Book
Binding: Paperback

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ASIN: 0130309532

Book Description

The book conveys a distinctive approach, stimulating readers to develop a broader, deeper understanding of mathematics through active participation—including discovery, discussion, and writing about fundamental ideas. It provides a series of interesting, challenging problems, then encourages readers to gather their reasonings and understandings of each problem.

Customer Reviews:

First-principle arguments: a great intro to theoretical geo........2004-06-19

Geometry has always come as a difficult subject for me. For some reason, it's hard for me to "see" the answers and proofs like some people can. But, that's where a book like this came in handy.

Basically, this book is good for two things: first-principle arguments and model building. The problems in here almost require you to make models and draw pictures to see for sure what's going on. So, for a person raised on the calculator like myself, having to make models is a good thing (although very challenging!). Also, the first principle arguments are good for thought and frustration: Almost every problem in this book takes a very long time to solve, with combined thinking time, study time with your notes, and actually writing out your solution. Only on a couple of the problems that my class worked on was I able to get away with using less than one sheet of paper to write out my answer. Basically, you start from the very beginning and look at concepts you know already, like straight lines, and try to explain them as clearly and concisely as you can, which isn't always easy. But, Henderson gives hints for almost every problem, gives background on each problem, and sometimes provides solutions to the problems (but rarely!); he also provides motivation for studying the concepts rather than to pass a math course, which is very good. He also does an excellent job of introducing the concept of hyperbolic geometry by first discussing what properties stay the same or differ between the sphere and plane, which is excellent because hyperbolic geometry is not as easy to grasp as the other two; this instructional method is actually carried out through the book, an excellent way to introduce two types of geometry that many people (including myself) may have never seen before.

Honestly, my only gripe is that the book is SO based on first principle arguments that, while it provides excellent framework for prospective elementary and high-school teachers, it won't do much for the applied mathematician who wants to see some computational examples because there are barely any! It forces a student to be held accountable for abstract thought and proof, but there are many aspects of geometry that can be handled with computations like arclength, areas of polygons, etc., which are not discussed in the book.

Basically, here's the short version: It's a great book for the student who wants to work through arguments based on first principles, but be prepared to work hard! Because almost none of the exercises come with answers, I don't find this book suitable for independent study, so I hope you have a good teacher! Also, get a ball, some rubberbands, and plenty of paper with some good erasers; you'll need them!

Geometry Teacher Likes It!!.......2001-01-02

I found this book caused me to rethink much of how I approach Geometry personally and in my classroom. From the first problem, "What is Straight?" which had me thinking about my own assumptions straight lines, I have been thinking of ways to approach Geometry differently with my students.

Book Description

This book is based on real inner product spaces X of arbitrary (finite or infinite) dimension greater than or equal to 2. With natural properties of (general) translations and general distances of X, euclidean and hyperbolic geometries are characterized. For these spaces X also the sphere geometries of Möbius and Lie are studied (besides euclidean and hyperbolic geometry), as well as geometries where Lorentz transformations play the key role. The geometrical notions of this book are based on general spaces X as described. This implies that also mathematicians who have not so far been especially interested in geometry may study and understand great ideas of classical geometries in modern and general contexts.

Proofs of newer theorems, characterizing isometries and Lorentz transformations under mild hypotheses are included, like for instance infinite dimensional versions of famous theorems of A.D. Alexandrov on Lorentz transformations. A real benefit is the dimension-free approach to important geometrical theories. Only prerequisites are basic linear algebra and basic 2- and 3-dimensional real geometry.

Hyperbolic Geometry (Springer Undergraduate Mathematics Series)

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Excellent book
Very good introduction
great book

Hyperbolic Geometry (Springer Undergraduate Mathematics Series)
James W Anderson
Manufacturer: Springer
ProductGroup: Book
Binding: Paperback

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ASIN: 1852339349

Book Description

The geometry of the hyperbolic plane has been an active and fascinating field of mathematical inquiry for most of the past two centuries. This book provides a self-contained introduction to the subject, providing the reader with a firm grasp of the concepts and techniques of this beautiful area of mathematics. Topics covered include the upper half-space model of the hyperbolic plane, Möbius transformations, the general Möbius group and the subgroup preserving path length in the upper half-space model, arc-length and distance, the Poincaré disc model, convex subsets of the hyperbolic plane, and the Gauss-Bonnet formula for the area of a hyperbolic polygon and its applications.

This updated second edition also features:

an expanded discussion of planar models of the hyperbolic plane arising from complex analysis;
the hyperboloid model of the hyperbolic plane;
a brief discussion of generalizations to higher dimensions;
many new exercises.

Customer Reviews:

Excellent book.......2007-08-22

This is an excellent introduction to hyperbolic geometry. It assumes knowledge of euclidean geometry, trigonometry, basic complex analysis, basic abstract algebra, and basic point set topology. That material is very well presented, and the exercises shed more light on what is being discussed. Plus, solutions to all the exercises are at the end of the book.

Very good introduction.......2007-08-03

I used this text along with Tristan Needham's "Visual Complex Analysis" to get a full dose of the geometric beauty inherent in studying complex variables. I found it to be a nice complement to the second year course in geometry at Cambridge University. Anderson does a wonderful job of working out in detail lots of examples so that you can get the algorithmic practice of solving problems. However this is not merely a cookbook. Rather, core elements of the theory are presented from the ground up, with plenty of time spent on understanding the group structure of Mobius transformations in various settings. Disc and upper-half plane models are treated as well as more general models. I recommend you buy both this book and Needham's if you want to appreciate the world of complex numbers.

great book.......2004-01-29

this is a really great introduction to hyperbolic geometry. especially if you want to study gammas acting on the upper half plane. it starts at a much lower level then any other text.

On Some Aspects of the Theory of Anosov Systems: With a Survey by Richard Sharp: "Periodic Orbits of Hyperbolic Flows" (Springer Monographs in Mathematics)

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On Some Aspects of the Theory of Anosov Systems: With a Survey by Richard Sharp: "Periodic Orbits of Hyperbolic Flows" (Springer Monographs in Mathematics)
Grigorii A. Margulis
Manufacturer: Springer
ProductGroup: Book
Binding: Hardcover

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ASIN: 3540401210

Book Description

In this book the seminal 1970 Moscow thesis of Grigoriy A. Margulis is published for the first time. Entitled "On Some Aspects of the Theory of Anosov Systems", it uses ergodic theoretic techniques to study the distribution of periodic orbits of Anosov flows. The thesis introduces the "Margulis measure" and uses it to obtain a precise asymptotic formula for counting periodic orbits. This has an immediate application to counting closed geodesics on negatively curved manifolds. The thesis also contains asymptotic formulas for the number of lattice points on universal coverings of compact manifolds of negative curvature.

The thesis is complemented by a survey by Richard Sharp, discussing more recent developments in the theory of periodic orbits for hyperbolic flows, including the results obtained in the light of Dolgopyat's breakthroughs on bounding transfer operators and rates of mixing.

Introduction to Complex Hyperbolic Spaces

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Introduction to Complex Hyperbolic Spaces
Serge Lang
Manufacturer: Springer
ProductGroup: Book
Binding: Hardcover

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ASIN: 0387964479

The Geometry of the Word Problem for Finitely Generated Groups (Advanced Courses in Mathematics - CRM Barcelona)

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The Geometry of the Word Problem for Finitely Generated Groups (Advanced Courses in Mathematics - CRM Barcelona)
Noel Brady , Hamish Short , and Tim Riley
Manufacturer: Birkhäuser Basel
ProductGroup: Book
Binding: Paperback

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Book Description

The origins of the word problem are in group theory, decidability and complexity, but, through the vision of M. Gromov and the language of filling functions, the topic now impacts the world of large-scale geometry, including topics such as soap films, isoperimetry, coarse invariants and curvature.

The first part introduces van Kampen diagrams in Cayley graphs of finitely generated, infinite groups; it discusses the van Kampen lemma, the isoperimetric functions or Dehn functions, the theory of small cancellation groups and an introduction to hyperbolic groups.

One of the main tools in geometric group theory is the study of spaces, in particular geodesic spaces and manifolds, such that the groups act upon. The second part is thus dedicated to Dehn functions, negatively curved groups, in particular, CAT(0) groups, cubings and cubical complexes.

In the last part, filling functions are presented from geometric, algebraic and algorithmic points of view; it is discussed how filling functions interact, and applications to nilpotent groups, hyperbolic groups and asymptotic cones are given. Many examples and open problems are included.

Hyperbolic Functions: with Configuration Theorems and Equivalent and Equidecomposable Figures (Dover Science Books)

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Hyperbolic Functions: with Configuration Theorems and Equivalent and Equidecomposable Figures (Dover Science Books)
V. G. Shervatov , B. I. Argunov , L. A. Skornyakov , and V. G. Boltyanskii
Manufacturer: Dover Publications
ProductGroup: Book
Binding: Paperback

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ASIN: 0486458865

Book Description

This single-volume compilation of three books centers on Hyperbolic Functions, an introduction to the relationship between the hyperbolic sine, cosine, and tangent, and the geometric properties of the hyperbola. Configuration Theorems concerns theories on collinear points and concurrent lines, and Equivalent and Equidecomposable Figures examines the dissection and reassembly of polyhedrons. 1963 edition.

Normally Hyperbolic Invariant Manifolds in Dynamical Systems (Applied Mathematical Sciences)

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Normally Hyperbolic Invariant Manifolds in Dynamical Systems (Applied Mathematical Sciences)
Stephen Wiggins , G. Haller , and I. Mezic
Manufacturer: Springer
ProductGroup: Book
Binding: Hardcover

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ASIN: 038794205X

Book Description

In the past ten years, there has been much progress in understanding the global dynamics of systems with several degrees-of-freedom. An important tool in these studies has been the theory of normally hyperbolic invariant manifolds and foliations of normally hyperbolic invariant manifolds. In recent years these techniques have been used for the development of global perturbation methods, the study of resonance phenomena in coupled oscillators, geometric singular perturbation theory, and the study of bursting phenomena in biological oscillators. "Invariant manifold theorems" have become standard tools for applied mathematicians, physicists, engineers, and virtually anyone working on nonlinear problems from a geometric viewpoint. In this book, the author gives a self-contained development of these ideas as well as proofs of the main theorems along the lines of the seminal works of Fenichel. In general, the Fenichel theory is very valuable for many applications, but it is not easy for people to get into from existing literature. This book provides an excellent avenue to that. Wiggins also describes a variety of settings where these techniques can be used in applications.

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