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A user-friendly introduction to metric and topological groups Topological Groups: An Introduction provides a self-contained presentation with an emphasis on important families of topological groups. The book uniquely provides a modern and balanced presentation by using metric groups to present a substantive introduction to topics such as duality, while also shedding light on more general results for topological groups. Filling the need for a broad and accessible introduction to the subject, the book begins with coverage of groups, metric spaces, and topological spaces before introducing topological groups. Since linear spaces, algebras, norms, and determinants are necessary tools for studying topological groups, their basic properties are developed in subsequent chapters. For concreteness, product topologies, quotient topologies, and compact-open topologies are first introduced as metric spaces before their open sets are characterized by topological properties. These metrics, along with invariant metrics, act as excellent stepping stones to the subsequent discussions of the following topics: Matrix groups Connectednesss of topological groups Compact groups Character groups Exercises found throughout the book are designed so both novice and advanced readers will be able to work out solutions and move forward at their desired pace. All chapters include a variety of calculations, remarks, and elementary results, which are incorporated into the various examples and exercises. Topological Groups: An Introduction is an excellent book for advanced undergraduate and graduate-level courses on the topic. The book also serves as a valuable resource for professionals working in the fields of mathematics, science, engineering, and physics.

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Customer Reviews (3)

I just couldn't read this.
Maybe I'm retarded, but this is a tough book for me to read.I have a basic knowledge of set theroy, but I couldn't get through chapter two of this book.

This book just blazes through stuff.Its proofs use as little writing as possible, which makes understanding the stuff more difficult.It takes me about 10 minutes to understand one sentence.At that rate it would probably take me 20 years to finish the book.You need to be part of a very, very special audience to enjoy this book.Get this only if your IQ is close to 200.

Hey, at least the price ain't bad compared to other college textbooks.

Excellent for Self-Teaching
I am teaching myself mathematics and I love this book. Not being an expert at math, I can't speak for everyone when I say that this book is at times very challenging. But, in my case, that is part of why I find it so enjoyable; unlike all of the other (math) books I presently own, its difficulty really inspires me to read further.

An oldy but a goody
A good succinct presentation of the basics of topology with topological groups folded in.The writing is easy to follow, but the book is definitely not light on rigor.

May not work well for a one-semester course, as the topology and group theory are interwoven fairly substantially.One does not need to know group theory before reading the book, but one cannot easily escape it and still get at all the topology. ... Read more

Product Description
Offering the insights of L.S. Pontryagin, one of the foremost thinkers in modern mathematics, the second volume in this four-volume set examines the nature and processes that make up topological groups. Already hailed as the leading work in this subject for its abundance of examples and its thorough explanations, the text is arranged so that readers can follow the material either sequentially or schematically. Stand-alone chapters cover such topics as topological division rings, linear representations of compact topological groups, and the concept of a lie group. ... Read more

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Topological Methods in Group Theory is about the interplay between algebraic topology and the theory of infinite discrete groups. The author has kept three kinds of readers in mind: graduate students who have had an introductory course in algebraic topology and who need a bridge from common knowledge to the current research literature in geometric, combinatorial and homological group theory; group theorists who would like to know more about the topological side of their subject but who have been too long away from topology; and manifold topologists, both high- and low-dimensional, since the book contains much basic material on proper homotopy and locally finite homology not easily found elsewhere.

The book focuses on two main themes:

1. Topological Finiteness Properties of groups (generalizing the classical notions of "finitely generated" and "finitely presented");

2. Asymptotic Aspects of Infinite Groups (generalizing the classical notion of "the number of ends of a group").

Illustrative examples treated in some detail include: Bass-Serre theory, Coxeter groups, Thompson groups, Whitehead's contractible 3-manifold, Davis's exotic contractible manifolds in dimensions greater than three, the Bestvina-Brady Theorem, and the Bieri-Neumann-Strebel invariant. The book also includes a highly geometrical treatment of Poincaré duality (via cells and dual cells) to bring out the topological meaning of Poincaré duality groups.

To keep the length reasonable and the focus clear, it is assumed that the reader knows or can easily learn the necessary algebra (which is clearly summarized) but wants to see the topology done in detail. Apart from the introductory material, most of the mathematics presented here has not appeared in book form before.

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Lie Groups Beyond an Introduction takes the reader from the end of introductory Lie group theory to the threshold of infinite-dimensional group representations. Merging algebra and analysis throughout, the author uses Lie-theoretic methods to develop a beautiful theory having wide applications in mathematics and physics. A feature of the presentation is that it encourages the reader's comprehension of Lie group theory to evolve from beginner to expert: initial insights make use of actual matrices, while later insights come from such structural features as properties of root systems, or relationships among subgroups, or patterns among different subgroups.

Topics include a description of all simply connected Lie groups in terms of semisimple Lie groups and semidirect products, the Cartan theory of complex semisimple Lie algebras, the Cartan-Weyl theory of the structure and representations of compact Lie groups and representations of complex semisimple Lie algebras, the classification of real semisimple Lie algebras, the structure theory of noncompact reductive Lie groups as it is now used in research, and integration on reductive groups. Many problems, tables, and bibliographical notes complete this comprehensive work, making the text suitable either for self-study or for courses in the second year of graduate study and beyond. ... Read more

Customer Reviews (1)

Review of Knapp's "Lie groups: beyond an introduction."
The short version:this is a superbly written and conceived book;if I had to learn this material (the basic theory of
structure and representation of Lie algebras and groups,
especially semimsimple ones) from a single book, this is
the one I'd choose, among those I've seen.If you know the
basics of abstract algebra and some very basic concepts from
topology and manifolds, and you want to learn this material,
use this book.It would be a good reference, too, as it is
easy to find things in it, and takes a fairly modern, sophisticated approach (without sacrificing motivation and
intuition).

The long version, if you want more convincing or details:

I have used several books recently in learning the structure and
representation theory of Lie algebras and groups (especially Humphreys' Introduction to Lie algebras and representation theory, Fulton
and Harris' "Representation Theory," Varadarajan's "Lie groups,
Lie algebras, and their representations.")Although I came to Knapp's book with a decent background from the others, I think it's the best pedagogically, for someone with a modicum of mathematical sophistication and some basics like abstract
algebra and an idea of what a smooth manifold is), and a smattering of Lie theory.Some examples of the book's strength:
Elementary but potentially confusing concepts (like complexification, real forms, field extensions)
are explained thoroughly but in a sophisticated way, rather
than viewed as obvious.Carefully chosen examples motivate and
clarify the general theory;consequently even though the book
is completely rigorous, and carefully delineates lemmas, proofs,
remarks, definitions, and the like, it seems less dry then some
others (e.g. Varadarajan, from my point of view).But the point
of the examples, and their relation to the general theory, is
made clear, so they do not provide an overload of detail or b
obscure the main structure.Thought is always given to the
reader's understanding, not just to logical correctness, though
the author also takes the point of view, with which I concur,
that logical clarity and sufficient detail are essential
to understanding.Relations between ideas, alternative
proofs, and the structure of the theory to come are discussed
thoroughly, but such discussion is clearly demarcated from
the main structure of the argument, so that the latter is never
obscured.This is a fantastic book, and exactly what I was
looking for.Whether you are learning the material for the
first time, or want to review it or refer to, it is a superb
source. ... Read more

Product Description
In this book, developed from courses taught at the University of London, the author aims to show the value of using topological methods in combinatorial group theory. The topological material is given in terms of the fundamental groupoid, giving results and proofs that are both stronger and simpler than the traditional ones. Several chapters deal with covering spaces and complexes, an important method, which is then applied to yield the major Schreier and Kurosh subgroup theorems. The author presents a full account of Bass-Serre theory and discusses the word problem, in particular, its unsolvability and the Higman Embedding Theorem. Included for completeness are the relevant results of computability theory. ... Read more

Product Description
Being the first monograph devoted to this subject, the book addresses the classification problem for semisimple Hopf algebras, a field that has attracted considerable attention in the last years. The special approach to this problem taken here is via semidirect product decompositions into Yetter-Drinfel'd Hopf algebras and group rings of cyclic groups of prime order. One of the main features of the book is a complete treatment of the structure theory for such Yetter-Drinfel'd Hopf algebras. ... Read more

Product Description

This book proceeds beyond the representation theory of compact Lie groups (which is the basis of many texts) and offers a carefully chosen range of material designed to give readers the bigger picture. It explores compact Lie groups through a number of proofs and culminates in a "topics" section that takes the Frobenius-Schur duality between the representation theory of the symmetric group and the unitary groups as unifying them.

Product Description
The study of homogeneous spaces provides excellent insights into both differential geometry and Lie groups. In geometry, for instance, general theorems and properties will also hold for homogeneous spaces, and will usually be easier to understand and to prove in this setting. For Lie groups, a significant amount of analysis either begins with or reduces to analysis on homogeneous spaces, frequently on symmetric spaces. For many years and for many mathematicians, Sigurdur Helgason's classic Differential Geometry, Lie Groups, and Symmetric Spaces has been--and continues to be--the standard source for this material.

Helgason begins with a concise, self-contained introduction to differential geometry. He then introduces Lie groups and Lie algebras, including important results on their structure. This sets the stage for the introduction and study of symmetric spaces, which form the central part of the book. The text concludes with the classification of symmetric spaces by means of the Killing-Cartan classification of simple Lie algebras over $\mathbf{C}$ and Cartan's classification of simple Lie algebras over $\mathbf{R}$.

The excellent exposition is supplemented by extensive collections of useful exercises at the end of each chapter. All the problems have either solutions or substantial hints, found at the back of the book.

For this latest edition, Helgason has made corrections and added helpful notes and useful references. The sequels to the present book are published in the AMS's Mathematical Surveys and Monographs Series: Groups and Geometric Analysis, Volume 83, and Geometric Analysis on Symmetric Spaces, Volume 39.

Sigurdur Helgason was awarded the Steele Prize for Differential Geometry, Lie Groups, and Symmetric Spaces and Groups and Geometric Analysis. ... Read more

Customer Reviews (3)

Superb Treatise and Indispensible Reference
The mere thought or mention of the name Helgason inspires respect and awe.This book gets five stars all the way on its merit alone, regardless of who wrote it.Difficult as it is, the book starts from the fundamentals and works up in a coherent logical manner, there are no gaps in his presentation.The negative review below is completely unjustified.If anyone would like to at least see some of what this book is like go to ocw.mit.edu and download Helgason's notes which use excerpts in this book.Some of the topics in this book are covered in a more easy going way in "Lie Groups, Lie Algebras, and Some of Their Applications" by Robert Gilmore.(If I'm not mistaken Gilmore was a student of Helgason.)This book is mathematical exposition at it's absolute finest and I don't think but 1 in 1,000 people reading this page need me to tell them that much less need a review to persuade them.This book has quite a reputation.

Unsurpassed, but demanding
As I reviewed this book at Amazon, I found only one review, which I considered to be too harsh.You should understand that Helgason is writing a graduate textbook.Students will learn about "modules" in their graduate algebra course.They will learn De Rham's theorem in an introductory analysis course or sometimes even in a topology course (yes, it can happen).So, most of the language for which another reviewer criticized him would usually be covered in other graduate courses.

Helgason writes tersely but extremely precisely.I know of no other author who gives similar sophistication of point of view and quick, to the point, proofs.He is a "best of breed," and I suppose that is part of the reason he has been a core member of the faculty at M.I.T. for such a long time.A serious student cannot really avoid reading the entire progression of these texts, particularly the "Groups and Geometric Analysis" title, perhaps second in the Helgason manuscripts.

Semisimple( Simple)->Bad
I certainly hate being cheated.
This book is advance as a textbook for a course in Lie Algebra.
I can picture the man who wrote this book lecturing to the future great minds of MIT
and putting them to sleep.
The fellow is the worst sort of pedant.
On page one he mentions one of the more difficult theorems in modern Mathematics,
De Rham's theorem, then drops it like it was too hot to handle.
On page three he introduces Hausdorff's difficult separation axiom
without any explanation at all.
Throughout the book he beats you over the head with terms like "module"
without adequate definition or explanation of terms.
He literally expects you to have learned
what he is supposed to be teaching
before you take his course?
In short , anyone taking the course with this book as a text book
will be hunting for a good text on Lie AlgebraSemi-Simple Lie Algebras and Their Representations (Dover Books on Mathematics) Lie Groups, Lie Algebras, and Some of Their Applications
and differential geometry,
since this one is entirely unreadable,
even by those who know and love the subjects. ... Read more

Product Description

This introduction to the representation theory of compact Lie groups follows Herman Weyl’s original approach. It discusses all aspects of finite-dimensional Lie theory, consistently emphasizing the groups themselves. Thus, the presentation is more geometric and analytic than algebraic. It is a useful reference and a source of explicit computations. Each section contains a range of exercises, and 24 figures help illustrate geometric concepts.

Customer Reviews (1)

Best there is, for now.
Most lie groups books fall into one of two categories: geometrically motivated ones, and algebraically motivated ones.I prefer the former.If you're perfectly happy with a more algebraic treatment, read no further.

There aren't very many geometrically flavored books on lie groups; I can think of only this one and "Compact lie groups" by Sepanski.His book has a nicer treatment of harmonic analysis, but this one beats his in almost every other respect.Sepanski is often too concise.

Still, this book has a long way to go to be really great.Though you're not assumed to have a prior knowledge of lie groups (just smooth manifold theory), you're going to crash and burn without it.That's why I suggest John Lee's Introduction to Smooth Manifolds before taking on this book.It has the best preparation in elementary lie groups you'll ever hope to find.It'll prepare you for Brocker and Dieck.Even then, the book isn't incredibly well written.

In short, there is no really well-written text on lie groups from a geometric perspective, but this one's the best there is.I hope someone writes a better one eventually. ... Read more

Product Description
This book addresses Lie groups, Lie algebras, and representation theory. In order to keep the prerequisites to a minimum, theauthor restricts attention to matrix Lie groups and Lie algebras.This approach keeps the discussion concrete, allows the reader toget to the heart of the subject quickly, and covers all of themost interesting examples. The book also introduces theoften-intimidating machinery of roots and the Weyl group in agradual way, using examples and representation theory asmotivation.

The text is divided into two parts. The first covers Lie groupsand Lie algebras and the relationship between them, along withbasic representation theory. The second part covers the theory ofsemisimple Lie groups and Lie algebras, beginning with a detailedanalysis of the representations of SU(3). The author illustratesthe general theory with numerous images pertaining to Liealgebras of rank two and rank three, including images of rootsystems, lattices of dominant integral weights, and weightdiagrams. This book is sure to become a standard textbook forgraduate students in mathematics and physics with little or noprior exposure to Lie theory.

Brian Hall is an Associate Professor of Mathematics at theUniversity of Notre Dame. ... Read more

Customer Reviews (8)

Abominable formatting
This review is only relevant to the Kindle edition, read on a well-functioning latest-edition (K3) Kindle e-reader.

Do not buy the Kindle edition of this book.

The default font used for the book is a struggle to read in normal font, and very nearly unreadable in italics. This is because portions of letters are missing, such as the lower half of most lowercase e's and the right half of many lowercase h's. The normal options to adjust the font are disabled, except for resizing. The Kindle version inserts spaces in the mi ddl e of wo rds, as demonst rate d in thi s rev ie w. To reduce the occurrence of this problem, it is necessary to shrink the font to its smallest size.

Entire lines in some paragraphs appear with drastically altered font size, even though in the original hardcover, these lines are normal and unaltered (all being part of the same paragraph).

This is a terrible, terrible, terrible conversion to Kindle format. Do not under any circumstances purchase this Kindle edition. The author ought to be infuriated, and the publisher ought to be ashamed. I intend to return the Kindle edition, if possible, and order a hard copy instead.

...

I should mention that, as a graduate student, I love this book. It requires surprisingly little familiarity with topology and algebra; I could have taken this course in my first year without being taxed by prerequisites. Its focus on specific examples, such as SU(2) and SO(3), match well with the situations in which I have previously encountered Lie groups outside of the course. The extensive discussion of examples also helps me to structure the big picture in my head, in that I feel more confident asserting why we are interested in, for example, the connectedness of a Lie group. Hall's discussion of the behavior of, and topological properties of, the most commonly encountered Lie groups is superb. The writing clarifies which details I ought to work out on my own as I read (which is another reason I would have been able to take this course in my first year, when my skill at actively reading and engaging with upper-level textbooks was still budding).

I don't know if this book is useful as a reference text for individuals who have a research interest in algebraic topology. I doubt it's extensive enough. However, as a learning text for physicists and mathematicians whose main research interest lies in analysis or PDE, this book has proven very satisfactory.

I would give the paper copy five stars, but Amazon does not allow me to distinguish between editions (I tried). Please bear this ratings dichotomy in mind if you wish to let my review influence your purchase.

Distracting focus on examples
I am a graduate student at UC Berkeley who used this book for an introductory course in Lie theory.

I found that Hall's book focuses too much on examples, often allowing the reader to lose sight of the underlying algebra. In the midst of hacking and slashing one's way through matrix computations, one fails to gain a deeper, and more valuable intuition for what Lie groups and Lie algebras are.

My classmates discovered and took a liking to Fulton and Harris. I wound up using Humphreys. Humphreys suffers the opposite failing, of being overly sparse and direct, but it succeeds in conveying an appreciation for the elegance of Lie algebras.

Lie Groups on Kindle
This review is for the Kindle edition. I have the hard copy, which I like, and downloaded a sample Kindle version, which I do not like. The Kindle edition occasionally breaks words up; sometimes, more than once per word. While not a deal breaker, editing should be better. The big problems are equations and italics. The equation characters, particularly exponents, are sometimes difficult to identify because they are not fully rendered, and the issue is not with the Kindle. I have a Kindle DX, and expanding the font size does not correct the problem. Missing segments remain missing. Italics have the same problem along slants.

Excellent introduction into the theory of Lie Groups
Brian Hall's book is a welcome addition to the material available for the study of Lie Groups.This book in particular provides a good basis for the study of Lie Groups without getting caught up in the study of Manifold Theory.The book is easy to access, requiring only a basic background in Modern and Linear Algebra and has many applications pertaining both to mathematics and physics.

Horrible
It doesn't take a lot of intelligence to figure
out how to present lie algebras and lie groups
if you are going to take the matrix route.
Namely, you give lots of concrete examples
(requiring nothing more than calculus as
background) and then just state what the general
case is.In this book, the author uselessly drags
the uninitiated through swamps of archaic notation
(save that for the real thing) and incomplete
proofs (where invariably the hard parts are just quoted)
so that you have to wonder what in the world is the point
of committing this mess to paper.It is ironic that the
very same publisher already has better books out on exactly
the same topics.Finally, if this really were an introduction
you wouldn't have to add 'elementary' to the title - so let's
call a spade a spade and leave the spin to the politicians. ... Read more

Product Description
The present book is a continuation of the three-volume workRepresentation of Lie Groups and Special Functions by the sameauthors. Here, they deal with the exposition of the main newdevelopments in the contemporary theory of multivariate specialfunctions, bringing together material that has not been presented inmonograph form before. The theory of orthogonal symmetric polynomials (Jack polynomials,Macdonald's polynomials and others) and multivariate hypergeometricfunctions associated to symmetric polynomials are treated.Multivariate hypergeometric functions, multivariate Jacobi polynomialsand h-harmonic polynomials connected with root systems andCoxeter groups are introduced. Also, the theory of Gel'fandhypergeometric functions and the theory of multivariate hypergeometricseries associated to Clebsch--Gordan coefficients of the unitary groupU(n) is given. The volume concludes with an extensivebibliography. For research mathematicians and physicists, postgraduate students inmathematics and mathematical and theoretical physics. ... Read more

Product Description
Graduate students in many branches of mathematics need to know something about topological groups and the Haar integral to enable them to understand applications in their own fields. In this introduction to the subject, Professor Higgins covers the basic theorems they are likely to need, assuming only some elementary group theory. The book is based on lecture courses given for the London M.Sc. degree in 1969 and 1972, and the treatment is more algebraic than usual, reflecting the interests of the author and his audience. The volume ends with an informal account of one important application of the Haar integral, to the representation theory of compact groups, and suggests further reading on this and similar topics. ... Read more

Product Description
This book is a revised and enlarged edition of "Linear Algebraic Groups", published by W.A. Benjamin in 1969. The text of the first edition has been corrected and revised. Accordingly, this book presents foundational material on algebraic groups, Lie algebras, transformation spaces, and quotient spaces. After establishing these basic topics, the text then turns to solvable groups, general properties of linear algebraic groups and Chevally's structure theory of reductive groups over algebraically closed groundfields. The remainder of the book is devoted to rationality questions over non-algebraically closed fields. This second edition has been expanded to include material on central isogenies and the structure of the group of rational points of an isotropic reductive group. The main prerequisite is some familiarity with algebraic geometry. The main notions and results needed are summarized in a chapter with references and brief proofs. ... Read more

Product Description
Throughout the text, the emphasis is on providing an approach that is accessible to readers equipped with a standard undergraduate toolkit of algebra and analysis. Although the formal prerequisites are kept as low level as possible, the subject matter is sophisticated and contains many of the key themes of the fully developed theory, preparing students for a more standard and abstract course in Lie theory and differential geometry. ... Read more

Customer Reviews (2)

Find a better introduction
It's about time that some decent attention is being given to matrix groups and Lie theory. I like the range of this book. However, there are better introductions. The main problem with this book is that its prerequisites are extremely unclear and its content is somewhat uneven. For example, it's hardly an undergraduate introduction if it needs a strong foundation in topology and a first reading doesn't seem to really require that or even some of the other concepts introduced. Sometimes when a concept is introduced, a theorem is stated or snatches of theory are brought in, the reader is left hanging without a more thorough contextual explanation or even a proof. For example, it's annoying to see a theorem stated and unproven without further comment and without any reference. The text might even be a little overwhelming for a first year graduate student.It would be difficult to teach a cogent and logically laid out course using the text. There are better introductory texts available now which are far more accessible to the intended audience. The author might want to do a little rewriting using Occam's Razor because it has the potential to be a more thorough intro than that available elsewhere.
On the plus side (more for the SUMS series than for any one book), having answers to the exercises helps the student immeasurably. In my opinion, any math text that has no answers to the exercises belongs to the last century and is either a reflection on the author's lack of industry or a basic fear that students would find little need to attend classes. But at this level of subject, we all know that attendance is directly proportional to the quality of the lecturer, not the quality of the text.

worst binding quality I've ever seen
I've been been reading it for only half an hour so I can't say much about the content of this book,however one thing is clear this book's life is going to be very very very short,in fact I don't think that tomorrow will be in one piece,I'm very disappointed to say the least,I don't understand how springer can produce such a poor quality book
now I could try to glue it wich is a mess or collect the pages as they fall apart and staple them,unfortunately I have to say that this is not a very rare case, cause many Springer paperbacks suffer from similar lack of binding quality

... Read more

Product Description

Foundations of Differentiable Manifolds and Lie Groups gives a clear, detailed, and careful development of the basic facts on manifold theory and Lie Groups. Coverage includes differentiable manifolds, tensors and differentiable forms, Lie groups and homogenous spaces, and integration on manifolds. The book also provides a proof of the de Rham theorem via sheaf cohomology theory and develops the local theory of elliptic operators culminating in a proof of the Hodge theorem.

Customer Reviews (5)

Worthless
I used this for a grad class and it sucked! Language is bad, no examples, seemed to give no real insight, i highly recommend getting a different book

Great book, but not perfect or ideal for every purpose
This is a solid introduction to the foundations (and not just the basics) of differential geometry. The author is rather laconic, and the book requires one to work through it, rather than read it. It presupposes firm grasp of point-set topology, including paracompactness and normality. The basics (Inverse and Implicit Function Theorems, Frobenius Theorem, orientation, and rudiments of de Rham cohomology) are covered in about 100 pages (Chapters 1, 2, and 4). This is not really suitable for an undergraduate course in differential geometry, but is great for a graduate course.

Chapter 3, 5, and 6 (self-contained introductions to Lie Groups, Sheaf Theory, and Hodge Theory, all from a geometric viewpoint) are a really nice feature. The book can't be covered in one semester, but these chapters are great for selft-study. In fact, the organization of Chapter 5 is more suitable for self-study than for being taught in class (lots of theory developed first, with all applications delayed until the end). The real jewel of the book is Chapter 6, a very clean introduction to Hodge Theory, with immediate applications.

The main drawback of the book in my view is that the author avoids vector bundles like the plague. These could have been very nicely incorporated into the book. No mention is made of Mayer-Vietoris or Kunneth formula, even though the former follows easily from the section on cochain complexes in Chapter 5 and the latter with some effort from Chapter 6. There is no mention of manifolds with boundary either, except as regular domains of manifolds for the purpose of Stokes Theorem.

The organization of the book could have been better as well. In particular, the section on cochain complexes could have been incorporated in the rather short de Rham Cohomology Chapter 4, so that MV could have been proved and used to compute the cohomology of spheres (beyond the circle). Some subsections, including in Chapter 1, appear out of order to me. There is a shortage of exercises in my view. Some of the author's notation (for tangent spaces, tangent bundles) is rather non-standard.

However, all-in-all, I can't think of a better differential geometry text for a graduate course. Spivak and Lee are quite wordy and do not have the same breadth. Either book would be preferable to Warner for an undergraduate course though. The price is a relative bargain too.

Don'twaste your money
This review refers only to the book printing quality not to the contents.

I had purchased some books from Springer in the past (Like Arnold Mathematical Methods of Classical Mechanics, Lang Algebra etc..) and found them beautifully edited: good binding, paper etc..

And to my surprise I was very disappointed with the overall quality of this book, poor binding -glued instead of sewn- bad quality paper -forming waves at the binding spine, etc..

You pay for a quality item, a book you can use for years, and you get a hardbound crap that you can not left open in a table without holding it tight risking to lose the pages after a few days of use in the process.

I find this unacceptable in books costing 60$+. Sadly I find this to occur very often, publishers should be more careful with their printings and custumers should demand a better quality.

Don't waste your money.

A reader.

Good, as long as you have enough background
I read this book at the very beginning of my studying in differential geometry and was striked. The definitions and methods used in this book seemed totally incomprehensible to me. However, after some development in this field, I found that this book is very concise. It is a very good surey on differential geometry but not a good book to start with. Definitions are given from the most "down to bottom" one. It is a very good attitude, yet, if you do not have much background in differential geometry, this book may takes you several days in order to understand the concept of tensor and exterior algebra.

A good book if you have some background
This book is a good introduction to manifolds and lie groups.
Still if you dont have any background ,this is not the book to start with.The first chapter is about the basics of manifolds:vector fields,lie brackts,flows on manifolds and more,
this chapter can help one alot as a second book on the subject.
The second chapter is about tensors, this introduction can be very hard for someone who didnt met the notion of tensors ,since the book tends to take a very general line with out down to earth examples.the 3ed chapter is about lie groups.It is avery good introduction ,to my point of view ,one of the best there is.
The 4th chapter is about integration on manifolds and is very good too.Chapters 5and6 are about De Rham cohomology theory and the hodge theorem.
If you have some knowledge on allthe above subjects this book can serve as a very good overview on the subject. ... Read more

Product Description
Discrete subgroups have played a central role throughout the development of numerous mathematical disciplines. Discontinuous group actions and the study of fundamental regions are of utmost importance to modern geometry. Flows and dynamical systems on homogeneous spaces have found a wide range of applications, and of course number theory without discrete groups is unthinkable. This book, written by a master of the subject, is primarily devoted to discrete subgroups of finite covolume in semi-simple Lie groups. Since the notion of "Lie group" is sufficiently general, the author not only proves results in the classical geometry setting, but also obtains theorems of an algebraic nature, e.g. classification results on abstract homomorphisms of semi-simple algebraic groups over global fields. The treatise of course contains a presentation of the author's fundamental rigidity and arithmeticity theorems. The work in this monograph requires the language and basic results from fields such as algebraic groups, ergodic theory, the theory of unitary representatons, and the theory of amenable groups. The author develops the necessary material from these subjects; so that, while the book is of obvious importance for researchers working in related areas, it is essentially self-contained and therefore is also of great interest for advanced students. ... Read more

Customer Reviews (2)

The Perfect Supplement to RJ ZIMMER!
As I read this masterpiece of 20th century mathematics, I couldn't help thinking of its relation to the book that defines modern group theory, RJ Zimmer's Ergodic Theory and Semisimple Groups.Coincidence?I think not.Robert Zimmer's book is simply the best there is, and Margoulis' brilliant work is an excellent supplement to that standard mathematical text.As the great Swiss mathematician Armand Borel once said, "Margoulis' masterpiece, while inferior to some of my own work, is an excellent supplement to Zimmer's classic.Zimmer's book is the standard to which all mathematicians should aspire (along with myself, bien sur).Other than my own work, Margoulis' comes as close as possible to reaching that standard."

The Perfect Supplement to RJ ZIMMER!
As I read this masterpiece of 20th century mathematics, I couldn't help thinking of its relation to the book that defines modern group theory, RJ Zimmer's Ergodic Theory and Semisimple Groups. Coincidence? I think not. Robert Zimmer's book is simply the best there is, and Margoulis' brilliant work is an excellent supplement to that standard mathematical text. As the great Swiss mathematician Armand Borel once said, "Margoulis' masterpiece, while inferior to some of my own work, is an excellent supplement to Zimmer's classic. Zimmer's book is the standard to which all mathematicians should aspire (along with myself, bien sur). Other than my own work, Margoulis' comes as close as possible to reaching that standard." ... Read more