Reviews (10)

Excelent exposition !
I used this book during my MsC in Math at University of Brasilia, Brazil. Manfredo gives a detailed exposition about the local and global properties of a surface and introduces the abstract surfaces at the end of the book. I really recommend this book for a first course in Differential Geometry and it gives a good knowledge to read the Riemannian Geometry(Manfredo).

Excellent introduction to differential geometry
I found this book to be a good introduction to differential geometry for undergraduates. All concepts are clearly and concisely developed and explained well. Also, the details of advanced calculus and analysis required for the book are covered very well in appendices. My only reservation is that the problems can be very difficult at first. For someone interested in really exploring and learning differential geometry, however, this is not a problem.

Very Difficult
Although some claim it is classic, don't expect it to be readable. The book's definitions cna be quite confusing, and it is often difficult to understand many of the definitions or problems without a great deal of effort. If you are using this book for a class, I would reccomend getting a more readable text for reference.

too wordy
Many people say that this is an excellent book on elementary Differential Geometry,so I borrowed one.I read most of chapter1-4,but I do not find it clear enough.Maybe I have been confused by so many examples.It is too wordy for me.It can be more concise.I recommend the book of Klingenberg,which is used as my textbook.Klingenberg'book is a little dense and does not contain enough exercises.But your professor will give you some.

Outdated classic in the field
While many consider this text the classic in the field, I think that it's time to find something a bit more modern. do Carmo often skips steps in his proofs or examples that he deems "obvious" but that my professor even had to spend a good deal of time trying to figure out how to bridge the gap. The writing style and terminology are beginning to show their age, and the exercises are particularly incomprehensible. Also, the few examples that are actually included don't give any insight into doing the exercises, which seem to be written primarily as a way for do Carmo to include topics that he couldn't squeeze into the body of the text. The hints in the back of the book are often too vague to provide any useful insight into solving the problem, especially for someone not familiar with geometry. Hope that you don't have to use this text for a first course in differential geometry, and if you do, hope that the professor writes his/her own exercises. ... Read more

Book Description

Book Series: INTERNATIONAL SERIES IN OPERATIONS RESEARCH AND MANAGEMENT SCIENCE : Volume 51

It is difficult to evaluate an organization's performance when there are multiple inputs and multiple outputs to the system. The difficulties are further enhanced when the relationships between the inputs and the outputs are complex and involve unknown tradeoffs. This book introduces DEA as a multiple-measure performance evaluation and benchmarking tool. The focus of performance evaluation and benchmarking is shifted from characterizing performance in terms of single measures to evaluating performance as a multidimensional systems perspective.

New DEA models and approaches are presented to deal with performance evaluation problems in a variety of contexts. A context-dependent DEA measures the relative attractiveness of similar operations/processes/products. Sensitivity analysis techniques can be easily applied and used to identify critical performance measures. Value chain efficiency models and DEA benchmarking models can be utilized to study the impact of information technology (IT) investments. These models can help organizations better understand the real impact of their IT investments and integrate technology more efficiently and effectively for the future.

Conventional and new DEA approaches are presented and discussed using spreadsheets - one of the most effective ways to analyze and evaluate decision alternatives. The user can easily develop and customize new DEA models based upon these spreadsheets.

This book also provides easy-to-use DEA software - DEA Excel Solver. This DEA Excel Solver is an Add-In for Microsoft® Excel and provides a custom menu of DEA approaches, which include more than 150 different DEA models. It is an extremely powerful tool that can assist decision-makers in benchmarking and analyzing complex operational efficiency issues in manufacturing organizations as well as evaluating processes in banking, retail, franchising, health care, e-business, public services and many other industries. The DEA Excel Solver does not set limit on the number of units, inputs or outputs. With the capacity of Excel Solver, the DEA Excel Solver can deal with large sized performance evaluation tasks. ... Read more

Reviews (2)

GREAT BOOK...THE BEST DEA CD
The book is excellent with a nice mix of theory and applications in DEA. I believe this is the best book on DEA. Of special interest is the CD that looks at many DEA models. The CD has no competitor and is by far the best around. I have used this CD in many of my academic papers on DEA. If you need DEA software this is the book.

very good DEA book and software
This is the DEA book has many DEA models that you can apply with the software supplied. The software is an Excel Add-In which can be easily used. The book also has good intro. on DEA. It is a book for both DEA beginner and advanced users. ... Read more

Reviews (7)

Flat spheres and more
Highly stimulating and extremely hard to read, written for mathematicians in physics. However, the chapter on Riemannian Geometry can be worked through, up to a point, without any knowledge of exterior differential forms, and is notable if for only one fact alone: a simple calculation is provided that explains explicitly that spheres in four and eight dimensions (3-spheres and 7-spheres) are flat with torsion! I don't know another reference that a physicist without special background in math can consult to understand this highly nonintuitive fact.

Just a "better than nothing" book
It's not the best way to learn geometry / topology for physics. It's better than nothing, though, if you are familiar with the topics already. There are many "holes" in Nakahara's book, which you would spend much more time and hard working in a "big" library. than you should to fill in. It's not worth that money and struggle. It's the last one you should consider about owning.

If you are a physics graduate who needs a nice guide to "understand" the aspects and skills of geo / top, I would recommend the following: (1) Milnor's Topology from the Differentiable Viewpoint, and (2) Kreysig's Differential Geometry. The first one was old, and so it does not assume much knowledge about the topic. The latter is a kind-of-Bible for the topic, and all solutions are provided for the problems. These two books will help you a lot if you care about the meaning, not only for those classroom exams or just showing off that you know something about it. Frankel is the next to put on your bookshelf as a detailed and rigorous development for your preparation to be a theoretical physicist.

If you have only a rough idea about topology, Hocking and Steen are the best choices, and they are Dover!!

Anyway, if I could find a cheap used Nakahara, I would get it as a reference.

Best in its genre
I suppose I should preface this by saying that I read this book *after* reading similar books, so my ability to understand this book is probably better than others, but that said, I think that my comparative evaluation is free from this bias...

There seem to be a few books on the market that are very similar to this one: Nash & Sen, Frankel, etc. This one is at the top of its class, in my opinion, for a couple reasons:

(1) It's written like a math text that covers physics-related material, not a book about mathematics for physicists. I prefer this; you may not. As a consequence, this book is more rigorous than its alternatives, it relies less on physical examples, and it cuts out a lot of lengthy explanation that you may not need. Of course, there are drawbacks to all of these "features" -- you need to decide what you need and what's best for you.

(2) It's most comprehensive, with Frankel coming in second, and Nash & Sen least comprehensive (though they have quite a bit on Fibre bundles and related topics). Nakahara has a chapter on complex manifolds, which is absent from the other two. Nakahara also concludes with a nice intro to string theory, which is absent from the other two as well (though nothing you couldn't find in Polchinski or the like). Actually -- I modify this slightly. Frankel covers less subjects than Nakahara, but with more depth (though also more wordiness -- I quit Frankel about 2/3 through because it wasn't succinct enough and I got tired of it).

Depending on your tastes, I would recommend this book before the other two.

It presupposes that you have an understanding of algebra (groups, rings, fields, etc.) but it has an introduction to the necessary components of topology within. Frankel has presupposes both algebra and topology; Nash & Sen presupposes only algebra.

Excellent book
A very nice blending of rigor and physical motivation with well chosen topics. Plenty of examples to illustrate important points. Especially noteworthy is its description of actions of lie algebras on manifolds : the best I have read so far.

Most of the topics are intepreted in terms of their topological/geomtrical structure (and the interplay between those two), but that's what the title of the book says. So you will learn things again in new ways, and gain a powerful new set of tools. If nothing else, it gives you a nice warm fuzzy feeling when you read other field/string theory books that glosses over the mathematics.

One minor rant : the notation of the book can be better. I personally uses indices to keep track of the type of objects (eg. greek index=components of tensors, no index=a geometrical object etc..), but Nakahara drops indices here and there "for simplicity". But that's my personal rant.

Good book. Buy it.

A must for any theoretical physicist
With an excellent balance between mathematical rigor and pedagogical simplicity, Nakahara remarkably captures in a single volume much of the mathematics a physicist will ever need. (If he wrote a few chapters on group theory, 'much' might be replaced with 'all'). Containing as much as it does, it is not something to breeze through. Depending on your mathematical background, you may only want to read a few chapters (and if the Homology chapter is tripping you up, just keep moving). But invest the time with it, and you will be rewarded with a solid grasp of the mathematical pictures underlying most modern physics. And once you read it and see physics from this perspective, you'll be amazed you had ever thought you understood the physics it describes. It should be said, though, that some of the latter chapters, in particular 12, are horribly sloppy. There are dozens upon dozens of errors, many at a deep conceptual level. Nonetheless, it is a monumental text, and I recommend it heartily. ... Read more

Book Description

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

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

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

Book Description

Theodore Frankel explains those parts of exterior differential forms, differential geometry, algebraic and differential topology, Lie groups, vector bundles and Chern forms essential to a better understanding of classical and modern physics and engineering. Key highlights of his new edition are the inclusion of three new appendices that cover symmetries, quarks, and meson masses; representations and hyperelastic bodies; and orbits and Morse-Bott Theory in compact Lie groups. Geometric intuition is developed through a rather extensive introduction to the study of surfaces in ordinary space. First Edition Hb (1997): 0-521-38334-X First Edition Pb (1999): 0-521-38753-1 ... Read more

Reviews (13)

over and over and over again
Having taken a course out of Frankel (over the first 7 chapters) and now having used it in my senior project (topology of circuit analysis) I have to say that I love this book more by the day.

Beforewarned it is not an easy text and you may have to read a section or a chapter over a hundred times. I have found that the material is dense and deep but in a way that welcomes effort. It is weak as far as rigor goes, but rigor can sometimes get in the way of understanding. Use this book alongside mathematics texts in topology, differential geometry and linear algebra and there is much to gain.

For an undergraduate in mathematical physics (which I am) I have come to love this book I highly recommend it to a serious student.

the geometry of physics
I just finished a class in mathematical physics, and the text we used was Bamberg & Sternberg. I found that books treatment muddled and shortsighted. I mean, most of the linear algebra in the book deals only with 2 dimensional vector spaces. And the book was entirely useless in teaching differential forms...

So i went looking for a better book to learn diferential forms. i didn t like flanders, it was too brief. this is the book for me. Don t expect to find any linear algebra here, but you d better know lin. alg. before you open this book.

it is a challenging book, mathematically speaking, to study on your own (for a senior ugrad phys major, anyway), but it s treatment of forms and tensors is comprehensive, thorough, and detailed. and it shows you all the applications to relativity and electrodynamics, etc... it also builds up all the theory in with a background of differential geometry and topology, which are developed in the first chapter (but wasn t i glad to have already studied those topics beforehand!)

this book prepared me for my mathematical physics class, plus gave me months of other material to study. it is difficult, so i read and reread each chapter.

Bad book.
Frankel's book is provbably the most confusing book I have ever looked into. As other readers noted, it is probably because of his approach not to define things properly. The book's style is extremely wordy, unnecessary wordy that is. The result - total confusion. Mr. Frankel probably thinks the readers are nearly morons, so he tries to re-express some (really simple) notions with words that supposedly will make things lucid. Well, he fails.
Alternative book by Nakahara is way better.I also recommend "Analysis, Manifolds and Physics" by Yvonne Cgiqyet-Bruhat, et al
2 stars for effort.

Good one, even if not the best, probably
This is a valuable reference for students pursuing a support or who want to get themselves deeper in the mathemathical part connected with QFT and GR. I particularly appreciated the first chapter about Manifolds and vector fields, the part about algebraic topology (chapter 13: chains, homology groups and De Rahm's theorem, Betti numbers) and the part about homotopy groups. On the other hand the first part about tensors, exterior forms, integration of differential forms and the Lie derivative seems to me a bit uneven compared to the one I've mentioned above. For this section I'd recommend: Aldrovandi - Pereira, "Introduction to geometrical Physics", or V.I. Arnold, "Classical Mechanics" (first part) which is not complete if compared to the other two books (this is a book about the symplectic formulation of CM and not strictly a matemathical book) but things that are contained are exposed in a beautiful way. Another valuable book is Nakahara (a classic one), but I still have to finish reading it so I'll leave a comment about it in the next. The level of T. Frankel is at last yr undergrad - 1st yr graduate.

There are better...
I have used this book in an independent study in Geometry of Differential Forms. It did not take me too long to start looking for other references. There is something about its content that makes it diffucult to follow. May be it's too wordy. There are several misprints in notation. After I few weeks of study, I turned to Morita's Geometry of Differential Forms. The mathematical presentation is much clear and it's only 300 pages. I really like Frankel's book mainly for its application to physics. But with respect to the math, I recommend Morita's and Thirring monographs. ... Read more

Book Description

Written primarily for readers who have completed the standard first courses in calculus and linear algebra, Elementary Differential Geometry, Second Edition provides an introduction to the geometry of curves and surfaces. Although the popular First Edition has been extensively modified, this Second Edition maintains the elementary character of that volume, while providing an introduction to the use of computers and expanding discussion on certain topics. Further emphasis has been placed on topological properties, properties of geodesics, singularities of vector fields, and the theorems of Bonnet and Hadamard. For readers with access to the symbolic computation programs, Mathematica or Maple, the book includes approximately 30 optional computer exercises. These are not intended as an essential part of the book, but rather an extension. No computer skill is necessary to take full advantage of this comprehensive text.

* Gives detailed examples for all essential ideas
* Provides more than 300 exercises
* Features more than 200 illustrations
* Includes an introduction to using computers, and supplies answers to computer exercises given for both Mathematica and Maple systems ... Read more

Reviews (6)

An Excursion into the Realm of Differential Geometry
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 differnetial geometry, at one of the University of California campuses . The class for me --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 class however, I decided to change direction and as time went on, I drifted more and more towards geometry as the field of further concentration. Before we proceed further, let me note that one main complaint that's rather well-known about this text is the issue of numerous typo's therein. What many may not know however is that the first edition from 1966 does not contain any noticeable typo's, unfortunately somehow all of them found their way in the 1997's second edition. This is very likely because of careless typesetting, but one good news is that many of these are noted on the errata sheet available from the author's UCLA web site. Moreover, the book uses a cumbersome section numbering format, and to make things worse, in quite a few places the reader is referred to one or more previous sections. This serves to disrupt the flow of reading by taking up some time and effort to locate the correct previous page number which is being referenced.

Within the eight chapters of the book (seven chapters in the first 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 shifts to 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 and some global theorems are also discussed, 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), the concepts of geodesics, complete surfaces, covering spaces, Jacobi fields, conjugate points, and a couple of constant curvature theorems for surfaces are explored. The appendices include help on using popular computer algebra systems, and another appendix providing solutions to most of the odd-numbered exercises in the book.

Again, looking on the downside, the book lacks a discussion of several essential tools, for example, the Schwarz-Christoffel symbols, tensors, and Lie derivatives, as well as 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 to less than 500 pages, Dr. O'Neill has preferred to skip some topics. One remedy is to back this text up with Manfredo Do Carmo's 1976 classic, which is mathematically more rigorous, and covers more of the above-mentioned topics (be aware though that Do Carmo is less accessible for the beginning students). Afterwards, one can certainly continue the study of the essentials by reading other advanced books such as Barrett O'Neill's (obscure) graduate-level 1983 treatise on Applications of the Semi-Riemannian Geometry to Relativity, or William Boothby's "An Introduction to Differentiable Manifolds and Riemannian Geometry". One other underrated source which is worthwhile to look into is Richard W. Sharpe's "Differential Geometry: Cartan's Generalization of Klein's Erlangen Program", from the Springer-Verlag GTM series.

Good low dimensional calculation
It's easy to read with enough examples. Suitable for self study after your advanced calculus (inverse function thm/implicit function thm should be covered here) and linear algebra classes. Tons of exercises will help you familiarize yourself with the calculation in low dimension. (Do I love the exercises on minimal surfaces and surfaces of revolution in chapter 5 and 6!) Most of them are workable. This is the strength of the book. Since the author limits the material to low dimensions, some definitions are a bit misleading, such as the definition of exterior derivative of 1-form in chapter 4, where another term to be added happens to be zero. I think there is a big gap in style and level of difficulty between this book and author's "Semi-Riemannian Geometry: With Applications To Relativity".

After this book, probably you want to read Hicks' "Notes on differential geometry", if you can find a copy in some lib. Darling's "Differential Forms and Connections" is also highly recommended. It is modern but not much topological stuff.
Company it with Warner's "Foundations of differentiable manifolds and Lie groups" for topology also much higher algebra.

Solid and Modern Introduction
I worked through the first edition of this book some years back. After finishing this book I was ready for more abstract treatments of Riemannian Geometry. For example, having seen covariant derivatives on 2-surfaces embedded in R^3 motivates the abstract definition of connections on manifolds.

Chapter 1 is a decent introduction to pullbacks and pushforwards of differntial forms and tangent vectors respectively. In fact, all the subsequent geometry is based on pullbacks and pushforwards.This itself motivates the more abstract definition of a differentiable manifold with its coordinate charts. True,tangent vectors are not described in the most abstract fashion (e.g. as derivations on the algebra of functions) but this is not appropriate for a first course.

Chapter 2 describes the language of frame field and connection forms and derives the Frenet-Serret equations in terms of moving frames and structure equations. We associate this with the methods of Elie Cartan, who used moving frames in a systematic manner.

Chapter 3 deals with isometries; frankly speaking I never understood the raison d'etre for such a long chapter on such a topic.

Chapter 4 discusses coordinate patches. Again, this is thoroughly modern, and you won't find this in Struik or Kreyszig. The idea of piecing together coordinate patches to get geometric or topological information is a twentieth-century conception.

Chapter 5 introduces the Shape Operator, which is subsequently used in Chapter 6 to derive the equations of surface theory. This is really moving frames again, in another guise.

Chapter 7 finally tries to put this in a more abstract setting by defining abstract surfaces with an intrinsically defined covariant derivative.Holonomy and the Gauss-Bonnet theorem are discussed.

After reading this book, one would be equipped to handle do Carmo's book on Riemannian geometry, or O'Neill's book on Semi-Riemanninan geometry, or the more recent book by Lee, again on Riemannian geometry.

Very useful, but lacking some abstraction
I like this book very much because it helps me frequently when I need to remember some definitions or formulas, but I think it could be improved if some topics were treated in a more abstract way, as all the material on differential forms, for example.

OK for undergrads, if you can put up with the typos
This book is, I suppose, an acceptable elementary introduction to the topic. However, I found that several important proofs were annoyingly incomplete, with the tag "the proof is too difficult to go into here." Many times, the only ingredient needed to complete the proof is the Inverse Function Theorem or the Implicit Function Theorem, which students taking differential geometry should know. Also, and more significantly, the exercises are riddled with typos; some are minor, others make the exercise incomprehensible. A final minor quibble is that the author uses a cumbersome numbering system for his sections. In sum: If your mathematical background is at least as strong as that of a senior honors undergraduate mathematics major, look elsewhere (perhaps "Differential Geometry of Curves and Surfaces," by do Carmo). If not, and you still would like an introduction to the field, this book may foot the bill--just beware of the typos in the exercises! ... Read more

Book Description

The geometry of lines occurs naturally in such different areas as sculptured surface machining, computation of offsets and medial axes, surface reconstruction for reverse engineering, geometrical optics, kinematics and motion design, and modeling of developable surfaces. This book covers line geometry from various viewpoints and aims towards computation and visualization. Besides applications, it contains a tutorial on projective geometry and an introduction into the theory of smooth and algebraic manifolds of lines. It will be useful to researchers, graduate students, and anyone interested either in the theory or in computational aspects in general, or in applications in particular. ... Read more

Book Description

This text is designed for a one-quarter or one-semester graduate couse in Riemannian geometry. It focuses on developing an intimate acquaintance with the geometric meaning of curvature and thereby introduces and demonstrates all the main technical tools needed for a more advanced course on Riemannian manifolds. The book begins with a careful treatment of the machinery of metrics, connections, and geodesics, and then introduces the Riemann curvature tensor, before moving on the submanifold theory, in order to give the curvature tensor a concrete quantitative interpretation. The remainder of the text is devoted to proving the four most fundamental theorems relating curvature and topology: the Gauss-Bonnet Theorem, the Cartan-Hadamard Theorem, Bonnet's Theorem, and a special case of the Cartan-Ambrose- Hicks Theorem. This unique volume will especially appeal to students by presenting a selective introduction to the main ides of the subject in an easily accessible way. The material is ideal for a single course, but broad enough to provide students with a firm foundation from which to pursue research or develop applications in Riemannian geometry and other fields that use its tools. Of special interest are the "exercises" and "problems" dispersed throughout the text. The exercises are carefully chosen and timed so as to give the reader opportunities to review material that hasjust been introduced, to practice working with the definitions, and to develop skills that are used later in the book. The problems that conclude the chapters are generally more difficult. They not only introduce new mateiral not covered in the body of the text, but they also provide the students with indispensable practice in using the ... Read more

Book Description

Students will find all the information covered in the standard textbooks--and more--explained clearly and concisely in this powerful study tool. Unusually detailed, it elucidates all the most difficult-to-grasp concepts that class studies and texts sometimes gloss over. The hundreds of problems with fully explained solutions illuminate important points and teach students sound problem-solving skills. Ideal, also, for independent study.

Reviews (1)

Differential Geometry - A Schaum's Outline Series
As with all of the Schaum's Outline Series, this book is particularly useful if the readers intent is to gain a working knowledge of the subject. The subject of Differential Geometry is no exception. Dr. Lipschultz has done an excellent job of communicating the essential aspects of differential geometry to the reader. The book assumes a fairly low level of mathematical ability having calculus as the primary prerequisite. From this humble beginning, Dr. Lipschultz takes the reader through the necessary discussions of vector functions, curvature, fundamental forms, and tensor analysis. Given the theoretical nature of the subject, Dr. Lipschultz has included most of the theorems and associated proofs necessary for a general understanding of the subject. However, this book is not a substitute for a serious study of differential geometry. In addition most of the problems are limited to two dimensional surfaces and this reader would have enjoyed a more adventurous investigation of higher dimensional spaces. Like all Schaum's series, the text is chock full of problems and their solution. I recommend this book for anyone interested in quickly gaining a working knowledge of the subject. ... Read more

Book Description

This book is a collection of 375 completely solved exercises on differentiable manifolds, Lie groups, fibre bundles, and Riemannian manifolds. The exercises go from elementary computations to rather sophisticated tools. It is the first book consisting of completely solved problems on differentiable manifolds, and therefore will be a complement to the books on theory. A 42-page formulary is included which will be useful as an aide-memoire, especially for teachers and researchers on these topics.Audience: The book will be useful to advanced undergraduate and graduate students of mathematics, theoretical physics, and some branches of engineering. ... Read more

Reviews (1)

This book rocks like Newton and Gauss....
Man, this is a very instructional manual that can help just about anyone having problems learning differential geometry. I give it two tangent vectors up. ... Read more

Book Description

Reviews (9)

Agree -- one of the best; elegant; beautiful
I had been seeking a book on differential geometry for self-study, as a preface to learning general relativity. A seasoned mathematics friend recommended Kreyszig.

So, I waded in, and patiently made my way through every page of the first six chapters, working the problems along the way, at a pace of a few pages per day. Now that the journey is behind me, I can say that I appreciated this book. It compares favorably to some other texts I had tried reading, with less success.

I realize that the author's approach is an old-style classical one, with a reliance on specific coordinate systems and transformations between coordinate systems. To work the problems requires a fair amount of paper and pencil work. Nonetheless, this approach worked well for me. On those occasions when my reading bogged down, inevitably there was a good reason. If I went back carefully, re-read and pondered, doodled on paper, and tried to visualize what Kreyszig was describing, it always worked! The light would soon go on, usually with a pleasurable sense of discovery.

I went back to re-read certain sections of the book to refresh my memory, and realized how elegant the writing is. Crystal clear, right to the heart, and always trustworthy. Everything follows in a gentle persuasive way; there are no jarring leaps or gaps.

Additionally, I had a nice sense of the different flavor brought to the field by the French geometers who made many of the key advances around the turn of the 19th-20th century.

Finally, the summary of key results and equations at the end is very smart and helpful.

Since finishing Kreysig, I did find it helpful to push on and try to grasp these same ideas from the standpoint of one-forms and the coordinate-free approach to tensors. But I'm not sorry I came at the subject this way first.

I do recommend this book, and think that a beginner needs only a moderate amount of stamina and patience here.

A postscript -- the book is also beautiful. I like that in a math book.

One of the best
Good, thorough, self-contained. Spend a little more time on the first three chapters, and the understanding will follow. What can I say? Just rush out and get it!

highly recommended
This is a wonderfully well written book. If you have a good background in calculus and analytic geometry, you will have no problems with understanding most of the book. (If you don't, you shouldn't be studying differential geometry anyway.) The last couple of chapters are more difficult. Make sure to do the problems after each chapter; they are very well designed to enhance your understanding, and as a huge bonus, their solutions can be found at the end of the book. Forget about those books with a fancy hard cover and cost ten times as much. Buy this book and enjoy!

Not for beginer
Not suitable for beginer if you try to study yourself, esecially from chapter IV on the prove is not quite straightforward.

By far the best intro to classical differential geometry
This is the sort of math book that you pick up, get something to drink, sit on the couch and read through as you would read a novel. I dont know if its possible to write a simpler or clearer treatment on differential geomerty. But be warned that it is still only "classical". Tensros are treated as objects that tranform in a certain way, rather than studied as general multilinear functions. However, after reading this book, any book on tensors is a breeeze to go through. Well worth having, especially considering the price. ... Read more

Reviews (2)

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

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

I would also hearitly recommend Burke's best book: Geometry, Spacetime and Cosmology which is out of print. It is much physical and the examples are clearer. He taught english majors and theater students general relativity with that book.

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

Book Description

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

Book Description

Here is the first modern introduction to geometric probability, also known as integral geometry, presented at an elementary level, requiring little more than first-year graduate mathematics. Klein and Rota present the theory of intrinsic volumes due to Hadwiger, McMullen, Santaló and others, along with a complete and elementary proof of Hadwiger's characterization theorem of invariant measures in Euclidean n-space. They develop the theory of the Euler characteristic from an integral-geometric point of view. The authors then prove the fundamental theorem of integral geometry, namely, the kinematic formula. Finally, the analogies between invariant measures on polyconvex sets and measures on order ideals of finite partially ordered sets are investigated. The relationship between convex geometry and enumerative combinatorics motivates much of the presentation. Every chapter concludes with a list of unsolved problems. ... Read more

Reviews (1)

a beautiful book
Everyone with any interest in probability or combinatorics should take alook at this book, and at least read Chapter 1, on the Buffon needleproblem.It gives a beautiful conceptual solution, quite different fromthe more well-known solution using integrals to get conditionalprobabilities. I find it hard to imagine anyone reading Chapter 1 and notdeciding to read the entire book.

I heard Rota lecture on this material,and the book has much the same feeling as his lectures: it is clear,elegant, and concise, full of illuminating examples.Relatively littlebackground is required, and it should be easily accessible to beginninggraduate students (or undergraduates with unusually strong backgrounds). ... Read more

Book Description

Ticciati's approach to quantum field theory falls between building a mathematical model of the subject and presenting the mathematics that physicists actually use. It begins with the need to combine special relativity and quantum mechanics and culminates in a basic understanding of the standard model of electroweak and strong interactions. The book is divided into five parts: canonical quantization of scalar fields, Weyl, Dirac and vector fields, functional integral quantization, the standard model of the electroweak and strong interactions, renormalization. This should be a useful reference for those interested in quantum theory and related areas of function theory, functional analysis, differential geometry or topological invariant theory. ... Read more

Reviews (2)

A Great Field Theory Book
Yes this book isn't perfect, but what book on physics is? That aside, there is no question this is an excellent field theory book with a rigorous approach. Physicists could learn from this style to produce better textbooks rather than following their usual mysterious approach to writing. This book is clearly laid out not only in mathematical style but also with clear and concise explanations of many physical concepts. It is in my opinion far better than Weinberg's book, written in a more readable style. It is also better than books like Peskin and Schroeder and Kaku which seem sloppily put together. Put the book together with Ryder and you will have the tools needed to get a good understanding of field theory. The title might be unfortunate, because it might keep physics professors from considering using it in their classes instead of the usual lousy standby's, which is too bad for the students.

fills a niche
This book is far from perfect, but I think it begins to fill an important niche in the world of QFT books: it presents most aspects of the theory, from basic principles to Feynman rules, gauge fields and renormalization, in a form that is unusually accessible to mathematicians. I'm coming at this from the perspective of a mathematician who has tried and failed to learn QFT from a variety of other books, and I wish I had discovered this one before even opening Weinberg or Peskin & Schroeder. Ticciati doesn't completely avoid the kind logical sleight of hand that is commonplace among physicists, but when doing manipulations whose mathematical basis is questionable, he's usually at least honest enough to point this out to the reader. I especially enjoyed the chapter on Lie algebra representation theory, which is closer to a mathematician's presentation of this subject than a physicist's, yet not without plenty of physical motivation. I'd criticize this book only for two things: (1) it's riddled with misprints (some obvious, some not) and (2) some topics are explained rather more concisely than they deserve, and not always in the most logical order; Ticciati has a tendency to use certain subtle concepts implicitly a few sections before he defines them precisely. One may hope that such errors will be corrected in a future edition. ... Read more

Book Description

Tensor Calculus and Analytical Dynamics provides a concise, comprehensive, and readable introduction to classical tensor calculus - in both holonomic and nonholonomic coordinates - as well as to its principal applications to the Lagrangean dynamics of discrete systems under positional or velocity constraints. The thrust of the book focuses on formal structure and basic geometrical/physical ideas underlying most general equations of motion of mechanical systems under linear velocity constraints.Written for the theoretically minded engineer, Tensor Calculus and Analytical Dynamics contains uniquely accessbile treatments of such intricate topics as:otensor calculus in nonholonomic variablesoPfaffian nonholonomic constraintsorelated integrability theory of FrobeniusThe book enables readers to move quickly and confidently in any particular geometry-based area of theoretical or applied mechanics in either classical or modern form. ... Read more

Reviews (1)

The definitive book on tensors in analytical mechanics
This book is not a text book. It is, in some sense, the final word on tensor formalism in finite degree of freedom (analytical) mechanics. It is one of the most scholarly books I have come across. The list of references is very exhaustive and the author is well read in the literature on the subject, not just in english, but also in russian, french, and german. The style is clear and concise, the notation is carefully chosen and summarized in a useful section where conventions, notation, and basic formulae are listed. ... Read more

Book Description

This book is an introduction to manifolds at the beginning graduate level. It contains the essential topological ideas that are needed for the further study of manifolds, particularly in the context of differential geometry, algebraic topology, and related fields. Its guiding philosophy is to develop these ideas rigorously but economically, with minimal prerequisites and plenty of geometric intuition.

A course on manifolds differs from most other introductory mathematics graduate courses in that the subject matter is often completely unfamiliar. Unlike algebra and analysis, which all math majors see as undergraduates, manifolds enter the curriculum much later. It is even possible to get through an entire undergraduate mathematics education without ever hearing the word "manifold." Yet manifolds are part of the basic vocabulary of modern mathematics, and students need to know them as intimately as they know the integers, the real numbers, Euclidean spaces, groups, rings, and fields.

In his beautifully-conceived Introduction, the author motivates the technical developments to follow by explaining some of the roles manifolds play in diverse branches of mathematics and physics. Then he goes on to introduce the basics of general topology and continues with the fundamental group, covering spaces, and elementary homology theory. Manifolds are introduced early and used as the main examples throughout.

John M. Lee is currently Professor of Mathematics at the University of Washington in Seattle. In addition to pursuing research in differential geometry and partial differential equations, he has been teaching undergraduate and graduate courses on manifolds at U.W. and Harvard University for more than fifteen years. ... Read more

Reviews (2)

Review of a non-mathematician
Being a physicist I've always been fascinated with the use of manifolds and differential geometry in mechanics, field theory, etc ... Most differential geometry books I've encountered only devote about 1 chapter to manifolds and smooth manifolds at that. However this text takes its time to teach the reader what the author states he thinks is the minimum amount of general knowledge about topological manifolds (no discussion of smooth/analytic manifolds is included). The author takes his time developing everything from scratch, not even assuming any experience with (point set) topology, so this book is particularly suited for those who shy away from the subject just because they're not mathematicians and don't know topology. The only prerequisites are advanced calculus and linear algebra, nothing too fancy. The writing itself is very clear and while rigorous this book does not get lost in the boring lemma-theorem-proof vicious cycle so many other math books fall flat at. Throughout the book are scattered exercises for the reader to do (about 10-20 each chapter) and there are problems at the end of each chapter (no solutions/hints included). All-in-all I feel this text has offered me a much greater understanding of manifolds and the general theory dealing with them. Highly recommended.

A very readable text
An excellent text for a beginning graduate level class. This is NOT a comprehensive text covering the material in exhaustive detail, but it is an excellent overview of surfaces, simplicial complexes, homotopy, homology, and the briefest peek at cohomology. The sequence is efficient, and the author does a good job of motivating the discussions, rather than simply dumping an abstraction into your lap. As always, one should be familiar with point-set and groups before jumping in. If you are looking for a text at an undergraduate level, see Armstrong's Basic Topology or Kinsey's Topology of Surfaces. ... Read more

Book Description

The book provides an introduction to stratification theory leading the reader up to modern research topics in the field. The first part presents the basics of stratification theory, in particular the Whitney conditions and Mather's control theory, and introduces the notion of a smooth structure. Moreover, it explains how one can use smooth structures to transfer differential geometric and analytic methods from the arena of manifolds to stratified spaces. In the second part the methods established in the first part are applied to particular classes of stratified spaces like for example orbit spaces. Then a new de Rham theory for stratified spaces is established and finally the Hochschild (co)homology theory of smooth functions on certain classes of stratified spaces is studied. The book should be accessible to readers acquainted with the basics of topology, analysis and differential geometry. ... Read more

Reviews (1)

WELL DONE!
This book is, as the author mentioned in the preface, a marriage of two parts. The first part provided more or less a self-contained introduction to the theory of Riemannian holonomy groups, which usually couldn't be found in differential geometry textbooks. The second part is a research monograph on exceptional holonomy groups, which is the subject that the author is famous at. This book contains lots of topics which are hard to be found in any other books. For example, it contains a proof of the Calabi conjecture, which I've never seen in anywhere else except Yau's original papers. It also has a concise introduction to Calabi-Yau manifolds, which includes lots of topics about CY manifolds that are hard to be found in just a single book. Overall, it's a great introduction to the theory of holonomy groups. And also provides a good start about differential geometric side of the theory of Calabi-Yau manifolds, together with a roughly complete list of further references. ... Read more