Customer Reviews (2)

A Comprehensive Introduction to Differential Geometry, Vol. 2, 3rd Edition
Hours of reading fun!Well paced and twice the fun of Volume 1. Michael does it again!A spellbinding thriller from cover to cover. You gotta love it.

Wonderful exposition of the foundations of Curvature and Connections
This book is the second volume of the 3rd edition in a five volume series on differential geometry. The focus here is on the foundations of curvature and connections.

The only prerequisite for volume II is a careful study of volume I. In particular, you'll need a good understanding of the Riemannian metric and you'll need to be comfortable with manipulating differential forms. Also pay attention to the differential equations material used to establish Frobenius Integrability in Chapter 6 of volume I. In addition, you'll need the main concepts from the Lie Groups study of Chapter 10 of volume I.

The author begins the study of curvature with a review of the classical theory of curvature of curves and surfaces in Chapters 1 and 2. These chapters are written in style that helps the reader anticipate more general results for Riemannian manifolds. For example, the reader will notice the rotation index of a planar curve can be represented in terms of its total curvature; a result which foreshadows the Gauss-Bonnet Theorem. Both Euler's Theorem and Meusnier's Theorem for surfaces embedded in Euclidean 3-space are studied.

Chapter 3 details the geometry of surfaces as developed by Gauss. Spivak's treatment here is very unusual, and, in Part A of this chapter, the author actually gives an English translation of original paper of Gauss. Reading this is a bit unusual as the author alternates the translation of Gauss on a page with comments by the author on the preceding page. Part B of the chapter gives the accounting of the Gauss Theory in modern notion. Part B is delightfully geometric and includes all of the 'greatest hits' from the theory, including the Theorema Egreguim and the Triangle Excess Theorem.

Chapter 4 studies Riemann's theory of curvature of manifolds, and contains 4 parts. Part A and Part C are English translations of Riemann's foundational work, while Part B and Part D cast this work in the light of more modern notion. Riemann's curvature tensor is built up from an intuitive study of the second-order terms in the Taylor series expansion of the Riemannian metric. The author also introduces what he calls the "Test Case" for curvature theory: Flat manifolds are locally isometric to Euclidean space. Spivak uses this "Test Case" repeatedly throughout the remainder of the text to reinforce the various notion of curvature as he studies the work of Riemann, Ricci, Kozul, Cartan and Ehresmann.

Chapter 5 (the Debauch of Indices) studies the work of Christoffel and Ricci in developing the covariant derivative. The aim of this work is to simplify the somewhat cumbersome formulas for Riemann's curvature tensor. The reader quickly sees that effort, called absolute differential calculus, is not altogether successful and leads to an veritable explosion of multi-indexed quantities and even harder-to-penetrate formulas.Clearly a better way is needed if we are to move forward with our study of differential geometry.

The "way forward" is Kozul's concept of the connection and this is introduced in Chapter 6. First, note that the connection here is one of the versions of the introduced by Kozul as a map of pairs of vector fields to a vector field. Another useful version, not studied in volume II, is to consider the connection as a Hessian which maps any smooth function to a bilinear form on the tangent space. Second, note that Chapter 6 is usually the starting point for most treatments of curvature in differential geometry (e.g Do Carmo's "Riemannian Geometry"). Without the motivating material from the previous chapters, it would be difficult to understand the need for(or the point of) Kozul's connection.

Cartan's theory of curvature via a study of moving frames is detailed in Chapter 7. The author is careful to intuitively motivate Cartan's deviation from Euclidean concept as represented in the structure equations. Cartan's curvature tensor is shown to agree with Riemann's tensor, the "Test Case" is revisited, and the well-known fact that the curvature determines the Riemannian metric is established.

Building on the orthonormal frames from the previous chapter, Spivak now considers Ehresmann's theory of connections in principal bundles in Chapter 8. The main results here introduce the Ehresmann connection on the frame bundle, and gives the Kozul connection as a Lie derivative, thought of as the Cartan connection obtained from the Ehresmann connection.

My only complaint is that the author didn't include any exercises in this second volume. This is a real shame as the exercises in the first volume were very well-designed and one of the highlights of that text.
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Book Description
Cartan geometries were the first examples of connections on a principal bundle. They seem to be almost unknown these days, in spite of the great beauty and conceptual power they confer on geometry. The aim of the present book is to fill the gap in the literature on differential geometry by the missing notion of Cartan connections. Although the author had in mind a book accessible to graduate students, potential readers would also include working differential geometers who would like to know more about what Cartan did, which was to give a notion of "espaces généralisés" (= Cartan geometries) generalizing homogeneous spaces (= Klein geometries) in the same way that Riemannian geometry generalizes Euclidean geometry. In addition, physicists will be interested to see the fully satisfying way in which their gauge theory can be truly regarded as geometry. ... Read more

Customer Reviews (3)

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

Customer Reviews (1)

The best brief intro to modern differential geometry
First, the binding: It's paperback.

This was Hicks' only book - he died young. It's a great concise intoduction to differential geometry, sort of the Schaum's Outline version of Spivak's epic "A Comprehensive Introduction to Differential Geometry" (beware any math book with the word "Introduction" in the title - it's probably a great book, but probably far from an introduction.) The ten chapters of Hicks' book contain most of the mathematics that has become the standard background for not only differential geometry, but also much of modern theoretical physics and cosmology. It thus makes a great reference book for anyone working in any of these fields. Make sure that you do, or at least read, all of the exercises - they're an important part of the book. The only caveat I have is that Hicks wasn't kidding when he included the word "Notes" in the title - as I said, the book is concise.

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Book Description
Differential geometry plays an increasingly important role in modern theoretical physics and applied mathematics. This textbook gives an introduction to geometrical topics useful in theoretical physics and applied mathematics, covering: manifolds, tensor fields, differential forms, connections, symplectic geometry, actions of Lie groups, bundles, spinors, and so on. Written in an informal style, the author places a strong emphasis on developing the understanding of the general theory through more than 1000 simple exercises, with complete solutions or detailed hints. The book will prepare readers for studying modern treatments of Lagrangian and Hamiltonian mechanics, electromagnetism, gauge fields, relativity and gravitation. Differential Geometry and Lie Groups for Physicists is well suited for courses in physics, mathematics and engineering for advanced undergraduate or graduate students, and can also be used for active self-study. The required mathematical background knowledge does not go beyond the level of standard introductory undergraduate mathematics courses. ... Read more

Customer Reviews (3)

The best book on the subject
Before discovering the new book my Marian Fecko I thought I know all that I need about differential geometry (I co-authored a monograph on this subject myself). I had my favorite books: Kobayashi-Nomizu, Bishop-Crittenden, Sternberg, Michor, Abraham and some more. Yet "Differential Geometry and Lie Groups for Physicists" was a completely new experience. It is written with a "soul" and covers topics that are important but missing in other books. As I was working on a paper dealing with torsion, I emailed the Author with some of my ideas and questions and got an instant answer.

Readers looking for explanations and geometrical interpretations of the abstract concepts will certainly find this book irreplaceable. Lie and covariant derivatives, parallel transport, Hodge operator, Cartan's moving frame method, Laplace-Beltrami operator, Lie groups, Maxwell equations, Clifford algebras and spin bundles, SL(2,C), Dirac operator, Momentum map etc. etc.- all introduced and explained in a concise yet clear way, with exmaples and exercises.

This book should find its place on the bookshelf of everyone interested in geometrical concepts required for understanding contemporary theoretical physics.

I recommend this book to all students and professionals. It should find its place in every university library.

Just one warning: certain mathematical symbols did not find their way to the "Index of frequently used symbols" at the end of the book. The reader trying to read the book starting from p. 600 may find it necessary to spent some time going through the earlier chapters to find out the meaning of a given symbol.

Differential geometry
Marian Fecko's textbook covers well fundamental elements of modern differential geometryand introduction to the Lie groups (not only) from geometrical point of view. Geometrical formulations of the classical mechanics, gauge theory and classical electrodynamics are discussed.

The textbook expects the reader to be familiar with mathematical analysis on the level of the standard course usual in the physics undergraduate study programs. Understanding of the parts dealing with physical applications (classical mechanics and electrodynamics) expects knowledge of fundamental principles of these subjects. Organization of the book allows the reader to concern on particular part, i. e. understanding of later parts doesn't require reading of all previous parts (reading of parts concerning on the classical dynamics does not require reading of parts dealing with electrodynamics). However, relations between different subjects of the theory are explained instructively.

The main advantage of this textbook is that reader "builds" the subject himself by solving the exercises usually appended by hints. It makes all the elements of the theory natural to the reader during study. This way is a little bit more time consuming when compared with other textbooks dealing with this subject. It provides good starting point for study of mathematical aspects of the general relativity and field theories. I recommend this book to everybody who wants to understand fundamental concepts in differential geometry in detail.

not for starter or self-learning
The book covers a good range of topics in Differnetial geometry with lots of exercises. One literarily has to do the exercises to develop the concept. Ecah chapter ends with a concise summary of the key equations. The problem is that all the exercises are mixed with the main context. It lacks any exposition or concept development for most of the topics, no definition, no prove, and every page is filled with exercises.This style make it difficult for someone to learn the subjects the first time or to use it as a reference.

Separately, there are too few graphs to assist the reader to visualize the ideas. The prints are also small making it hard to read.

Nakahara's book (Geometry, topology and physics) is a much better choice on the same subject.



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Book Description

Customer Reviews (4)

Outdated
This is a very outdated book. It deals mainly with geometry of curves and (hyper)surfaces in euclidean space. "Modern" elements like connections and rieman metrics are barely touched upon. If you are a mature geometer though, you could skim it for useful material that it surely does contain.

If you are looking for an introduction to current differential geometry go elsewhere, for instance Spivaks' A Comprehensive Introduction to Differential Geometry, Volume 1, 3rd Edition.

If you need a book on classical differential geometry take a look at Struiks' Lectures on Classical Differential Geometry: Second Edition.

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.

if interested in this topic
Guggenheimer gives a good introduction to the topic, although a goodbackground in mathematics (analysis) is very helpful.While not one of theeasiest books to read, it serves as a good supplemental text to anyonebrowsing through other texts on differential geometry. ... Read more

Book Description
Since the times of Gauss, Riemann, and Poincaré, one of the principal goals of the study of manifolds has been to relate local analytic properties of a manifold with its global topological properties. Among the high points on this route are the Gauss-Bonnet formula, the de Rham complex, and the Hodge theorem; these results show, in particular, that the central tool in reaching the main goal of global analysis is the theory of differential forms.

The book by Morita is a comprehensive introduction to differential forms. It begins with a quick introduction to the notion of differentiable manifolds and then develops basic properties of differential forms as well as fundamental results concerning them, such as the de Rham and Frobenius theorems. The second half of the book is devoted to more advanced material, including Laplacians and harmonic forms on manifolds, the concepts of vector bundles and fiber bundles, and the theory of characteristic classes. Among the less traditional topics treated is a detailed description of the Chern-Weil theory.

The book can serve as a textbook for undergraduate students and for graduate students in geometry. ... Read more

Customer Reviews (2)

Self contained introduction to techniques of classifying manifolds.
This text is phenomenally easy to read and well organized. The author starts you on a journey by first explaining the importance and power of classifying manifolds namely by certain invariants preserved by certain mappings ( diffeomorphisms ).

For example, like Euler, we could count the number of holes in the surface and using this combinatorial method we are led to homology theory.

Or like Gauss, we could use a differentiation and integration to come up with the idea of curvature as an intrinsic feature of the surface.

Modern approaches use differential forms to represent homology and cohomoly groups.

The author also deals with fibre bundles demonstrating their importance in analyzing manifolds specifically how the number of fibre bundles possible with given Lie groups as structure groups over the manifold can be answered by characteristic classes such as the Chern and Pontrjagin classes. The use of differential forms is indispensible.

Perhaps the most satisfying aspect of this book is that it clarifies the notions of connection, connection form, curvature, curvature form for manifolds and fibre bundles.

There are plenty of exercises to boot.

A very good book.
This is probably the most clearly written self-contained book on the basics of differential geometry. The author does a great job explaining the ideas behind purely mathematical 'dry' constructions. On the other hand, everything is defined correctly and precisely. A very readable and useful book with the perfect combination of formal math. and intuition.I would recommend it to students in theoretical physics area, together with the Nakahara's fantastic book. ... Read more

Book Description

Book Description
This introductory text develops the geometry of n-dimensional oriented surfaces in Rn+1. By viewing such surfaces as level sets of smooth functions, the author is able to introduce global ideas early without the need for preliminary chapters developing sophisticated machinery. the calculus of vector fields is used as the primary tool in developing the theory. Coordinate patches are introduced only after preliminary discussions of geodesics, parallel transport, curvature, and convexity. Differential forms are introduced only as needed for use in integration. The text, which draws significantly on students' prior knowledge of linear algebra, multivariate calculus, and differential equations, is designed for a one-semester course at the junior/senior level. ... Read more

Customer Reviews (3)

Introduces differential geometry to advanced-calc students
As a math undergrad at Kent State University some twenty-odd years ago, I took a course in differential geometry. This was the text; I still have my copy. (Autographed by the author, in fact; I met him on a visit to his university, where I subsequently attended grad school.)

The title of this book states, accurately, that its subject matter is 'elementary topics _in_ differential geometry'. This is one of those 'transition' books that introduces students familiar with Subject A to a more-or-less-systematic smattering of elementary topics in Subject B. Here, Subject A is multivariate calculus and Subject B is, of course, differential geometry.

Since that's what this book is for, there are way more numbers and pictures in it than you'll ever see in a modern graduate-level differential geometry text. The idea is to show the student the geometric meaning behind all the advanced calculus and help him/her understand _both_ words in the name 'differential geometry'. In short, much of the motivation here is geometric.

I liked it a lot and I am still grateful for its highly accessible introduction to a fascinating field. However, I must also add that its approach is not representative of any graduate-level math course I ever took. Of course this is an undergraduate text and isn't supposed to represent graduate-level coursework. Nevertheless, it _may_ give a student the wrong idea about what to expect in more advanced treatments. (Is there some personal history lurking behind that remark? You guess.)

An excellent 'transitional' book, then, and highly recommended to readers who want to connect their knowledge of multivariate calculus to the geometry of Euclidean space. It's also a fine example of an expository work on mathematics that remembers its target audience. However, as other reviewers have commented, it needs some answers to the exercises in order to be really useful for self-study.

A good start
This book could be considered as the second semester of an advanced calculus course and serves as an excellent introduction to differential geometry. The approach is rigorous, but the author does employ a great deal of illustrations to explain the relevant concepts. The first five chapters cover vector fields on curves and surfaces. The many concrete examples given by the author illustrate effectively the normal and tangent vector fields. The Gauss map is then appropriately introduced in Chapter 6 and shown to be onto for compact, connnected, oriented n-dimensional surfaces in n+1-dimensional Euclidean space.

This is followed by a discussion of geodesics and parallel transport in the next two chapters. The important concept of holonomy is introduced in the exercises along with the Fermi derivative. These ideas are extremely important in physical applications and must be understood in depth if the reader is to go into areas such as general relativity and high energy physics.

The next chapter considers the local behavior of curvature on an n-surface via the Weingarten map. The important concept of the covariant derivative is introduced. The concept of a geodesic spray, so important in the theory of differential equations, is introduced in the exercises.The curvature of plane curves is treated in Chapter 10 with the circle of curvature introduced. The Frenet formulas, which relate the tangent and normal vectors to the curvature and torsion, are discussed in the exercises. The curvature of surfaces is discussed later in Chapter 12 with the first and second fundamental form introduced, along with the very important Gauss-Kronecker curvature. And in this chapter the author introduces the idea of local and global properties of an n-surface. Although not rigorous, the discussion is helpful for students first introduced to these concepts.

After a nice overview of convex surfaces, the parametrization of surfaces is discussed in the next two chapters, where the inverse function theorem for n-surfaces is proved. This is followed by a consideration of focal points with Jacobi fields discussed in the exercises.

More measure-theoretic concepts are discussed in the next chapter on surface area and volume. Partitions of unity are brought in so as to define the integral of an n-form over a compact oreinted n-surface. Exterior products of forms are introduced in the exercises.

Soap bubble enthusiasts will appreciate the discussion on minimial surfaces in Chapter 18. Although very short, the author's treatment does bring out the important ideas. Minimal surfaces have taken on particular important in the new membrane theories in high energy physics recently. This is followed by a detailed treatment of the exponential map in Chapter 19. Once again, techniques with a variational calculus flavor are used to characterize geodesics as shortest paths.

After a discussion of surfaces with boundary in Chapter 20 the Gauss-Bonnet theorem is proved in Chapter 21 using Stoke's theorem. The discussion of this important result is crystal clear and should prepare the reader for more advanced statements of it in the general context of differentiable manifolds. This is followed by a brief discussion of rigid motions and isometries in the next two chapters. The book ends with ta discussion of Riemannian geometry, a topic of upmost importance in physics and discussed here with care.

A very good book and one that will be useful to beginning students of differential geometry, and also physics students going into the areas of gravitational physics or high energy physics.

Another Differential Geometry Book
I bought this book as a supplement, and I wish I hadn't. It's more archaic and has a large amount of 'hidden' steps than most mathematical books. It has problems, but no solutions. Not recommended for the physics, applied physics or self-learner. It's really aimed at the 'hard-core' mathematicians, and even they would have to have some experience/guidance in differential geometry.

I have an MS in physics, and found this book to be very difficult to get information out of. It has a few nuggets, but can only be seen after going through other books.It might go well with a good lecturer, but as a self-studied person, this is not the way to go. ... Read more

Book Description

The study of curves and surfaces forms an important part of classical differential geometry. Differential Geometry of Curves and Surfaces: A Concise Guide presents traditional material in this field along with important ideas of Riemannian geometry. The reader is introduced to curves, then to surfaces, and finally to more complex topics. Standard theoretical material is combined with more difficult theorems and complex problems, while maintaining a clear distinction between the two levels.

Book 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

Book Description

Intended for a one year course, this volume serves as a single source, introducing students to the important techniques and theorems, while also containing enough background on advanced topics to appeal to those students wishing to specialize in Riemannian geometry. This is one of the few works to combine both the geometric parts of Riemannian geometry and the analytic aspects of the theory, while also presenting the most up-to-date research. This book will appeal to readers with a knowledge of standard manifold theory, including such topics as tensors and Stokes theorem. Various exercises are scattered throughout the text, helping motivate readers to deepen their understanding of the subject.

Book Description
The second edition of this text has sold over 6,000 copies since publication in 1986 and this revision will make it even more useful. This is the only book available that is approachable by "beginners" in this subject. It has become an essential introduction to the subject for mathematics students, engineers, physicists, and economists who need to learn how to apply these vital methods. It is also the only book that thoroughly reviews certain areas of advanced calculus that are necessary to understand the subject.

Line and surface integrals
Divergence and curl of vector fields ... Read more

Customer Reviews (4)

This is a book for REAL mathematicians
This book is an wonderful introduction to Differential Geometry for the serious student of mathematics. However, it is not aimed at engineers, physicists or even applied mathaticians.
The author assumes the reader has an extensive knowledge of abstract algebra and at least one course in analysis. Likewise, he has chosen to emphasis applications of the subject to Lie Groups, homotopy theory, and group actions, rather than the physical applications that applied mathematicians are looking for. But, for the student of pure mathematics, this text is a great starting point into the rich world of differential geometry.
Also, while this book is an introduction and requires no previous knowledge of the subject, it covers enough ground to be followed up by such topics as the Gauss-Bonnet Theorem, the Cartan-Hadamard Theorem, Bonnet's Theorem, or Morse Theory.

When accountants and soldiers take interest in geometry.....
One day, accountants and soldiers may take an interest in differential geometry. If and when such a day comes to pass, this book will have a role to play. Until then, engineers, physicists and mathematicians alike have better alternatives, such as the inspiring texts, with complementary qualities, by Burke, "Applied Differential Geometry"; by do Carmo, "Riemannian Geometry", or by Spivak, "A Comprehensive Introduction to Differential Geometry".

Even more advanced books such as Lang's or Petersen's are more readable: in them the extra formalism brings the reward of more powerful results. Here the retentive attention to the trees at the expense of the forest is merely a barrier to entry for the uninitiated. This text's popularity in some areas of engineering must have played a role in the slow acceptance of Riemannian geometric methods.

Manuel Tenide

great introductory text
My first course on manifolds was based on this book,and I believe that it is the best introduction to the subject (especially for beginners). I thoroughly enjoyed it! I should also recommend Conlon's 'Differentiable Manifolds' (2ed, Birkhauser), as it is the perfect follow-up to Boothby. --A

Very Nice Nontrivial Introduction
This book is a careful treatment of the subjects in the title. It is an introduction, but it manages to cover quite a bit of ground with lots of examples to illustrate. One of it's distinguishing pointsis the way inwhich the concrete, coordinate based calculations are emphasized even whileusually presenting the more abstract, coordinate free approach as well.

The book does a good job at stimulating those studying it to developintuition. I found the book helpful when I was first studying the subject. ... Read more

Book Description
This edition of the invaluable text Modern Differential Geometry for Physicists 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.

These lecture notes are the content of an introductory course on modern, coordinate-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 these days 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.

The volume is divided into four parts: (i) introduction to general topology; (ii) introductory coordinate-free differential geometry; (iii) geometrical aspects of the theory of Lie groups and Lie group actions on manifolds; (iv) 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. This is done with awareness of the difficulty which physics graduate students often experience when being exposed for the first time to the rather abstract ideas of differential geometry. ... 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

Book 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

Book Description
Using a self-contained and concise treatment of modern differential geometry, this book will be of great interest to graduate students and researchers in applied mathematics or theoretical physics working in field theory, particle physics, or general relativity.The authors begin with an elementary presentation of differential forms.This formalism is then used to discuss physical examples, followed by a generalization of the mathematics and physics presented to manifolds. The book emphasizes the applications of differential geometry concerned with gauge theories in particle physics and general relativity.Topics discussed include Yang-Mills theories, gravity, fiber bundles, monopoles, instantons, spinors, and anomalies. ... Read more

Customer Reviews (3)

NOTHING NEW
THIS BOOK IS JUST ANOTHER EXAMPLE OF THE MANY PUBLICATIONS IN THIS SUBJECT MATTER THAT MISSES THE WHOLE POINT.

Concise, big picture treatment of the subject
This text, while lacking in rigour and detail, is an ideal supplement for self-study or lectures on modern mathematical methods in physics.In fact, it is precisely its lack of detail that allows it to act as the yin to the yang of other, weightier texts.Most books on this subject obscure the big picture behind their equations, reducing pleasant geometry to the grimy level of analysis.No such crime is committed here, and the reader is much the better for it.To be sure, this is not a stand-alone text - to not delve into the details would only enter the reader into the false security of ignorance.However, it is most definitely a must-have book for anyone interested in modern physics and mathematics.

Recommended texts to accompany this one are: 1) Geometry of Physics, Frankel 2) Intro to Lie Algebras & Rep. Th., Humphreys 3) Geometry, Topology,& Physics, Nakahara (another useful survey) 4) Spin Geometry, Lawson & Michelson

excellent introduction to relevant topics!
This is a concise introduction to applications of differential geometry on some improtant topics in physics, such as gauge theories, gravity...etc. Despite its size (which is rather comfortable for readers who prefer lessabstract definitions and theorems), nothing essential to the spirit of thetopics has been missed, I personally think it sure is one of the excellentbooks on these subjects and am glad to recommand it to you all who love andwant to discover the geometric aspects of physics! ... Read more

Book 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

Book Description
Differential Geometry: A First Course is an introduction to the classical theory of space curves and surfaces offered at the Graduate and Post- Graduate courses in Mathematics. Based on Serret-Frenet formulae, the theory of space curves is developed and concluded with a detailed discussion on fundamental existence theorem. The theory of surfaces includes the first fundamental form with local intrinsic properties, geodesics on surfaces, the second fundamental form with local non-intrinsic properties and the fundamental equations of the surface theory with several applications. ... Read more

Customer Reviews (1)

THIS IS HOW A FIRST COURSE IS DONE
This beautiful, incisive and up-to-date treatment of classical differential geometry shows that D Somasundaram is blessed with a profound mathematical thought. I have particularly appreciated some smart and agile procedures which give the reader an illuminating insight into the essence of Mathematics. Obviously the inexperienced reader need to be willingly interactive if he wants to reach a full comprehension of the matter, and in my opinion this is the greatest merit of the book. ... Read more

Customer Reviews (2)

The Definitive Reference for Four Decades
The two-volume set by Kobayashi and Nomizu has remained the definitive reference for differential geometers since their appearance in 1963(volume 1) and 1969 (volume 2).Over the decades, many readers have developed a love/hate relationship with these difficult, challenging texts.For example, in a 2006 edition of a competing text, the author remarked that "every differential geometer must have a copy of these tomes," but followed this judgment by observing that "their effective usefulness had probably passed away," comparing them to the infamously difficult texts of Bourbaki.

As a practicing differential geometer, I would argue that Kobayashi and Nomizu remains an essential reference even today, for a number of reasons.
Volume 1 still remains unrivalled for its concise, mathematically rigorous presentation of the theory of connections on a principal fibre bundle---material that is absolutely essential to the reader who desires to understand gauge theories in modern physics.The essential core of Volume 1 is the development of connections on a principal fibre bundle, linear and affine connections, and the special case of Riemannian connections, where a connection must be "fitted" to the geometry that results from a pre-existing metric tensor on the underlying manifold, M.
Volume 2 offers thorough introductions to a number of classical topics, including submanifold theory, Morse index theory, homogeneous and symmetric spaces, characteristic classes, and complex manifolds.

The influence of the texts by Kobayashi and Nomizu can be seen in most of the subsequent differential geometry texts, both in organization and content, and especially in the adoption of notation.If there was a particularly fine point in your favorite introductory differential geometry text that you never completely understood, the odds are good that you will find the answer, fully developed and presented at an entirelydifferent mathematical level, in Kobayashi and Nomizu.It is not an unreasonable analogy to say that learning differential geometry without having your own copy of Kobayashi/Nomizu is like studying literature in the complete ignorance of Shakespeare.

Let there be no mistake about the advanced level of these texts.The Preface to Volume 1 clearly states that the authors presume the reader to be familiar with differentiable manifolds, Lie groups, and fibre bundles, as developed in the (now classical) texts by Chevalley, Montgomery-Zippin, Pontrjagin, and Steenrod.Today's reader is far more likely to have studied these subject from more recent books like those by Boothby, Hall, and Husemoller, but whatever the source, a familiarity IS presumed.The "lightning review" provided in Chapter I of Volume 1 will be extremely tough going for the reader who is new to these topics.It should also be noted that in 329 pages of Volume 1 and 470 pages of Volume 2, not a single diagram or picture is to be found!Those drawn to geometry for its visual aspects will find Kobayashi/Nomizu totally lacking in visual aids.

As with so many classic references in mathematics, the hardbound edition of Kobayashi and Nomizu is no longer in print.Copies appear sporadically on the used book market at absolutely obscene prices.The Classics Library paperback edition is still available, but the serious student willfind that the paperbacks simply do not fare well under serious, sustained use.

A good book for advanced learner
This book is an a good reference for advanced study in differential geometry. It display an overall view of the subject. However, the symble used is might be too much for beginners. ... Read more

Book Description
Accessible, concise, and self-contained, this book offers an outstanding introduction to three related subjects: differential geometry, differential topology, and dynamical systems. Topics of special interest addressed in the book include Brouwer's fixed point theorem, Morse Theory, and the geodesic flow.Smooth manifolds, Riemannian metrics, affine connections, the curvature tensor, differential forms, and integration on manifolds provide the foundation for many applications in dynamical systems and mechanics. The authors also discuss the Gauss-Bonnet theorem and its implications in non-Euclidean geometry models.The differential topology aspect of the book centers on classical, transversality theory, Sard's theorem, intersection theory, and fixed-point theorems. The construction of the de Rham cohomology builds further arguments for the strong connection between the differential structure and the topological structure. It also furnishes some of the tools necessary for a complete understanding of the Morse theory. These discussions are followed by an introduction to the theory of hyperbolic systems, with emphasis on the quintessential role of the geodesic flow.The integration of geometric theory, topological theory, and concrete applications to dynamical systems set this book apart. With clean, clear prose and effective examples, the authors' intuitive approach creates a treatment that is comprehensible to relative beginners, yet rigorous enough for those with more background and experience in the field. ... Read more

Customer Reviews (2)

Excellent book
It was a great pleasure to read the book Differential Geometry and Topology With a View to Dynamical Systems by Keith Burns and Marian Gidea. The topic of manifolds and its development, typically considered as very abstract and difficult, becomes for the reader of this outstanding book tangible and familiar.This joyful aspect of the book was achieved by the authors by setting the advanced material of differential geometry and topology as if on a mobile bridge or a crossroad that associates a(n) (primarily) unfamiliar abstract part of the text with elementary math theories. The latter pedagogical approach was mostly carried out through carefully prepared examples, in which, for essentially abstract structures and mathematical topics, well known familiar elementary settings serve as obvious motivations, which make the transition to a higher level of an abstraction smooth. Nevertheless, the scope of the main topic in this book, differential geometry and topology, is pretty far advanced.Besides the basic theory, centered around analytical properties of manifolds (mostly endowed with additional, in particular Riemannian, structures and vector or tensor fields defined on them) and their applications, it also provides a good introductory approach to some deeper topics of differential topology such as Fixed Points theory, Morse theory, and hyperbolic systems throughout the rest of the book.
The main stream of the applications that always follow or motivate the theoretical context is dynamical systems. Excellent examples reveal the close ties ofthisbeautiful mathematical theory with common problems intheoreticalphysics, classical and fluid mechanics, field theory, and, most importantly, the theory of general relativity.
The book by Burns and Gidea is also be strongly recommended for those readers who wish to enhance their mathematical tools to make possible a deeper insight into these fascinating physical theories.

Jerzy K.Filus

A very good book
A very clear and very entertaining book for a course on differential geometry and topology (with a view to dynamical systems).

First let me remark that talking about content, the book is very good. Each of the 9 chapters of the book offers intuitive insight while developing the main text and it does so without lacking in rigor. The first 6 chapters (which deal with manifolds, vector fields and dynamical systems, Riemannian metrics, Riemannian connections and geodesics, curvature and tensors and differential forms) make up an introduction to dynamical systems and Morse theory (the subject of chapter 8). Chapter 7 is devoted to fixed points and intersection numbers. The last chapter is an introduction to hyperbolic systems.

This enjoyable and highly instructive book contains a large number of examples and exercises. It is an incredible help to those trying to learn dynamical systems (and not only). It teaches all the differential geometry and topology notions that somebody needs in the study of dynamical systems.
The authors, without making use of a pedantic formalism, emphasize the connection of important ideas via examples. It completely enhanced my knowledge on the subject and took me to a higher level of understanding.

... Read more