Product Description
An introductory textbook on the differential geometry of curves and surfaces in three-dimensional Euclidean space, presented in its simplest, most essential form, but with many explanatory details, figures and examples, and in a manner that conveys the theoretical and practical importance of the different concepts, methods and results involved. With problems at the end of each section, and solutions listed at the end of the book. 99 illustrations.
... Read more

Customer Reviews (16)

Excellent, but Challenging, Introduction
I strongly recommend this book to anyone looking for an introduction to differential geometry. This book restricts its coverage to curves and surfaces in three dimensional Euclidean space, which is highly appropriate for a first book on the subject. Beyond that, nothing is held back. This book includes a self-contained introduction to tensorial methods in chapter two, and tensors are used heavily in the remainder of the book, which makes this book much more suitable for anyone interested in studying general relativity than a book that tries to limp through the same subject matter using only vector methods.

In fact, all of the basic elements that are necessary for the study of general relativity are introduced in this book and in the simplest possible setting.

This book includes exactly 99 figures and a large number of examples which are extremely helpful in understanding the material and as other reviewers have remarked has numerous exercises with full solutions in the back of the book. There is also a collection of formulae at the end which makes for a good review and enhances the book's usefulness as a reference.

The definitions are explicit and the proofs are quite clear. However, the proofs do make references to the theory of differential equations and to results in complex variable theory in a couple of places.

Downsides? While the exposition is excellent, it is a bit terse. Towards the end, there is a lot of flipping back to look at referenced earlier formulas. In addition, small steps are omitted from many derivations. Also, there is a section on the Bergman metric that seemed completely tangential to the rest of the material in the book.

Here's a breakdown of the contents:

Chapter 1 is preliminaries. It provides a quick review of vector methods and fixes notation.

Chapter 2 is the theory of curves in the three dimensions. Topics include: arc length, the tangent vector, the principal normal vector, curvature, binormal vector, torsion, Frenet's formulas, spherical images of curves, the canonical representation of curves, orders of contact between curves, natural equations for curves, involutes and evolutes, and more.

Chapter 3 introduces surface theory and covers the first fundamental form, normals to surfaces, and an introduction to tensorial methods. This introduction is good, self-contained, and covers only the tensor calculus that is required for the rest of the book. Tensors are presented using index notation rather than the more modern -- and for me at least usually less clear -- abstact notation. The Einstein summation convention is introduced immediately and used throughout except in formulas where it is explicitly suspended.

Chapter 4 covers the second fundamental form, gaussian and mean curvature for a surface, Gauss' Theorema Egregium, and Christoffel symbols.

Chapter 5 is about geodesics and also covers the Gauss-Bonnet theorem.

Chapter 6 studies mappings and provides good coverage of various types of mappings of a sphere into a plane such as conformal and equiareal. It also covers conformal mappings of three space.

Chapter 7 discusses absolute differentiation and parallel transport. It also has a section on connections in general. Absolutely key material for understanding general relativity.

Chapter 8 tackles special surfaces such as minimal surfaces, modular surfaces of analytic fucntions of one complex variable, and surfaces of constant gaussian curvature.

This book absolutely requires a strong background in multivariable calculus and differential equations. In addition, some exposure to complex variables is recommended.

I strongly recommend this book for any scientist or engineer looking for an introduction to differential geometry. If this book proves to be too much, then I'd suggest looking at a book that makes ues of only vector methods for some additional background before returning to this book. Finally, the price is hard to beat!

a question about 2nd fundamental form mixed tensor
In the book 'differential geometry' by Kreyszig, a result is frequently used
about the principal curvature k1, k2. For example, we know that gaussian
curvature K=k1*k2.

When lines of curvature (curves with principal curvature as tangents)
coincide with coordinate curves, it can be shown k1 = b_1^1, the first
element of a mixed tensor with degree 2 and 1 covariance indice. (p.131)

The author then equate k1 = b_11/g_11, k2=b_22/g_22. And this result is used
in several places. Here is what I am having trouble with.

b_u^v = b_uc g^cv, by summation rules, my result is b_1^1 = b_11*g^11 = b_11
/g_22, not b_11/g_11...

The relevant text can be found on page 131, 137-139. Can someone help me out
with this? It doesn't look like the author is wrong but I can't figure out
why..

----

I am giving this book a 5 star because of the sheer intellectual wealth it possesses. There are 2 complaints, 1) typos, at times they really made me scratch my head and I had to find references to make sure it's a problem with the book, e.g. the definition of 'asymptotic curve'; 2) terseness, I wished the author could be more verbose at times to explain the geometrical implication of certain theorems, e.g. the Gaussian-Bonnet theorem, too much formula not enough explanations...

The best book of Differential Geometry ever written !!!
It is written in a clear cut and concise manner.
It is unlikely that another author will be able to write a better book in Differential Geometry. Guggenheimer's, also by Dover, is the second best but by far.

I have bought Advanced Engineering Mathematics by Kreyszig just because I was impressed by the way this author writes, but I was dissapointed, it was not that good.

I am an Engineer.
A fellow Engineer who doesn't like theoretical books may not share my point of view.
A mathematitian who is -as always mathematitians are- theoretical may also not share my point of view.

Old Fashioned
I read the first couple chapters, trying to learn differential geometry on my own.The approach that this text uses seems a bit dated.Most of the terminology used isn't frequent in modern math texts.If you're an undergrad and interested in the subject, I found that the Springer book by Andrew Pressley is a much nicer option for self-teaching.It has a modern feel to it, and all the exercises have hints or solutions in the back, so you can check your work, or get help when stuck.

Very well written and informative
If I hadn't seen much worse and somewhat better,
I would have given this five stars.
What it lacks is a good classification of curvature types, a discussion of Willmore surfaces, and solitons, but as an introduction it is pretty complete
and the price is very good. As a contrast to how bad such books can be I give the Link:Differential Geometry, Lie Groups, and Symmetric Spaces (Graduate Studies in Mathematics) ... Read more

Product Description

Curves and surfaces are objects that everyone can see, and many of the questions that can be asked about them are natural and easily understood. Differential geometry is concerned with the precise mathematical formulation of some of these questions. It is a subject that contains some of the most beautiful and profound results in mathematics yet many of these are accessible to higher-level undergraduates.

Elementary Differential Geometry presents the main results in the differential geometry of curves and surfaces suitable for a first course on the subject. Prerequisites are kept to an absolute minimum – nothing beyond first courses in linear algebra and multivariable calculus – and the most direct and straightforward approach is used throughout.

New features of this revised and expanded second edition include:

• a chapter on non-Euclidean geometry, a subject that is of great importance in the history of mathematics and crucial in many modern developments. The main results can be reached easily and quickly by making use of the results and techniques developed earlier in the book.
• Coverage of topics such as: parallel transport and its applications; map colouring; holonomy and Gaussian curvature.
• Around 200 additional exercises, and a full solutions manual for instructors, available via www.springer.com

Praise for the first edition:

"The text is nicely illustrated, the definitions are well-motivated and the proofs are particularly well-written and student-friendly…this book would make an excellent text for an undergraduate course, but could also well be used for a reading course, or simply read for pleasure."

Australian Mathematical Society Gazette

"Excellent figures supplement a good account, sprinkled with illustrative examples."

Times Higher Education Supplement

Customer Reviews (7)

Definately a good begineers book
If you want a very general introduction of Differential Geometry, this is the book to start. Very nicely written text. Understandable examples. Broad coverage of materials . Explains space curves and surface properties with amazing quality. Recommended as a beginners level introduction

Very appropriate for self-study
It's a very good book overall, especially if you like to spend more time reading on your own than in a classroom.

Written to teach rather than to impress
I have purchased hundreds of technical books and really treasure the ones that seem to have been written in order to really convey the material rather than impress the reader with how smart the author is. This is such a book. The material is remarkably clear and the author's style strikes me as a notable example of the mathematical writing styles put forth in the articles comprising the text "How to Write Mathematics." For example, the material proceeds in a logical chain such that the reader is never confronted with a term or concept before it has been explained. The notation is defined meticulously and repeatedly so the reader is not forced to continually refer backwards through the text to remember the meaning of the symbols. This also is a boon for "grasshopper readers" who will use the text as a reference, as opposed to a linear reader. Symbols don't change meaning, are not overloaded, and seem to have been chosen for intuitive appeal. For example, a lower-case gamma denotes a parametric function for a curve and, to me, the shape of the gamma suggests the sorts of curves being discussed. In my experience, this book is best in class.

An enjoyable text on the subject!
I've been looking for a decent book on differential geometry for years now.Most of the good ones are fairly pricey, or require the reader to have a deep knowledge of mathematics.This fits in neither category.You only need multi variable calculus, linear algebra, and some experience with reading/writing proofs.This book will also appeal to those who want to learn on their own, as every problem has a hint/solution in the back.

Dissapointing
The book starts ok, but very quickly deteriorates into the classical boring math style of theorem-proof. There are a million books on the subject matter, and I don't see the need of another one which is pretty much identical. It is not a bad book, but has absolutely no added value - just pick any of the differential geometry books out there, and they will be the exact same thing. Why do they bother writing the same book over and over?? ... Read more

Product Description
Intended for upper undergraduate or beginning graduate students, this book introduces students to the modern theory of manifolds. Assuming a basic knowledge of the differential geometry of curves and surfaces the focus is on differentiable manifolds and the study of Riemannian manifolds. The book concludes with applications of manifolds to physics. Exercises at the end of each section and appendices on topology and linear algebra make this book ideal for self-study or as a textbook. ... Read more

Customer Reviews (1)

Nice! Very clear, concise, rigorous reader-friendly introduction to differential manifolds
Lovett provides a very nice introduction to the differential geometry of manifolds useful for self-study.It is very clearly written, rigorous, concise yet reader-friendly.The difficulty level is midway between O'Neill's Elementary Differential Geometry, Revised 2nd Edition, Second Edition and Tu's An Introduction to Manifolds (Universitext) (Volume 0).

The pace is nice.As you can see in more detail from the "search inside this book" function: Ch. 1 Analysis of Multivariable Functions [pp. 1-36] provides some background math; Ch. 2 [pp. 37-78] Coordinates, Frames, and Tensor Notation discusses some more applied topics needed for physics applications; Ch. 3 Differential Manifolds [pp. 79-124] and Ch. 4 Analysis on Manifolds [pp. 125-184] discuss essential standard topics including differential maps; immersions, submersions and submanifolds; vector bundles; differential forms; integration and Stokes' Theorem;Ch. 5 [pp. 185-248] provides an introduction to Riemannian Geometry, including vector fields, geodesics and the curvature tensor; and finally Ch. 6 [pp. 249-294] provides very brief discussions of some applications to physics including Hamiltonian mechanics, electromagnetism, string theory and general relativity.

My main gripe is that there are no answers to problems, which detracts from its value for self-study (but to fill that gap, cf. Analysis and Algebra on Differentiable Manifolds: A Workbook for Students and Teachers). This is especially annoying because Lovett refers to answers to some problems in his mathematical exposition, e.g., on p. 234 (section 5.4.1), he refers to problem 5.2.17 on page 217 in his discussion of connections that are not symmetric; moreover answers to some exercises depend on material in other problems, e.g., the answer to problem 5.2.17 refers to problem 5.2.14.This is a common practice I dislike because it seriously degrades from a book's value for self-study. It could well be that one star should be deducted for this despicable practice.Nevertheless, I have given it 5 stars because I like the fact that it covers Riemannian Geometry (including an exposition of Pseudo-Riemannian metrics in section 5.1.4 and 5.3.3) and in section 6.4, a short introduction to general relativity but mostly because it's the only book I know that can help one make the leap from very elementary books like O'Neill's Elementary Differential Geometry, Revised 2nd Edition, Second Edition, Pressley's Elementary Differential Geometry (Springer Undergraduate Mathematics Series) or Banchof and Lovett's Differential Geometry of Curves and Surfaces to graduate level books like Tu's An Introduction to Manifolds (Universitext) (Volume 0), John Lee's Introduction to Smooth Manifoldsor Jeffrey Lee's massive [[ASIN:0821848151 Manifolds and Differential Geometry (Graduate Studies in Mathematics), all of which I also recommend after Lovett.

All in all, this text is a welcome addition to the many books on differential geometry because of its refreshing, "no nonsense" clarity, rigor and conciseness as well as the various topics covered. ... Read more

Product Description
Written primarily for students who have completed the standard first courses in calculus and linear algebra, ELEMENTARY DIFFERENTIAL GEOMETRY, REVISED SECOND EDITION, provides an introduction to the geometry of curves and surfaces.

The Second Edition maintained the accessibility of the first, while providing an introduction to the use of computers and expanding discussion on certain topics. Further emphasis was placed on topological properties, properties of geodesics, singularities of vector fields, and the theorems of Bonnet and Hadamard.

This revision of the Second Edition provides a thorough update of commands for the symbolic computation programs Mathematica or Maple, as well as additional computer exercises. As with the Second Edition, this material supplements the content but no computer skill is necessary to take full advantage of this comprehensive text.

*Fortieth anniversary of publication! Over 36,000 copies sold worldwide
*Accessible, practical yet rigorous approach to a complex topic--also suitable for self-study
*Extensive update of appendices on Mathematica and Maple software packages
*Thorough streamlining of second edition's numbering system
*Fuller information on solutions to odd-numbered problems
*Additional exercises and hints guide students in using the latest computer modeling tools ... Read more

Customer Reviews (10)

O'Neil
I found the product in excellent condition.
The mailing was also superb.

Thanks

Introduce Some Of The Main Ideas Of Differential Geometry
"This book is an elementary account of the geometry of curves and surfaces.
It is written for students who have completed standard courses in calculus and linear algebra, and its aim is to INTRODUCE SOME OF THE MAIN IDEAS OF DIFFERENTIAL GEOMETRY....."
[from the preface to the second edition, by Barrett O'Neill]

Definitely worth checking out
I found the approach in this book on the stuff touched on in most diff. geometry books at the level I was hunting for (I'm a mech. engineering major) very nice. I liked the notation used and could follow it very well. Everything I've seen so far is in E3 which I like for intuitive grasp. I feel like this is one of those bridging books to be able to understand higher level abstract books, especially if you are trying to learn on your own. The first 50 or so pages hammer out pretty much anything I've seen at the general level in other books of this subject, and I think this book does it better than most. There are also some solutions (well, just answers) in the back.

Introductory level text with empasis on intuition examples and exercise.
If you are looking for abstraction with little in the way of intuition I suggest Conlan " differential manifolds"

If you are an applied mathematician or physicist this book is for you.

I have always beleived that to truly grasp mathematics one must be provided with a reason for WHY things are the way they are and WHAT IDEAS the expression must express. This is best done with examples and exercises.

I digress.

The book restricts is exposition to two and three dimensions. Some of the topics can readily be bootstrapped to higher dimensions.

The book starts with basic ideas of curve, directional derivative and tangent vector in Euclidean space with a sprinkling of differential forms to wet the appetite.

It then moves into the notion of frame fields along curves resulting in the Frenet formulas which express how the frame fields change along the curve. These are expressed in terms of the frame field themselves giving ideas of curvature and torsion.

The book then abstracts these concepts to show how we can talk about change of frame fields along arbritrary directions not just along the curve. The tools used to do this are the covariant derivative and connection forms which can then be used to develop connection equations ( abstracted analogue of frenet formulas ) and then the cartan structural equations.

The book talks about isometries and defines euclidean geometry as those properties preserved by isometries. It then abstracts once again to surfaces in R3 using patches and appropriate conditions on the overlap without introducing manifolds although these are briefly mentioned later.

We then see how calculus in euclidean space can be adapted to surfaces using these patches. The corresponding concepts of function, differentiability and tangent vectors on these objects is introduced. Forms on these surfaces are introduced and their application to integration theory on these surfaces is developed showing how forms on the surface are " pulled back" to euclidean space using the idea of differential of a map and integrated there. The integration gives the volume ( area ) of that surface. Stokes theorem is introduced.

We now move into the idea of shape operators on the surface and show how these describe how the normal vector on the surface move in various directions giving ideas of mean and gaussian curvature . We see a very nice interplay of algebraic analysis leading to a geometric analysis.

The book then deals with studying geometrical properties on surfaces using the Cartan methods described earlier.

We then see how to define intrinsic geometry of any surface. Namely those properties of the surface that are preserved by isometries. From the definition of isometry we see that these rely on on the concepts of tangent vector and inner products. Shape operators and mean curvature are not intrinsic.

We now study the geometry of surfaces specifically the intrinsic geometry without reference to an imbedding space ( R3). An abstract "surface" is endowed with an inner product. A different inner product gives a different geometry. We talk about gaussian curvature and covariant derivative which are intrinsic.

Geodesics are introduced as is the gauss bonnet theorem which relates a geometric property to a topological one.

The book concludes with a chapter on global properties ( 2 d surfaces ) especially how gaussian curvature influences geodesics and how the two completely determine the geometry of the surface.

Cartan's formulation of differential geometry taken up here.
My first encounter with this book was during the academic year of 2000-2001, when it was used as the main text for an upper division course on differential geometry. The class --taught by a distinguished scholar-- was only meant to be a brief excursion into the realm of continuous math, beyond analysis and topology. After finishing the term however, I decided to change direction and as time went on, I drifted more and more towards geometry as the field of further concentration. The original second edition (from 1997) contained numerous typos, but luckily, the revised 2006 issue takes care of these and also streamlines the section numbering formats which had made the referencing and following through with the material a bit cumbersome. As some of the other reviewers have mentioned, the emphasis here is on the low (= 2 and 3) dimensional geometry, formulated in the language of differnetial forms (Cartan's early 20th century approach).

Within the eight chapters of the book (seven in the 1966 edition), the reader is first introduced to some preliminaries such as tangent vectors, directional derivatives, and differential forms. In chapter two, the author presents the Frenet frame formulas, covariant derivatives, connection forms, and Cartan's structural equations, which are generalizations of the Frenet frame formulas for surfaces. In chapters three and four, there is a healthy dose of Euclidean geometry and calculus on surfaces. In chapter five, discussion is on the study of the shape operators and normal and Gaussian curvatures, where also some useful computational examples have been presented. Geometry of surfaces is the subject of chapter six, where the crucial Gauss' egregium theorem is proved, and in chapter seven students are introduced to the basics of the Riemannian geometry, culminating in the famous Gauss-Bonnet theorem. In chapter eight (which is highly topological) complete surfaces, covering spaces, Jacobi fields, and the subject of classification of surfaces are explored. The appendices include help on using popular computer algebra systems (with updates in the latest revised edition), and another appendix providing solutions to many of the odd-numbered exercises in the book.

Please note that the author leaves out a discussion of several essential tools, for example, the Schwarz-Christoffel symbols, tensors, and Lie derivatives. The exposition does not fully explore some other important topics such as the first and second fundamental forms, and parallel translation, which only show up in the exercises. Then again, perhaps to keep the level of exposition elemantary and the size limited, Dr. O'Neill has preferred to skip some topics. One remedy is to back his text up with Manfredo Do Carmo's 1976 classic, which is mathematically more rigorous, and covers more of the above-mentioned topics. Afterwards, one can certainly continue the study of the essentials by reading other advanced material such as William Boothby's "An Introduction to Differentiable Manifolds and Riemannian Geometry". There is also a somewhat obscure title by Richard W. Sharpe with the title "Differential Geometry: Cartan's Generalization of Klein's Erlangen Program", from the Springer-Verlag GTM series that's worth checking into. Finally, other elemantary-level sources to keep in mind for a beginning student are the recent texts by Andrew Pressley (2001) and Wolfgang Kuhnel (2002) both available on amazon.com's catalog.

[Review updated in May 2006] ... Read more

Product Description
Excellent brief introduction presents fundamental theory of curves and surfaces and applies them to a number of examples. Topics include curves, theory of surfaces, fundamental equations, geometry on a surface, envelopes, conformal mapping, minimal surfaces, more. Well-illustrated, with abundant problems and solutions. Bibliography.
... Read more

Customer Reviews (5)

The most consistent reliableand readable so far
With this book, I hope I have finally broken the code and reached a critical mass in advanced mathematical understanding. These Dover Series books allow "it all to hang out." It is "old school" in the best sense of that phrase: that is, in the sense that they do no "sugar coat" their explanations. They do not "dumb it down" or "fancy it up to" ease the pain. One knows what one is up against when one picks up a book from the "Dover Series." They are always clean and sparse in their explanations.

In this regard, this book is no exception. Professor Struik begins at the beginning and goes straight through to the end without skipping any steps and without passing go to collect his $200. He gives the fundamental conceptions of the theory of curves and surfaces, introducing all of the machinery necessary to understand them in a graduated fashion suitable only to the requirements of the topic itself. Elementary calculus will serve the reader well, especially with a smattering of Linear Algebra thrown in. The author wastes no time with sexy side issues or superfluous explanations: Just the basic facts of the fundamental elements here. Those looking for more advanced topics, should consult those books that use this one as their background.

Explanations are sparse, but never deficient; the same is true of the equations. Notation is straightforward and always clear and economical. It is easy to see that(and why) other books on the same topic have used this one as background, but oddly, those other books have been unable to improve upon this one. Other than the fact that the graphics need updating, and more modern topics are missing, this is a splendid effort. Just what I needed.

Five Stars.

Very Readable Work on Classical Differential Geometry
While it is quite true Dirk Struik's work is on classical differential geometry, the older methods and treatment do not necesarily imply obsolescence or mediocrity as some readers or reviewers suggest in their evaluations.Classical Analysis is still an important branch of Mathematical Analysis.So classical approaches and topics should not be dismissed as a waste of time, useless, outdated or even invalid.Remember Andrew Wiles' recent attack on Fermat's Last Theorem and his ultimate proof of its validity, an event that made headline news.That is a quintessential classical problem in mathematics (i.e., in number theory), only recently resolved.So remember: CLASSICAL Differential Geometry is part of the title.

First of all, this book is very readable, being that it requires no more than 2 years of calculus (with analytic geometry and vector analysis) and linear algebra as prerequisites.Exposure to elementary ordinary and partial differential equations and calculus of variations are highly desirable, but not absolutely necessary.There are numerous carefully drawn diagrams of geometric figures incorporated throughout the book for illustration and, of course, better understanding.Topological methods are not used in the book, and the concept of manifolds not mentioned, much less treated.So this is an older work that bridges the very foundational and applied aspects of differential geometry with vector analysis, a field and body of knowledge widely used nowadays in the sciences and engineering and exploited in applications such as geodesy.For those insisting on modern approaches and want to omit studying foundations and historical development, please read up on other books such as O'Neill and Spivak.(Also, there are tons of other newer works, i.e., on "modern differential geometry", I am unfamiliar with.They are probably availble for browsing in college bookstores.)

The author begins by leading the reader from analytic geometry in 3-dimensions into theory of surfaces, done the old fashion or classical way, i.e., utilizing vector calculus and not much more.Along the way, he takes the reader through subjects such as Euler's theorem, Dupin's indicatrix and various methods for surfaces.Then he continues with developing important fundamental equations underlying surfaces, e.g., Gauss-Weingarten equations, looks at Gauss and Codazzi equations, and proceeds to geodesics and variational methods.He includes a somewhat detailed treatment of the Gauss-Bonnet theorem as he progresses.He ends up with introducing concepts in conformal mapping, which plays an important role in differential geometry, minimal surfaces and various applications, one of which is geodesic mapping useful in geodesy, surveys and map-making.He does all of it with clarity and focus, including problems or "exercises" as he calls it, in under 240 pages - brevity that is rare in many mathematical books and works these days.

For those with a mind for or bent on applications, e.g., applied physics (geophysics), applied mathematics, astronomy, geodesy and aerospace engineering, this book would be an excellent introduction to differential geometry and the classical theories of surfaces - being that one need not worry about abstract analysis and topological aspects of mathematics.Perhaps the title should be "Topics in Classical Differential Geometry" or "Introduction to the Theory of Surfaces in Classical Differential Geometry".But one must keep in mind that Dirk Struik is an old MIT hand and contemporary of Norbert Wiener, also at MIT, and Richard Courant (and many great German-educated mathematicians) who lived and worked in the early to mid-20th century, a long time ago and before computers became commonplace, an era in which total abstraction in mathematics and physics was not quite widely emphasized, but clear concrete thinking was important.A good friend of mine and co-worker who studied at the University of California, Berkeley, told me he had great respect for the classical geometers such as Struik and Eisenhart, understanding that they built ideas from a scatch and wrote in such a way that readers can discern the physical origins of geometry, in particular of differential geometry, a subject that supposedly started with Gauss during the early or mid-19th century when he performed survey work for his government in Germany.(The term "torsion" introduced and sed by Struik in the first few chapters of the book comes from classical mechanics, and is commonly employed in mechanical structures/structural engineering nowadays.)

I for one am an aerospace engineer.There were one or more occasions where I consulted the book for formulas and expressions of curved surfaces and spheroids in my work of flight navigation (flying over the ellipsoidal Earth, as one example).I am sure that are other areas, e.g., space engineering, where classical methods of differential geometry embodied in Struik's book can come in handy.

The only problem I have with the book is that the "exercises" do not come with solutions, but I do not think that is a major drawback unless one uses it as a textbook for a course that requires assignments and drill exercises.

Judge for yourself by borrowing this book to read, i.e., if you are interested, can tell whether you like or dislike it on the first pass, and for what reasons one way or another.Find out for yourself.

classical
This is a survey of classical i.e. early 20th century differential geometry and not a more "modern" abstract treatment.

Good treatment of classical differential geometry
Struik's book provides solid coverage of curve and surface theory from the classical point of view, i.e. the kind of stuff Monge, Serret, Frenet and Gauss did. I agree that the book should be on the shelves of mathematicians. A number of classical topics are simply not in vogue these days, and one can find them discussed at length in Struik, or in the exercises. In this sense the book certainly has a more geometric flavor than a number of contemporary texts.

However, Struik can't be used to understand what is happening today. For these purposes,books by O'Neill and do Carmo would be more appropriate. The discussion of manifolds and coordinate charts, the discussion of connection forms, differential forms, covariant derivatives, exterior derivatives, pullbacks and pushforwards can be found in these texts. This is the language of modern geometry.It leads on naturally to tensors, fibre bundles, de Rham cohomology and so on and so forth.The emphasis in modern geometry is on global phenomena, the interaction between local and global (e.g. Morse theory or De Rham cohomology), and the attempt to do everything in an algebraic setting (projective modules, spectral sequences, categories etc.) For this purpose, Struik is useless, though he does have some coverage of forms (he calls them by their earlier name of 'pfaffians').

The price of the book makes it an attractive purchase.

Struik's book - a classic on classical differential geometry
I simply cannot believe I am the first reviewer of this book!This book should be on the shelf of every mathematician interested in geometry, every computer graphics specialist, everyone interested in solid modelling.Forten bucks, you get a great summary of a wide range of topics in"classical differential geometry" -- the stuff geometers wereinterested in one hundred years ago.Today it's gauge and string theory --but the topics discussed in this book are timeless, and many have seenremarkable renaissances in recent years. It is a wonderful little book ...I am using it to teach a basic differential geometry course next year. ... Read more

Product Description
Differential geometry is a major field of mathematics that uses tools from calculus, in particular integrals and derivatives, to study problems in geometry. Differential geometry has applications in several fields, including physics, economics, engineering, and computer vision. This book focuses on the geometric properties of curves and surfaces, one- and two-dimensional objects in Euclidean space. The problems generally relate to questions of local properties (the properties observed at a point on the curve or surface) or global properties (the properties of the object as a whole). Some of the more interesting theorems look at how local properties relate to global ones. A special feature is the availability of accompanying online interactive java applets coordinated with each section. The applets allow students to investigate and manipulate curves and surfaces to develop intuition and to help analyze geometric phenomena. Each section includes numerous interesting exercises that range from straightforward to challenging. ... Read more

Product Description

This volume covers local as well as global differential geometry of curves and surfaces.

... Read more

Customer Reviews (21)

Highly recommend
I really like Do Carmo's style of presentation, I can just imagine what type of lecturer he was. Though there are other books on the subject that are more concise a/o cover more topics, I believe Do Carmo's treatment is excellent for beginners to the field.

Another plus is that he gives hints on how to solve some of the exercise problems, which is essential for someone like me who very often studies a subject on my own.

A must of its kind
This is an excellent book. Anyone who wants to have a global view on classical differential geometry must have it (and read it!). As an undergraduate course's textbook, I find it quite difficult for the not determined student. Nevertheless, it gives the teacher the grounds to present a rigorous course. I fully recommend it to both students and teachers.

Good content, cheaply manufactured
The content is well-presented and instructive. My main gripe with this book is the very low quality paperback edition. After just a month of (careful) reading, many pages already falling out. The cover and content look as if they have been scanned and re-printed.

Update: by the end of the semester, my copy has now completely fallen apart. I wish this hadn't been manufactured so cheaply.

classical
It's a classical book on diff. geo. But it seems that the torsion in this book is different by sign from the notion in many other books

Great Potential - If Only a 2nd Edition Were in the Works
An earlier review said this book has few errors, and even then only typographical ones. Are we talking about the same book? The text is pockmarked - nay, cratered - with scads of dire gaffes. The skeptical empiricist should go to Google and enter these keywords: bjorn carmo errata. The first hit will be a link to a 7-page pdf file a U.C. Berkeley professor and his students created a few years ago which compiles errata they turned up. Seven full pages, and they only covered a third of the text! A sample item in the list: "p. 97, definition of domain: It is not clear whether the boundary is the boundary as a subset of R^3 or the boundary as a subset of S. Either way, we run into trouble..." The Heine-Borel theorem on page 124 is so botched up it's beyond repair, and even the basic definition of what it means for a function to be continuous on a set is faulty (p. 123).

The author claims a student should be able to hack the material with"only the most basic concepts" from linear algebra and multivariable calculus. Largely but not entirely true. For example, you better be up to speed on linear mappings defined by NON-square matrices - something no undergraduate-level linear algebra book in my library discusses (though I only possess a handful). Many of those tidy little results for linear operators from R^n into R^n you might know from Linear Algebra 301 become worthless when one of those n's becomes an m. I don't really fault the author for this, but anyone thinking about acquiring this text should know it is not by any stretch "self-contained" as one previous reviewer stated.

The biggest irritant with this text is the constant abuse of notation. When you're just starting out trying to learn this stuff, it most emphatically does NOT help when the author keeps butchering or truncating the notation in the interests of "brevity". For example, entire derivations are often carried out using only the names of functions and not their arguments. Maybe I have a screw loose, but sometimes I find it really helps knowing that f is really f(x) and g is really g(y). And then in a single section the same symbol, N, is used to denote three different functions. Okay: this N really means N composed with alpha, and that N is N composed with the parametrization x, and this other N is really N all right, but when you stick q into it you really mean x inverse of q because N's domain is a plane, not three-dimensional space...yeah...oh wait...

Over the course of a semester I wasted uncounted hours unraveling DoCarmo's infuriating and uncalled-for notational "short-cuts".

So why do I give this text 3 stars? Well, in using it I still managed to master the core concepts quite nicely, so it had to be doing something right. Yes, the exercises are challenging, but I was able to crack most of the ones I attempted. Maybe a problem would take me 8 hours to do, but I could do it. I would not say this book is "dated", as one reviewer put it. In fact, I found most of the minimalist, non-computer generated figures to be refreshing and adequate. Anyone seeking cookbook methods for computing things will be sorely disappointed: this text is written largely for students of mathematics, and I have no qualms with that (being a math student myself).

Truly, if this text were given a buff and polish, it could become 5-star material (in my opinion anyway). Alas, that's probably not in the cards. ... Read more

Product Description
This is the new edition of Serge Lang's "Differential and Riemannian Manifolds." This text provides an introduction to basic concepts in differential topology, differential geometry, and differential equations, and some of the main basic theorems in all three areas: for instance, the existence, uniqueness, and smoothness theorems for differential equations and the flow of a vector field; the basic theory of vector bundles including the existence of tubular neighborhoods for a submanifold; the calculus of differential forms; basic notions of symplectic manifolds, including the canonical 2-form; sprays and covariant derivatives for Riemannian and pseudo-Riemannian manifolds; applications to the exponential map, including the Cartan-Hadamard theorem and the first basic theorem of calculus of variations. ... Read more

Customer Reviews (1)

Complete but lacking unity
This book looks more like a big collection of essays than really a treatise on one subject. If you have read other books by Lang you will notice that some material included in previous works appears reproducedhere word by word. Also, there are some inconsistencies in the notation ofdifferent sections, making it obvious that different parts of the book werewritten at different times, and perhaps by different persons.

All theseshortcomings don't mean that the book is bad. Quite the opposite: It is avery complete survey on modern differential geometry, including from thefundamentals up to recent results. The graduate student and the workingmathematician will find it very useful. ... Read more

Customer Reviews (3)

Long Journey - no damages
+: Product arrived without any damage.
+: Arrival date long before the promised, specified date
-: none

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.
... Read more

Product Description

The modern subject of differential forms subsumes classical vector calculus. This text presents differential forms from a geometric perspective accessible at the sophomore undergraduate level. The book begins with basic concepts such as partial differentiation and multiple integration and gently develops the entire machinery of differential forms. The author approaches the subject with the idea that complex concepts can be built up by analogy from simpler cases, which, being inherently geometric, often can be best understood visually.

Each new concept is presented with a natural picture that students can easily grasp. Algebraic properties then follow. This facilitates the development of differential forms without assuming a background in linear algebra. Throughout the text, emphasis is placed on applications in 3 dimensions, but all definitions are given so as to be easily generalized to higher dimensions. A centerpiece of the text is the generalized Stokes' theorem. Although this theorem implies all of the classical integral theorems of vector calculus, it is far easier for students to both comprehend and remember.

The text is designed to support three distinct course tracks: the first as the primary textbook for third semester (multivariable) calculus, suitable for anyone with a year of calculus; the second is aimed at students enrolled in sophomore-level vector calculus; while the third targets advanced undergraduates and beginning graduate students in physics or mathematics, covering more advanced topics such as Maxwell's equations, foliation theory, and cohomology.

Containing excellent motivation, numerous illustrations and solutions to selected problems in an appendix, the material has been tested in the classroom along all three potential course tracks.

Customer Reviews (7)

Kindle version is acceptable, even if not perfect
The Kindle edition is indeed not ideal, but I would still recommend it. The math fonts are not perfect, but I was able to read it just fine. Occationally there are formulas that are not clear, but I was able to interpret them. I read this entire book on an IPOD touch with no real problem, so it should be better on larger devices.

The book itself (in any format) is good and I recommend it. Personally I used it to get a better physical understanding of differential forms to aid in my study of differential geometry. The book delivers on its promise to provide clear descriptions and explanations. It does provide a real geometric (and physical) understanding of the subject.

Not on the Kindle
DO NOT buy the Kindle edition of this book.You will be wasting your money.The mathematical fonts are bitmapped and almost unreadable.Amazon needs to fix this problem.Buy the print edition.

A painless path to some important results
I think the usefulness of this book will depend a lot on the reader's background and goals.

The subject of differential forms was one of the gaps in my otherwise strong math background.More than once I started reading a differential geometry text and found myself bogged down in definitions.I chose this book in the hope of being quickly "brought on board".I was not disappointed -- in a few weeks, I finally understood what the generalized Stokes Theorem was.

In my case I had a background in multivariate calculus, so skipped the initial chapters.It is not clear to me how useful this book would be to someone without that background.

I feel there is one big point that the author does not adequately emphasize: a large part of the motivation for differential forms is their independence of coordinate system.The large number of numerical examples, while quite helpful, tend to obscure this point.On the other hand, to elaborate this point might have involved so much formalism as to lose me like the other books did.

A very nice introduction to forms
This, and another book I will mention shortly, is where I learned about differential forms. The author spells out the fundamentals of differential forms in a very friendly and very geometric and intuitive way. I'm one of those people who managed to collect a ton of books on the subject and couldn't really made sense of forms - that is, until I started this one (Flanders, Bishop & Goldberg, do Carmo, Cartan, etc... sound familiar?).

First of all, it must be noted - this is a very simplified version of what Valdimir Arnold covers in his book titled "Mathematical Methods of Classical Mechanics" - the other book where I learned about forms. Bachman takes chapter 7 of Arnold's book and translates it into "english" ... or math that the rest of us can understand (Arnold's book is even cited in the bibliography of Bachman's book). Arnold can be a little confusing at first - this book is a very very welcome addition to my library and a very welcome companion to Arnold's book.

All the problems in this book either have answers in the back or enough hints for you to get through it painlessly. It took me about 3 - 4 days of non-stop reading to get through all the problems (I was on summer break when going through this book).

As one reviewer mentioned - this isn't a thorough book on forms, you won't learn all the algebraic details. You will get a hint of it's application to manifold calculus - for these I might recommend Morita's book titled "Geometry of Differential Forms". For the physicist I might recommend (after this one and Arnold's book) Frankel's "The Geometry of Physics", which goes into much more. I also am enjoying Schutz's "Geometrical Methods of Mathematical Physics" after learning what forms are and how to use them - the former is shorter and gets to the gist much quicker and not entirely as rigorous.

After reading this book, and then Arnold's, I felt gypped! This stuff is so simple and an almost obvious extension of multivariable calculus! Why are people complicating forms so much?

A good place to start and objective accomplished
I am a graduate physics student who as such, has got a prior and long exposure to vector calculus and I was searching for a good intuitive exposition to the subject of differential forms. I also had this objective: To finally understand how the fundamental theorem of calculus, Green's theorem on the plane, Gauss Theorem of the divergence and the stoke theorem of the curl in vector calculus all arise and where diferent faces of ONE SINGLE FORMULA, namely: The "generalized" Stoke's Theorem of differential forms. I must say before anything else that after reading this book the objective was accomplished.
I have found this text to be a very nice introduction to differential forms. I read it in just two weeks starting from chapter 3 to 9 (The book has 9 chapters and an Appendix), I didn't bother with the first two chapters which are a review of multivariable calculus (Calculus III).
The chapters are as follow:
1-Multivariable calculus, 2-Parameterizations, 3-Introduction to forms, 4-Forms, 5-Differential forms, 6-Differentiation of forms, 7-Stokes' Theorem, 8-Applications, 9-Manifolds, A-Non-linear forms
I list now some of the good features that I have found about this book:
i)-The author does a very clear presentation of each topic and gives plenty of intuitive explanations.
ii)-It is suited for undergraduates.
iii)-The book and therefore, the chapters are short making it an even easier reading.
The bad features (reasons for why I gave it only four stars):
iv)-Chapter 9 is different from all previous chapters, is harder and explanations aren't clear, the only drawback of the book (or perhaps is just me). To cite an example of this: The definition of a pull-back of a differential form or the section on quotient spaces.
But nevermind all in all, a great place to start. ... Read more

Product Description
Presenting theory while using Mathematica in a complementary way, Modern Differential Geometry of Curves and Surfaces with Mathematica, the third edition of Alfred Gray’s famous textbook, covers how to define and compute standard geometric functions using Mathematica for constructing new curves and surfaces from existing ones. Since Gray’s death, authors Abbena and Salamon have stepped in to bring the book up to date. While maintaining Gray's intuitive approach, they reorganized the material to provide a clearer division between the text and the Mathematica code and added a Mathematica notebook as an appendix to each chapter. They also address important new topics, such as quaternions.

The approach of this book is at times more computational than is usual for a book on the subject. For example, Brioshi’s formula for the Gaussian curvature in terms of the first fundamental form can be too complicated for use in hand calculations, but Mathematica handles it easily, either through computations or through graphing curvature. Another part of Mathematica that can be used effectively in differential geometry is its special function library, where nonstandard spaces of constant curvature can be defined in terms of elliptic functions and then plotted.

Using the techniques described in this book, readers will understand concepts geometrically, plotting curves and surfaces on a monitor and then printing them. Containing more than 300 illustrations, the book demonstrates how to use Mathematica to plot many interesting curves and surfaces. Including as many topics of the classical differential geometry and surfaces as possible, it highlights important theorems with many examples. It includes 300 miniprograms for computing and plotting various geometric objects, alleviating the drudgery of computing things such as the curvature and torsion of a curve in space. ... Read more

Customer Reviews (3)

Modern Differential Geometry of Curves and Surfaces with Mathematica, Third Edition
I have purchased the third edition of Modern Differential Geometry of Curves and Surfaces with Mathematica.
I have seen the second edition before and I think that second edition is better than the third edition.
If you plan to buy this book and I could give you an advice: I would say buy the second edition.
The second edition has some chapters, Mathematica algorithms and appendices, which are not in the third.

Eu comprei a terceira edição do livro "Modern Differential Geometry of Curves and Surfaces with Mathematica".
Eu vi a segunda edição antes e penso que a segunda está melhor que a terceira.
Se você planeja comprar este livro eu poderia dar-lhe um conselho: diria para comprar a segunda edição.
A segunda edição tem alguns capítulos, algoritmos do Mathematica e appendices que não estão na terceira edição.

Very comprehensive, modern textbook and reference
I have nine differential geometry textbooks, and this one is my favorite. Compared to the other classics (such as those by do Carmo and Kreyszig), this book is far more comprehensive, practical. and as the title of the book suggests, modern. It is more computational oriented than almost all other differential geometry books. It covers some interesting subjects (such as canal surfaces) that other classics are lacking.

Posted notebooks not current.
The book has merit, but, as of 7/1/2008, the notebooks posted on the publisher's web site have not been updated for Mathematica 6. ... Read more

Customer Reviews (3)

10 stars ....
I've read (attempted to read) a number of books on general relativity, including the elementary texts (Schutz, others), the introductions for mathematicians (Lee), books on tensors, web intros, .... and the only one that I've been able gain any traction with is this book.And, I'm not exactly unprepared, as I have a MS in math and have worked a number of years as an engineer, much of the time doing inertial nav and other presumably related work.Yet, the other GR books have completely baffled me.Warning ... if you start a book and there is a long section on tensors .... look out... or you'll soon be playing a game of 'find the pea (relativity)' and it will be under plenty of thick mattresses (tensor analysis being but one).This book develops all the machinery using a 2-d surface embedded in 3-d.This is the way to go, no question in my mind.There is a degree of concreteness to counter the endless poliferation of new symbols and notation.Then there is a good (could have been better see the Calculus Without Tears web page) intro to special relativity.And then .... before you've finished breakfast ... field equations... which I'm working on now. If not for this book, I might have given up on my latest hobby, GR.

For example ... spacetime is curved?Yes.Hard to visualize?No.Throw a ball ... look at the arc .... there it is.Calculate the curvature?Easy... no tensors, field equations, differentials.Of course, even Faber waits till page 274 for this little demonstration, and he could have put it on page 1... still, it's there.

Highly recommended. Shame about the price
The first part of this book is a lucid introduction to classical differential geometry from the Frenet formulas to Riemannian manifolds, via Gauss curvature, metric coefficients, connections, geodesics and the curvature tensor, with many well-motivated examples and exercises. If you have a working knowledge of basic linear algebra and multivariable calculus you should have no trouble with any of this.

The remainder (two-thirds of the book) provides one of the most readable introductions to special and general relativity that I have ever come across. Because of the geometric approach, modest prerequisites and limited space, there is no treatment of relativistic mechanics, electrodynamics or the 'matter' field equations, and actually no formal development of tensor calculus. Within these constraints, it is amazing what the author does manage to cover. You will be led easily through the vacuum field equations, Schwarzschild solution, perihelion precession, light-bending ...

If this book had been twice the length - more comprehensive but in the same style - it could have been a classic. As it is, it does not seem to be anywhere near as widely used as it deserves to be .. Surely this is a prime candidate for a ..Dover edition? Until then, get it from the library.

A very accessible book - well worth a look
This book comprises two sections. The first section develops the tools of classical differential geometry with a thorough treatment of 2D surfaces embedded in a 3D space. First and second fundamental forms are introduced and their relationships analysed. This leads into the usual realms - Christoffel symbols, connections and the Riemann Curvature Tensor.A very readable account of Gauss's Theorem Egregium is presented - it's great!. Finally, more abstract manifolds are presented.I worked thoroughly through the first half of this book.There are many good exercises.This volume is cited a number of times In Gregory L. Naber's 'Spacetime and Singularities'- a more difficult book. I found Faber's volume of real help when studying the latter(which has loads of exercises and problems but no hints/partial solutions at all). The second section is an introduction to Special and General Relativity. I have to confess I did not study this half as I had already covered the material presented in this section elsewhere. The first half of the book gives a very good feel for curved space and is fitting preparation for the second half if the material presented there is new to you.. A good little book for independent study. ... Read more

Product Description
Differential geometry plays an increasingly important role in modern theoretical physics and applied mathematics. This 2006 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 (5)

Excellent reference for self study
An excellent reference for self-study. Four stars not five, because contrary to its claim, a reader with an undergraduate physics background cannot read it from the start to end without referring to other books. I decided to learn some General Relativity after hearing Smolin talk better smack than Triple H, and encountering Penrose's intriguing Road to Reality. Fecko logically and succintly weaves together many possible views of each subject he discusses. He clarified for me, for example, the links between the approaches taken by the texts of d'Inverno and Ludvigsen. Many of these links are given as well-structured exercises, so the book is best used when one has an uneasy suspicion that something might be true. Fecko also gives outstanding motivations and intuitive pictures for many definitions. Even after I hadunderstood pull-backs and differentials, it was a delight to discover that putting a shoe on my foot was as good as putting my foot in the shoe.

Fairly good content, very bad exposition
There's no doubt about it: the material in this book is incredibly interesting and important for an ambitious physics student. The organization of the book is fairly good: informal passages relating the necessary theory alternate with exercises which are all written as "Check that..." or "Prove that...", which allows you to choose which results to prove and which to take as given facts if you -- for any reason -- don't feel like proving them.

However, the book also has some serious shortcomings. The most important one seems to be the horrid style. A book of mathematics for physicists should not be written just like a standard math textbook without proofs -- and this is exactly what this book is like. The definitions that are given are "mathematical" at heart; very rarely can one find an intuitive picture of what is going on immediately after a concept is introduced. On the other hand, the propositions that are not left as excercises are never proven. Granted, they might be intuitively clear, but that doesn't mean that their proofs are obvious. Due to all this, I have always felt a bit confused and certainly not comfortable with new concepts. The author's occasional attempts to "raise morale" by inserting jokes would always backfire because these jokes are so trivial that they seem offensively condecending. Take, for example, the sentence that finishes the introduction of a vector as the equivalence class of tangency of curves:

"And a good old arrow, which cannot be thought of apart from the vector, could be put at P in the direction of this bunch, too (so that it does not feel sick at heart that it had been forgotten because of some dubious novelties)." (p. 25)

So... first of all, this is probably not particularly funny. But more seriously: are we to conclude that the notion of vectors as "directed lines" is important only because otherwise the "good old arrow" (and the reader alike) would feel "sick at heart"? This is an example of a concept so intuitive that a joke like this is generally harmless; however, trouble arises when the same kind of explanation is applied to more abstract concepts (e.g. why not study non-Hausdorff spaces? The explanation given on p. 4 relates to Amazon Basin Indians).

Another important issue is that a large part of this book teaches you the principles of the mathematics behind the physics. This is fine, provided you learn how to operate with these principles; however, the book seldom teaches you how to *work* with the most basic concepts, and that's what the author promises to deliver in the preface.

Unfortunately, there are other issues as well. Introducing new, vital ideas in exercises *only* is one of them. Also, one would desire to know which ideas are crucial or well-worth meditating upon, and this is generally not given in the text. Finally, the excessively informal style prevents this book from being even a good reference.

All in all -- it is possible to learn a lot of new things from this book, but the effort probably isn't worth it.

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.



... Read more

Customer Reviews (10)

Vol 1 of a series -- sets a high mark to follow!
I see we have 3rd edition now. I got 1st.

SO, I got it while it was hot!

So many pages that some folks will be put off. Exquisite writing however. See also his elementary book called Calculus. Bingo, you'll learn to appreciate him!

One of the best for Differential Geometry
Spivak's explanation to hard stuff is awesome. Sometimes daily chat language is used, he gave satisfactory and very good examples. I like it.

Charming
A lively, terribly ambitious tome on differential geometry.It was meant as a guided tour through the jungles of geometry, from a historical perspective.It is neither easy to read nor altogether successful in it's aim, but it IS comprehensive, masterful, and absolutely unlike all the others.It's kind of a legend since virtually every mathematician seems to own a copy.Full of pictures and history.Reads like a novel.

Not the best
Spivak's text gets a lot of good reviews, and it is a fine text.In fact, it's one of the best I've ever seen.Read a few other books on the subject, and you'll agree that this is a massive improvement on them.So why only 3 stars?Because there's a much better text on the subject:John Lee's "An Introduction to Smooth Manifolds".This book outshines Spivak's in so many ways.Sure, Spivak is great at motivating major developments in the theory (for instance, he really helps you understand why we need to define a tangent space and why it is the way it is), but he fails pretty bad when it comes to developing some actual theory.

Reading Spivak's text is like taking a stroll, a fresh break from the usual mathematics textbook style.But you also hit a bunch of brick walls on this stroll.It'll be a great discussion, and then you'll come to a theorem.You'll have no idea what its for (some of the time) and you'll struggle to work through its proof (most of the time).Furthermore, the organization is... well, there is no organization!As a result, Spivak can seem to droll on.Lee isn't as good at giving the overall big picture as well as Spivak, but he does everything else exceptionally.Leave Spivak for bed time reading, but do your real studying out of Lee.

Great book for amatures
If you want a book that is rich with examples then this is it.The proofs are, for the most part, clear and concise, thus a person who is learning the material without the aid of an instructor can follow the logic.However, the author could have spent some more time developing topological ideas (thought he does have an appendix section that does a fair job of it) within the flow of the first chapter.I personally find appendices to be too distracting and tend to slow down the flow of the material in a particular chapter.Other than that, this is a great book if you want to learn differential geometry and the theory of smooth manifolds. ... Read more

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

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

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

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

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

Customer Reviews (3)

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

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

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

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

Product Description
Our first knowledge of differential geometry usually comes from the study of the curves and surfaces in $I\!\!R^3$ that arise in calculus. Here we learn about line and surface integrals, divergence and curl, and the various forms of Stokes' Theorem. If we are fortunate, we may encounter curvature and such things as the Serret-Frenet formulas. With just the basic tools from multivariable calculus, plus a little knowledge of linear algebra, it is possible to begin a much richer and rewarding study of differential geometry, which is what is presented in this book. It starts with an introduction to the classical differential geometry of curves and surfaces in Euclidean space, then leads to an introduction to the Riemannian geometry of more general manifolds, including a look at Einstein spaces. An important bridge from the low-dimensional theory to the general case is provided by a chapter on the intrinsic geometry of surfaces. The first half of the book, covering the geometry of curves and surfaces, would be suitable for a one-semester undergraduate course. The local and global theories of curves and surfaces are presented, including detailed discussions of surfaces of rotation, ruled surfaces, and minimal surfaces. The second half of the book, which could be used for a more advanced course, begins with an introduction to differentiable manifolds, Riemannian structures, and the curvature tensor. Two special topics are treated in detail: spaces of constant curvature and Einstein spaces. The main goal of the book is to get started in a fairly elementary way, then to guide the reader toward more sophisticated concepts and more advanced topics. There are many examples and exercises to help along the way. Numerous figures help the reader visualize key concepts and examples, especially in lower dimensions. For the second edition, a number of errors were corrected and some text and a number of figures have been added. ... Read more

Customer Reviews (5)

elegant work
The author gives a clean and wise introduction to the three major parts in differential geometry-curves-surfaces-manifolds. The important concepts in classic results were introduced by short but fully content paragraphs.

The author wrote no gossip in the context and always touch the ideas with a niddle; therefore I should follow that:

This is the best book for introducing differential geometry.

Fast moving
This is a very fast moving book, covering a huge amount of material at a fairly sophisticated level in under 380 pages. For example, differential forms are introduced in about 2 pages so that the Maurer-Cartan structural equations can be defined.The first 4 chapters makes up a very concise course in curves and surfaces, while the last 4 chapters cover Riemannian geometry.In comparison, do Carmo's two books take 500 pages for the former and 320 pages for the latter.

For this reason I think the claim that this could be used as an undergraduate text is overly optimistic.For that I would use a more self-contained text like Millman & Parker (ISBN: 0132641437).But it would make an excellent text for a graduate survey, or as a second text for someone wanting to make the transition from classical theory (learned from, say, one of the Dover books like Struik, ISBN: 0486656098) to more modern methods.Also, you'll probably want to supplement with a gentler introduction to differential forms.

Of interest to students of physics, the book covers curves and surfaces in Minkowski space, as well as Einstein spaces.

A excellent introduction for the 21st century
While there is exist many classic texts on differential geometry, I have particularly appreciated this book for its up-to-date treatment, numerous well-done figures, broad coverage, elegant type-setting, and clear expositions. The book covers all the basics expected from an introduction to differential geometry, including curves and 2-D surfaces, but with a look towards the more advanced material in the second half of the book. It alternates between Ricci style notation and Koszul style notation, often carefully explaining the relation between the two and giving examples (I found this particularly helpful). There are, however, some sections where the english is a bit rough (perhaps the fault of the translator). It is also quite brisk throughout, often mentioning advanced topics before they are treated in detail. For example, it already mentions submanifolds, tangent spaces, and tangent bundles in the first chapter on "Notations and Prerequisites from Analysis." It will require serious attention, especially if one has not encountered a good dose of abstract mathematics before. Nonetheless, I have found myself returning to it over several years as an excellent reference and source of many additional topics that I skipped on a first reading. For example, the final chapter on Einstein spaces is a valuable, though demanding, bonus. Thanks to the AMS for publishing a fine edition of a top-notch German author's work.

A beautiful geometry
This book is very useful for students who are interested in geometry. The book is organized from elementary facts to advanced geometry very well. This book provides to students thereason why they study the geometry. This book explains very easily that the geometry of curves and surfaces can be generalized to high dimensional Riemannian manifolds naturally.
Moreover, the edition of this book is very beautiful and helpful for readers. For example, the important results are placed in boxes.

Attractive book on differential geometry
Differential geomety is perhaps the most beautiful part of higher mathematics. It combines geometry, analysis and intuition in a wonderful way. This attractive book is a concise and modern book that manages to be both pedagogical and accurate in a pleasant way. In only 350 pages most of the differential geometry that a non-expert will ever need is outlined. Illustrations and notation seem optimal for their purpose. The book is a worthy successor of classics like Struik, Stoker, and Kreyzig. ... Read more

Product Description
This text has been adopted at:

University of Pennsylvania, PhiladelphiaUniversity of Connecticut, StorrsDuke University, Durham, NCCalifornia Institute of Technology, PasadenaUniversity of Washington, SeattleSwarthmore College, Swarthmore, PAUniversity of Chicago, ILUniversity of Michigan, Ann Arbor

"In the reviewer's opinion, this is a superb book which makes learning a real pleasure."

- Revue Romaine de Mathematiques Pures et Appliquees

"This main-stream presentation of differential geometry serves well for a course on Riemannian geometry, and it is complemented by many annotated exercises."

- Monatshefte F. Mathematik

"This is one of the best (if even not just the best) book for those who want to get a good, smooth and quick, but yet thorough introduction to modern Riemannian geometry."

- Publicationes Mathematicae

Contents: Differential Manifolds * Riemannian Metrics * Affine Connections; Riemannian Connections * Geodesics; Convex Neighborhoods * Curvature * Jacobi Fields * Isometric Immersions * Complete Manifolds; Hopf-Rinow and Hadamard Theorems * Spaces of Constant Curvature * Variations of Energy * The Rauch Comparison Theorem * The Morse Index Theorem * The Fundamental Group of Manifolds of Negative Curvature * The Sphere Theorem * Index

Series: Mathematics: Theory and Applications ... Read more

Customer Reviews (8)

Excelent book.
I studied the portuguese version of this book during the master degree program in mathematics at University of Brasilia, 1999. The book is very well written with beautiful results. Manfredo is an excellent mathematician, a great professor, and I had the chance to be present in many colloquiuns where he was the speaker. This is an very good exposition for those interested to learn more about the subject.

Needs a table of symbols
This is another well-written text by Do Carmo. I browsed through it and found I could not understand several passages because I did not know what the special symbols meant and there was no table of symbols. I plead with the publisher to add such a table to the next edition or printing.

Concise and clear
This is really a very good book to start Riemannian Geometry (RG). Exposition of key concepts of RG (affine connection, riemannian connection,geodesics, parallelism and sectional curvature, ...) are well motivated and concisely explained with numerous motivating and not so difficult execises. The book is self contained convenient for self study. It contains an introductory chapter on mathematical background explaining basic concepts as differentiable manifolds, immersion, embedding and so on, which are necessary to deal with RG. I have essentially one basic remark about this book. Formulation of RG as presented in it, is a little bit dated. Now, with the development of geometric algebra and Geometric calculus most, if not all, mathematical concepts needed to study RG like covariant derivative, curvature, and general tensors can be formulated without ressort to coordinates and in a manner to highlight their essential geometric features. Moreover derivation of certain formulae can be much easier and natural. For example the author defines the formula for |x^y| as sqrt(sqr(|x|).sqr(|y|)-sqr(inner product(x,y))). Then explains that it is the area of two dimensional parallelogram determined by the pair of vectors x and y. The reader might be puzzled as to how this formula is obtained. In the context of geometric algebra this is derived very naturally from basic concepts. Anyway, this remark does not diminish the value of this book.

Best 1st semester Riemannian Geometry book after 1 semester DG
This is the best Riemannian Geometry book after students have finished a semester of differential geometry.It gives geometric intuition, has plenty of exercises and
is excellent preparation for more advanced books like Cheeger-Ebin.

Students should already know differential geometry (Spivak "Calculus on manifolds" and Spivak "Differential Geometry Volume I" might be used there)

Warning: the curvature tensor is defined backwards as compared to Cheeger-Ebin.

Definitely a good start
This book is definitely a solid way to start in Riemannian geometry. The topics chosen give a glimpse of more advanced topics that the reader can venture to next, and the order covered leaves little confusion. The book is to the point, with little conversation about the concepts except at the very beginning of each chapter.

I only have two complaints, but neither would cause me to lower the rating to 4 stars.

1. There could be more "deep" exercises that allow the reader to explore more of the subtleties of the subject. And for what exercises there are, the author sometimes gives far too much away in "hints."

2. The book does not take a unified approach to the subject that fits nicely with the full generality of the theory. This is probably what makes the book good to start with, but there is still going to be a somewhat difficult transition from this book to a typical differential/riemannian geometry book. Namely, the basic language of vector bundles, pull backs/push forwards, tensors and tensor fields are either covered in a very specific framework or not at all. ... Read more

Product Description
Differential geometry began as the study of curves and surfaces using the methods of calculus. In time, the notions of curve and surface were generalized along with associated notions such as length, volume, and curvature. At the same time the topic has become closely allied with developments in topology. The basic object is a smooth manifold, to which some extra structure has been attached, such as a Riemannian metric, a symplectic form, a distinguished group of symmetries, or a connection on the tangent bundle. This book is a graduate-level introduction to the tools and structures of modern differential geometry. Included are the topics usually found in a course on differentiable manifolds, such as vector bundles, tensors, differential forms, de Rham cohomology, the Frobenius theorem and basic Lie group theory. The book also contains material on the general theory of connections on vector bundles and an in-depth chapter on semi-Riemannian geometry that covers basic material about Riemannian manifolds and Lorentz manifolds. An unusual feature of the book is the inclusion of an early chapter on the differential geometry of hypersurfaces in Euclidean space. There is also a section that derives the exterior calculus version of Maxwell's equations. The first chapters of the book are suitable for a one-semester course on manifolds. There is more than enough material for a year-long course on manifolds and geometry. ... Read more

Customer Reviews (1)

FINALLY, AN EXCELLENT BOOK IN MODERN DIFFERENTIAL GEOMETRY!
Finally (!), an excellent book in Modern Differential Geometry that is here to stay! For people who struggled with the old-fashioned style and notations of Kobayashi and Nomizu, or with the happy-go-lucky style of Spivak's book, this new book is a must read! You will find it very well written, and you will regret that a book like this was not available 10 years ago! It leaves the pre-existing graduate textbooks behind, thanks to several attributes: 1. it is methodical and full of rigor, but at the same time easy to read; 2. it builds up all the concepts gradually, from a beginner's first-year graduate level .... up to the advanced level of a researcher who wants to learn about Lorentz manifolds and the Geometry of Physics; 3. it is smart, modern and user-friendly. As a professor and a researcher, I consider this book to be one of the best choices for your graduate students with an interest in differential geometry, topology and related areas. ... Read more