Product Description
This is a self-contained introductory textbook on the calculus of differential forms and modern differential geometry. The intended audience is physicists, so the author emphasises applications and geometrical reasoning in order to give results and concepts a precise but intuitive meaning without getting bogged down in analysis. The large number of diagrams helps elucidate the fundamental ideas. Mathematical topics covered include differentiable manifolds, differential forms and twisted forms, the Hodge star operator, exterior differential systems and symplectic geometry. All of the mathematics is motivated and illustrated by useful physical examples. ... Read more

Customer Reviews (3)

Thinking geometrically...
A unique book. Changes the way one thinks about geometry. The concepts and tools become second nature. I strongly recommend it for engineers who need differential geometry in their research (they do, whether they know it or not).

To give an example from page 134: "Vector fields that do not commute are called anholonomic. If two transformations commute, then the system would never leave a 2-surface. This obvious results is called the Frobenius Theorem."

Now after reading about the Frobenius Theorem elsewhere, few people would call in "obvious." Nonetheless, when you read Burke, you will agree. (Granted, it will not happen at first reading unless you are already familiar with the material. So you will read the book several times, which only adds to the pleasure.) Afterwards, you will be happy to consult the proof elsewhere.

Caveat: this book is not the place to go for a formal presentation. It may cause conniptions in the more ideological bourbakistes. Nothing should prevent one from also reading some of the excellent texts that present the material in a precise way, for instance those by Manfredo Perdigão do Carmo, Spivak, or Lang. Nonetheless, Burke is the one to go for the intuition.

The man was a complete loon, but in a good way.
The previous review is amazingly perceptive into Bill Burke's personality and thinking.He was not the most discplined writer or lecturer, (I had no less than 4 courses from him) but his insight and intuition could beamazing.I would recommend this book as a companion to something moretraditional.If you are interested in General Relativity, which is whatthe book was suppose to be a precursor for, get Schutz or Misner, Thorneand Wheeler, or Wald.

Also, if you do want this book, get the errata fromBurke's webpage,...is quite helpful.

I wouldalso hearitly recommend Burke's best book: Geometry, Spacetime andCosmology which is out of print.It is much physical and the examples areclearer.He taught english majors and theater students general relativitywith that book.

It's a lot of work but I like it.
I'm not a physicist or mathematician but I play one on TV.So I am more qualified to review a book on differntial geometry than either of the above professionals.This book is a very good introduction to all the hairy squibbles that theoretical physicists are writing down these days.In particular if you are perplexed by the grand unification gang then this book will help you understand the jargon.However, having only had physics when advanced vector calculus was enough to get by, it is a bit hard going due to the frequent errors and glosses the author makes.Burke gives a very hip and entertaining introduction to some of the most beautiful ideas in physics.It is enjoyable to read if you like sinking your teeth into something more rewarding than Ann Rice.I gave it a six rating because the errors and glosses are so annoying. I suspect Burke's puckishness is responsible;the book has no actual problem sets but he does work out problems that don't always work out.So the reader really has to work at understanding by correcting the possibly(?) intentional errors.Very sly of him.I am on my second reading and suspect that several readings down the line I will probably get the message.The book deserves loving attention. ... Read more

Product Description
These lecture notes are the content of an introductory course on modern, co-ordinate-free differential geometry which is taken by first-year theoretical physics PhD students, or by students attending the one-year MSc course, "Fundamental Fields and Forces" at Imperial College. The book is concerned entirely with mathematics proper, although the emphasis and detailed topics have been chosen bearing in mind the way in which differential geometry is applied to modern theoretical physics. This includes not only the traditional area of general relativity but also the theory of Yang-Mills fields, nonlinear sigma models and other types of nonlinear field systems that feature in modern quantum field theory. This edition of the text contains an additional chapter that introduces some of the basic ideas of general topology needed in differential geometry. A number of small corrections and additions have also been made. The volume is divided into four parts.The first provides an introduction to general topology, the second covers introductory co-ordinate-free differential geometry, the third examines geometrical aspects of the theory of Lie groups and Lie group actions on manifolds, and the fourth provides an introduction to the theory of fibre bundles. In the introduction to differential geometry the author lays considerable stress on the basic ideas of "tangent space structure", which he develops from several different points of view - some geometrical, others more algebraic. ... Read more

Customer Reviews (2)

Good, with problems
Wow! What a great Table of Contents.It has all the stuff I've been wanting to learn about.So I bought the book in spite of seeing only one review of it.After one day, I'm now only at page 26, but I already have read enough to make some comments about it.
The main point about this book is that it is, as the author specifically states, LECTURE NOTES, not, I repeat, not a textbook. What are the implications of this (outside of a somewhat more chatty style than a textbook)?["chatty" isn't quite what I mean; "smooth" might be a better word'] There are two which are noticable to me.1) A lot of math knowledge is taken for granted.2) It has a somewhat sloppy style to it.

Regarding point one, make sure you have a lot of math under your belt before picking up this book.By page 18 the author uses these terms without defining them: Differentiable Manifold,semigroup, Riemannian Metric, Topological Space, Hilbert Space, the "" notation, vector space, and Boolean Algebra.Fortunately for me, I have a fairly extensive math education, and self-studied Functional Analysis, so I wasn't thrown for a loop;but for many others -- brace yourselves!

Regarding point two, Here are two examples:
1) Here is a quote: "The collection of all open sets in any metric space is called the topology associated with the space."Sounds like a definition to me!Fortunately the author gives a (sloppy) definition a few lines later.By the way, the only thing the reader learns about what an 'open set' is, is that it contains none of its boundary points.All the topology books I have read define open sets to be those in the topology.This is another point of confusion for the reader.In fact, points of confusion abound in that portion of the book.
2) On page, 17, trying somewhat haphazardly to explain the concept of a neighborhood, the author defines N as "N := {N(x) | x is an element of X}"This is already a little disconcerting: x is already understood to be an element of X.So he is saying that N is defined as N(x) (which he defines to be a collection of subsets of X).This is all he has to say on the matter until, on page 26, he writes "each N, an element of N(x)".Now N isn't bothN(x) and an element of N(x).This is a point which the author does not clear up.He then starts using N all over the place, yet the reader isn't sure of what he's refering to.

A couple of other things:
-When he defines terms, they is not highlighted, and are embedded in a sentence, making it difficult to find them later.
- The index is pitifully small. Typical for English texts, I know; but this *is* the 3rd millinium!

On the other hand, I have good things to say about the book, too.
I like his style of writing. If it were just more precise, it would be fine for me.I like it better than the normal higher math texts, which tend to be too laconic for me. Notice that I make a distinction between the somewhat chatty style, which I like, and the sloppiness, which is confusing. One can be chatty, yet clear. So far, the undefined math terms which I listed above were not central to the text; and one would not miss much by just reading past them. The author includes many 'comments' sections throughout the book.These are wonderful so far. They are full of comments and examples which really clear up a lot of points.His examples are very good, too, although he is very terse in stating them.The paperback is nice looking.The paper, font, etc. make for easy reading (except for the sub/super-script font, which is too small for me).

To wrap this review up, I had already pretty much learned the stuff covered in the book so far, but judging from what I have read, I will be able to learn a lot from the rest of it; and, unlike some other math books I have studied, the experience won't be too painful.
p.s. See other reviews of it on the UK Amazon site.

Very readable presentation of diff. geometry
I have found Isham's treatment of differential geometry very clear, while maintaining quite an abstract nature. Ishamtakes care to motivate hisdefinitions and include comments where comments are due. No problems areincluded but the book sometimes omits the simpler results and lets you workthem out by yourself. A very readable introduction indeed. ... Read more

Product Description
This is a translation of an introductory text based on a lecture series delivered by the renowned differential geometer, Professor S.S. Chern in Beijing University in 1980. The original Chinese text, authored by Professor Chern and Professor Wei-Huan Chen, sought to combine simplicity and economy of approach with depth of contents. The present translation is aimed at a wide audience, including (but not limited to) advanced undergraduate and graduate students in mathematics, as well as physicists interested in the diverse applications of differential geometry to physics. In addition to a thorough treatment of the fundamentals of manifold theory, exterior algebra, the exterior calculus, connections on fibre bundles, Riemannian geometry, Lie groups and moving frames, and complex manifolds (with a succinct introduction to the theory of Chern classes), and an appendix on the relationship between differential geometry and theoretical physics, this book includes a new chapter on Finsler geometry and a new appendix on the history and recent developments of differential geometry, the latter prepared specifically for this edition by Professor Chern to bring the text into perspective. ... Read more

Customer Reviews (6)

Great Summary
Let me begin by saying that I am biased.I worked as Mr. Chern's assistant in a differential geometry class when I was a grad student.He was a great person to work for and his lectures were well organized.This book is a NOT aimed at the typical undergraduate.It is a major advance in comprehensability from the books from which I learned the covered material.Modern differential geometry does not yet have a great, easy for the novice, self-study friendly text that really covers the material - this book and the Russian trilogy by Dubrovin, et al. are major steps along the way.

an excellentbook!
As many professors in China recommend, itis an excellent book by a great Geometrician. Though it may not be a beginning book, it should appear on your shelf as a classic one!

Be Careful!
This book is written by famous authors alright. It may have their reason in the way they choose those materials and the way they are presented.

The point is: as an introductory text, the various ideas and structures are not well motivated. They may be economical in the way of the presentation. However, it never seems natural from the point of view of a beginner. It is more natural to start with Riemannian geometry and then proceed to the more general concept of vector bundles and connections. It is in Riemannian geometry, that it is natural to first introduce the concept of a geodesic, and this leads, though a lot of books dont do it this way, to the concept of Levi -Civita connection and therefore holonomy and curvature. The general concept of vector bundles and connections before introducing the Riemannian geometry, makes a complex subject even more abstract and though maybe economical from the point of view of the writers, are formidable for a reader.

Even the presentation of specific facts, the book should emphassize, for the benefit of the reader, the structrual (pictorial) aspects more than it does, to illuminate the essence of the formulas, for example, the way it introduces the theta forms on frame bundle omits entirely in mentioning that the essence of thse forms is simply the concept of a coframe. It merely constructs these forms using local coordinates, which seems to be quite tricky to get to its bottom.

To readers of this book
I am reading this book now. It is as the other reviewers said,
rather condensed. However, it would not be beyond comprehension
if the crucial pictures are established. It is my personal opinion that the first crucial place where it should be understood without any compromise is the section on the frame bundle. Later chapters build on this. Previous chapters are
synthesized here. To any readers who are interested, you are invited to discuss this book. My email address is topollogy@hotmail.com (Notice there are two "l" in "topollogy")

dense book
This book contains most important material in differential geometry in about 330 page. No exercise, few exaples make this book very dense, which is just the style of Chinese professors.
It deserves cautions that in chapter 8, Chern introduce the connection proposed by himself in 1948, which is the proper tools for finsler geometry. ... Read more

Product Description

Discrete differential geometry is an active mathematical terrain where differential geometry and discrete geometry meet and interact. It provides discrete equivalents of the geometric notions and methods of differential geometry, such as notions of curvature and integrability for polyhedral surfaces. Current progress in this field is to a large extent stimulated by its relevance for computer graphics and mathematical physics. This collection of essays, which documents the main lectures of the 2004 Oberwolfach Seminar on the topic, as well as a number of additional contributions by key participants, gives a lively, multi-facetted introduction to this emerging field.

Product Description
This text presents the systematic and well motivated development of differential geometry leading to the global version of Cartan connections presented at a level accessible to a first year graduate student. The first four chapters provide a complete and economical development of the fundamentals of differential topology, foliations, Lie groups and homogeneous spaces. Chapter 5 studies Cartan geometries which generalize homogenous spaces in the same way that Riemannian geometry generalizes Euclidean geometry. One of the beautiful facets of Cartan Geometries is that curvature appears as an exact local measurement of "broken symmetry". The last three chapters study three examples: Riemannian geometry, conformal geometry and projective geometry.Some of the topics studied include:- a complete proof of the Lie group - Lie algebra correspondence- a classification of the Cartan space forms- a classification of submanifolds in conformal geometry- Cartan's "geometrization" of an ODE of the formy"=A(x,y)+B(x,y)y'+C(x,y)(y')^{2}+D(x,y)(y')^{3}Topics included in the five appendices are a comparison of Cartan and Ehresmann connections, and the derivation of the divergence and curl operators from symmetry considerations. ... Read more

Customer Reviews (4)

Amazing!
The book "Differential Geometry: Cartan's Generalization of Klein's Erlangen Program", purchase from NRVBOOKSPLUS via AMAZON.COM, exceed my expectations! It's completely new. It seems that no one has ever opened it. Also, I received the product very quickly. I'm so pleased with this purchase ande really recommend this seller.

Great Book
I was fortunate enough to have Sharpe as my supervisor at University of Toronto just when his book was published.His highly abstract thinking is very impressive and I have enjoyed immensely his first chapter on differential topology, which is my specialized area.Though his book branches off into realms that don't particularly suit me, the beginnings of his book had given me great inspiration in my discipline in differential topology

Differential Geometry: Cartan's Generalization of Klein's Erlangen Program
This is definitely a graduate school text. Though I believe the text can be read by a eager undergraduate. The text is about Differential Geometry.
The subject matter demands that the reader read more than 1 book on the subject. This is a good introduction to a difficult but useful mathematical discipline.

Everything via principal bundles.
Sharpe's book is a detailed argument supporting the assertion that most of differential geometry can be considered the study of principal bundles and connections on them, disguised as an introductory differential geometrytextbook.

Some standard introductory material (e.g. Stokes' theorem) isomitted, as Sharpe confesses in his preface, but otherwise this is a trulywonderful place to read about the central role of Lie groups, principalbundles, and connections in differential geometry.The theme is that whatone can do for Lie groups, one can do fiberwise for principal bundles, toyield information about the base.

The informal style (just look at thetable of contents) and wealth of classical examples make this book apleasure to read.While its somewhat nonstandard approach and preferencefor classical terminology might confuse those who have never beenintroduced to the concepts, this is a perfect *second* place to read andmarvel about differential geometry. ... Read more

Product Description
ContentsAnomalous diffusion and stability of Harnack inequalities(by M.T. Barlow)From isoperimetric inequalities to heat kernels via symmetrization(by G. Besson)Discrete isoperimetric inequalities(by F. Chung)An excursion into geometric analysis(by T.H. Colding & W.P. Minicozzi II)Eigenvalues of elliptic operators and geometric applications(by A. Grigor'yan, Y. Netrusov & S.-T. Yau)Spectral gap, logarithmic Sobolev constant, and geometric bounds(by M. Ledoux)Discrete analytic functions: An exposition(by L. Lovász)Conformal properties in classical minimal surface theory(by W.H. Meeks III & J. Pérez)Analysis of the cut locus via heat kernel(by R. Neel & D. Strook)Analysis on Riemannian co-compact covers(by L. Saloff-Coste)Functoriality and small eigenvalues of Laplacian on Riemann surfaces(by F. Shahidi)The inverse spectral problem(by S. Zelditch) ... Read more

Product Description

The study of CR manifolds lies at the intersection of three main mathematical disciplines: partial differential equations, complex analysis in several complex variables, and differential geometry. While the PDE and complex analytic aspects have been intensely studied in the last fifty years, much effort has recently been made to understand the differential geometric side of the subject.

This monograph provides a unified presentation of several differential geometric aspects in the theory of CR manifolds and tangential Cauchy–Riemann equations. It presents the major differential geometric acheivements in the theory of CR manifolds, such as the Tanaka–Webster connection, Fefferman's metric, pseudo-Einstein structures and the Lee conjecture, CR immersions, subelliptic harmonic maps as a local manifestation of pseudoharmonic maps from a CR manifold, Yang–Mills fields on CR manifolds, to name a few. It also aims at explaining how certain results from analysis are employed in CR geometry.

Motivated by clear exposition, many examples, explicitly worked-out geometric results, and stimulating unproved statements and comments referring to the most recent aspects of the theory, this monograph is suitable for researchers and graduate students in differential geometry, complex analysis, and PDEs.

Product Description
In the Spring of 1984, the authors gave a series of lectures in the Institute for Advanced Studies in Princeton. These lectures, which continued throughout the 1984-1985 academic year, are published in this volume. Lectures on Differential Geometry was originally printed in Chinese, and widely circulated in China. This greatly anticipated English translation is an essential reference tool for Differential Geometry. Differential Geometry has progressed rapidly in this century. This book describes the major achievements in this field. ... Read more

Product Description
Ever since the introduction by Rao in 1945 of the Fisher information metric on a family of probability distributions there has been interest among statisticians in the application of differential geometry to statistics. This interest has increased rapidly in the last couple of decades with the work of a large number of researchers. Until now an impediment to the spread of these ideas into the wider community of statisticians is the lack of a suitable text introducing the modern co-ordinate free approach to differential geometry in a manner accessible to statisticians. This book aims to fill this gap. The authors bring to the book extensive research experience in differential geometry and its application to statistics. The book commences with the study of the simplest differential manifolds - affine spaces and their relevance to exponential families and passes into the general theory, the Fisher information metric, the Amari connection and asymptotics. It culminates in the theory of the vector bundles, principle bundles and jets and their application to the theory of strings - a topic presently at the cutting edge of research in statistics and differential geometry. ... Read more

Product Description

This text is intended for an advanced undergraduate (having taken linear algebra and multivariable calculus). It provides the necessary background for a more abstract course in differential geometry. The inclusion of diagrams is done without sacrificing the rigor of the material.

For all readers interested in differential geometry.

Customer Reviews (5)

understandable, clear differential geometry book
There are many differential geometry books out there. Some are very rigorous others not. This book walks the road in the middle. Intuition is developed in the first few chapters by discussing familiar surfaces in R^n, and then a discussion on more abstract manifolds follow.

The book requires some very basic knowledge of linear algebra and some multivariate calculus knowledge. So basically every undergrad in the sciences should find this book easy to understand, and a good introduction to differential geometry.

Took the class and the book
I had the class from Prof. Parker ~20 years ago. (BS Mathematics 83 from SIU)It was a wonderful class and this is a wonderful book.I still have my signed! copy.I am now a professor of EE and a large research university and this is still a subject that I love.Credit that to the Book and Prof. Parker.

A solid introduction
It is hard to disagree with the idea that one must pursue the learning of mathematics in way that might be at odds with its axiomatic structure. One can pursue the study of differentiable manifolds without ever looking at a book on classical differential geometry, but it is doubtful that one could appreciate the underlying ideas if such a strategy were taken. Some background in linear algebra, topology, and vector calculus would allow one to understand the abstract definition of a differentiable manifold. However, to push forward the frontiers of the subject, or to apply it, one must have a solid understanding of its underlying intuition.

Thus a study of classical differential geometry is warranted for someone who wants to do original research in the area as well as use it in applications, which are very extensive. Differential geometry is pervasive in physics and engineering, and has made its presence known in areas such as computer graphics and robotics. In this regard, the authors of this book have given students a fine book, and they emphasize right at the beginning that an undergraduate introduction to differential geometry is necessary in today's curriculum, and that such a course can be given for students with a background in calculus and linear algebra. They also do not hesitate to use diagrams, without sacrificing mathematical rigour. Too often books in differential geometry omit the use of diagrams, holding to the opinion that to do so would be a detriment to mathematical rigour. Much is to be gained by the reading and studying of this book, and after finishing it one will be on the right track to begin a study of modern differential geometry.

A Perfect Introduction
A Must !!!After reviewing a few dozen books in the subject, this is without any doubt one of the best. It it written with rare clarity, and givesenough motivation and examples to understand the more abstract and difficult aspects of the field. The book is intended for advanced undergraduate (with good understanding of linear algebra and calculus III) and should be read prior to an abstract course in differential geometry (such as is covered in the books of Warner and Hicks).

Another Differential Geometry Book - So So
This book I also purchased as a resource for studying differential geometry. It's a little bit better than the one by Thorpe, but not by much. The text is dedicated to the 'hard-core' mathematical, and even they would have to have some experience/guidance in this subject.I'm a self-learning type of guy, with an MS in physics. Too many questions arise to justify this book for the self-learner. There are problems, and a FEW examples. ... Read more

Product Description
The link between the physical world and its visualization is geometry. This easy-to-read, generously illustrated textbook presents an elementary introduction to differential geometry with emphasis on geometric results. Avoiding formalism as much as possible, the author harnesses basic mathematical skills in analysis and linear algebra to solve interesting geometric problems, which prepare students for more advanced study in mathematics and other scientific fields such as physics and computer science. The wide range of topics includes curve theory, a detailed study of surfaces, curvature, variation of area and minimal surfaces, geodesics, spherical and hyperbolic geometry, the divergence theorem, triangulations, and the Gauss-Bonnet theorem. The section on cartography demonstrates the concrete importance of elementary differential geometry in applications. Clearly developed arguments and proofs, colour illustrations, and over 100 exercises and solutions make this book ideal for courses and self-study. The only prerequisites are one year of undergraduate calculus and linear algebra. ... Read more

Product Description

This is the first systematic account of the main results in the theory of lightlike submanifolds of semi-Riemannian manifolds which have a geometric structure, such as almost Hermitian, almost contact metric or quaternion Kähler. Using these structures, the book presents interesting classes of submanifolds whose geometry is very rich.

The book also includes hypersurfaces of semi-Riemannian manifolds, their use in general relativity and Osserman geometry, half-lightlike submanifolds of semi-Riemannian manifolds, lightlike submersions, screen conformal submersions, and their applications in harmonic maps.

Basic constructions and definitions are presented as preliminary background in every chapter. The presentation explores applications and suggests several open questions.

This self-contained monograph provides up-to-date research in lightlike geometry and is intended for graduate students and researchers just entering this field.

Product Description
Presenting an introduction to the mathematics of modern physics for advanced undergraduate and graduate students, this textbook introduces the reader to modern mathematical thinking within a physics context. Topics covered include tensor algebra, differential geometry, topology, Lie groups and Lie algebras, distribution theory, fundamental analysis and Hilbert spaces. The book also includes exercises and proofed examples to test the students' understanding of the various concepts, as well as to extend the text's themes. ... Read more

Customer Reviews (5)

Provides an excellent foundation for advanced studies
I started this book with very little mathematical background (just an electrical engineer's or applied physicist's exposure to mathematics). By the end of this book, I had an advanced exposure to foundational modern mathematics. Now, I am planning to start on "Differential Topology and Quantum Field Theory" by Charles Nash (with other mathematics reference books to complete the proofs in it).

This book also provides a good amount of material showing the application of mathematical structures in physics - Tensors and Exterior algebra in Special relativity and Electromagnetics, Functional Analysis in Quantum mechanics, Differentiable Forms in Thermodynamics (Caratheodory's) and Classical mechanics (Lagrangian, Hamiltonian, Symplectic structures etc), General Relativity etc.

A fast introduction to mathematics in physics
The book does not assume prior knowledge of the topics covered. However, the reader will find use of prior knowledge in algebra, in particular group theory, and topology. Compared to texts, such as Arfken Weber, Mathematical Methods for Physics, A Course in Modern Mathematical Physics is different, and emphasis is on proof and theory. The text is reasonably rigorous and build around stating theorems, giving the proofs and lemmas with occasional examples. The style is not the strictest, although making the text more reader friendly, it is easy to get confused with which assumptions have been made, and the direction of the proof. Sometimes only the "if" part is proven.

Students familiar with algebra will notice that the emphasis is on group theory, interestingly the concept of ideals is left mostly untouched. For more on representation theory a good reference is Groups Representations and Physics by H.F. Jones where solutions to some of the exercises can be found, and examples of the use of the fundamental orthogonality theorem applied to characters of represenations.

The first 6 chapters are relatively straight forward, but in chapter 7 Tensors the text becomes much more advanced and difficult. Chapter 10 on topology offers some lighter material but the reader should be careful, these consepts are to re-appear in the discussion of differential geometry, differentiable forms, integration on manifolds and curvature. These are not the most simple subjects and it is clear that they deserve entire courses of their own.

The book has insight and makes many good remarks. However, chapter 15 on Differential Geometry is perhaps too brief considering the importance of understanding this material, which is applied in the chapters thereinafter. The book is suitable for second to third year student in theoretical physics.

Jumping over the Gap
Most physicists avoid mathematical formalism, the book attacks this by exposing mathematical structures, the best approach I've ever experience. After reading the first chapter of this books I can assure is a must for everyone lacking mathematical formation undergraduate or graduate.

It surely jumps over this technical gap experienced by most physics opening the gate for advanced books an mathematical thinking with physic intuition.

Unfortunately is very expensive, i hope i could have it some day.

A serious, wide spectrum introduction to modern mathematical physics
This book covers almost every subject one needs to begin a serious graduate study in mathematical and/or theoretical physics. The language is clear, objective and the concepts are presented in a well organized and logical order. This book can be regarded as a solid preparation for further reading such as the works of Reed/Simon, Bratteli/Robinson or Nakahara.

Not a review, only a little more information
Since I don't yet have this book, I cannot review it; however, I have found the contents of this book on the publisher's web site in case it would help anyone decide to purchase it or not.

Contents

Preface
1. Sets and structures
2. Groups
3. Vector spaces
4. Linear operators and matrices
5. Inner product spaces
6. Algebras
7. Tensors
8. Exterior algebra
9. Special relativity
10. Topology
11. Measure theory and integration
12. Distributions
13. Hilbert space
14. Quantum theory
15. Differential geometry
16. Differentiable forms
17. Integration on manifolds
18. Connections and curvature
19. Lie groups and lie algebras

I will return at a later date to properly review it in case I need to change the rating I gave it. ... Read more

Product Description
Designed for advanced undergraduate or beginning graduate study, this text contains an elementary introduction to continuous groups and differential invariants; an extensive treatment of groups of motions in euclidean, affine, and riemannian geometry; and development of the method of integral formulas for global differential geometry.
... Read more

Customer Reviews (6)

Very unsatisfactory
I ordered this book on July 18, 2010. They charged to my credit card and then send me a confirm email saying that the estimated delivery date is August 9, 2010. By, August 11, 2010, I still did not receive the book. I send an email to ask and then they told me that they did not have that book. I don't know why they could not tell me that earlier. If I don't send email to ask, I even don't know when they could let me know and refound me.

Don't the judge the book by the title
I had in the past bought another book with the same title from the same publisher (Dover books):
Differential Geometry. They were even published first in the same year
1963 adding to confusion.
This first book was sort of a standard text with very little imagination or
in the long term worth. The book I'm reviewing in contrast gives
tools for development and a catalog of surface types by their differential geometry.
I think the most important thing is the development of the algebra
and calculus of the affine geometry of surfaces.
Some of the results here are very important in the study of
general relativity and chaotic systems theory ( both).
The difference between the two books is that the first
I never go back to and this one I will spend some time
trying to get more out of.
I'm grateful to Heinrich W. Guggenheimer for writing this text:
he gives me hope for mathematics.

A Different Approach
Guggenheimer's book is a very solid introduction to differential geometry which emphasizes the Cartan moving-frame approach. This approach is used to produce invariants for surfaces under affine transformations, etc. Also included are some integral formulas which are used to show that spheres are the only star-shaped surfaces with constant mean curvature.

Lots of math for the serious differential geometry student to chew on.
I think this must be the least expensive differential geometry book that uses Cartan's orthonormal frame method.Though more than 40 years old, the notation is essentially modern (there are a few typographical oddities which aren't really bothersome).

This is a very rich book, with fascinating material on nearly every page.In fact, I think it's a bit too rich for beginners, who should probably start with a more focused text like Millman & Parker or Pressley.

Table of Contents for Differential Geometry
Preface
Chapter 1. Elementary Differential Geometry
1-1 Curves
1-2 Vector and Matrix Functions
1-3 Some Formulas
Chapter 2. Curvature
2-1 Arc Length
2-2 The Moving Frame
2-3 The Circle of Curvature
Chapter 3. Evolutes and Involutes
3-1 The Riemann-Stieltjès Integral
3-2 Involutes and Evolutes
3-3 Spiral Arcs
3-4 Congruence and Homothety
3-5 The Moving Plane
Chapter 4. Calculus of Variations
4-1 Euler Equations
4-2 The Isoperimetric Problem
Chapter 5. Introduction to Transformation Groups
5-1 Translations and Rotations
5-2 Affine Transformations
Chapter 6. Lie Group Germs
6-1 Lie Group Germs and Lie Algebras
6-2 The Adjoint Representation
6-3 One-parameter Subgroups
Chapter 7. Transformation Groups
7-1 Transformation Groups
7-2 Invariants
7-3 Affine Differential Geometry
Chapter 8. Space Curves
8-1 Space Curves in Euclidean Geometry
8-2 Ruled Surfaces
8-3 Space Curves in Affine Geometry
Chapter 9. Tensors
9-1 Dual Spaces
9-2 The Tensor Product
9-3 Exterior Calculus
9-4 Manifolds and Tensor Fields
Chapter 10. Surfaces
10-1 Curvatures
10-2 Examples
10-3 Integration Theory
10-4 Mappings and Deformations
10-5 Closed Surfaces
10-6 Line Congruences
Chapter 11. Inner Geometry of Surfaces
11-1 Geodesics
11-2 Clifford-Klein Surfaces
11-3 The Bonnet Formula
Chapter 12. Affine Geometry of Surfaces
12-1 Frenet Formulas
12-2 Special Surfaces
12-3 Curves on a Surface
Chapter 13. Riemannian Geometry
13-1 Parallelism and Curvature
13-2 Geodesics
13-3 Subspaces
13-4 Groups of Motions
13-5 Integral Theorems
Chapter 14. Connections
Answers to Selected Exercises
Index

Not only for pure mathematician
I find the book very interesting: it's a very good presentation of"classical problems with modern methods" in DifferentialGeometry. It's appreciable for the selection of topics and their logicalorder, the clarity of their exposition (based on the use of modernterminology), the set of proposed problems and the relative results and thelist of references at the end of each chapter. ... Read more

Product Description
Compiling data on submanifolds, tangent bundles and spaces, integral invariants, tensor fields, and enterior differential forms, this text illustrates the fundamental concepts, definitions and properties of mechanical and analytical calculus. Also offers some topology and differential calculus. DLC: Geometry--Differential ... Read more

Product Description
Sigurdur Helgason's Differential Geometry and Symmetric Spaces was quickly recognized as a remarkable and important book. For many years, it was the standard text both for Riemannian geometry and for the analysis and geometry of symmetric spaces. Several generations of mathematicians relied on it for its clarity and careful attention to detail. Although much has happened in the field since the publication of this book, as demonstrated by Helgason's own three-volume expansion of the original work, this single volume is still an excellent overview of the subjects. For instance, even though there are now many competing texts, the chapters on differential geometry and Lie groups continue to be among the best treatments of the subjects available. There is also a well-developed treatment of Cartan's classification and structure theory of symmetric spaces. The last chapter, on functions on symmetric spaces, remains an excellent introduction to the study of spherical functions, the theory of invariant differential operators, and other topics in harmonic analysis. This text is rightly called a classic. Sigurdur Helgason was awarded the Steele Prize for Groups and Geometric Analysis and the companion volume, Differential Geometry, Lie Groups and Symmetric Spaces. ... Read more