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Affine differential geometry has undergone a period of revival and rapid progress in the past decade. This book is a self-contained and systematic account of affine differential geometry from a contemporary view. It covers not only the classical theory, but also introduces the modern developments of the past decade. The authors have concentrated on the significant features of the subject and their relationship and application to such areas as Riemannian, Euclidean, Lorentzian and projective differential geometry. In so doing, they also provide a modern introduction to the latter. Some of the important geometric surfaces considered are illustrated by computer graphics. ... Read more

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This classic work is now available in an unabridged paperback edition. Stoker makes this fertile branch of mathematics accessible to the nonspecialist by the use of three different notations: vector algebra and calculus, tensor calculus, and the notation devised by Cartan, which employs invariant differential forms as elements in an algebra due to Grassman, combined with an operation called exterior differentiation. Assumed are a passing acquaintance with linear algebra and the basic elements of analysis. ... Read more

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Differential geometry has become a standard tool in the analysis of statistical models, offering a deeper appreciation of existing methodologies and highlighting the issues that can be hidden in an algebraic development of a problem. This volume is the first to apply these techniques to econometrics. An introductory chapter provides a brief tutorial for those unfamiliar with the tools of differential geometry. The following chapters offer applications of geometric methods to practical solutions and offer insight into problems of econometric inference. ... Read more

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In view of the significance of the array manifold in array processing and array communications, the role of differential geometry as an analytical tool cannot be overemphasized. Differential geometry is mainly confined to the investigation of the geometric properties of manifolds in three-dimensional Euclidean space R3 and in real spaces of higher dimension.

Extending the theoretical framework to complex spaces, this invaluable book presents a summary of those results of differential geometry which are of practical interest in the study of linear, planar and three-dimensional array geometries. ... Read more

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

Customer Reviews (4)

Diff Geo Great textbook
This is a great introduction to differential geometery in n dimensional approach. It is much more general treatment than the one provided by other textbooks I saw. Having the problem section in each chapter is very useful. It would be even more useful having the solutions available as well.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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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. ... Read more

Customer Reviews (6)

nice seller
The Seller was very nice. He communicated me quickly and he does not leave a problem or a miscommunication behind. Thanks

A practicalelementary introduction to classical differential geometry
After so many years, this book continues to be a valuable introduction to the differential geometry (DG) of curves and surfaces in euclidean 3 dimensional space, quite clear and efficient for self study, since it combines theory and problems. It reviews the necessary calculus needed.Then it goes into curves and the Frenet equations (little attention is given to plane curves) and continues with surfaces. There one finds an excellent introductory exposition of curvature and assymptotic lines, (including Meusnier, Euler, Rodrigues andBeltrami-Enneper theorems) as well as geodesic curvature, geodesics and Gauss curvature. No mention of parallel transport though (this you can find in Stoker Differential Geometry (Wiley Classics Library),in Goetz Introduction to Differential Geometry (Addison-Wesley Series in Mathematics),Millman-Parker Elements of Differential Geometry's, do Carmo Differential Geometry of Curves and Surfaces or Klingenberg's A Course in Differential Geometry (Graduate Texts in Mathematics), all of them introductory books on DG too. No global properties of curves are given, but we find a clean proof of Liebmann's theorem characterising compact connected surfaces of constant curvatureas spheres (without assuming its orientabilty) and a rather sketchy proof of Gauss-Bonnet theorem. Many proofs of theoretical properties appear as problems. Practical questions are easy or not too hard to solve. If you really don't know the subject, this is a perfect start, alone or combinedwith those previously cited works or withStruik's classicalLectures on Classical Differential Geometry: Second Edition,or Oprea Curves and Surfaces (Graduate Studies in Mathematics) (Applied DG) or Montiel-Ros' recent book Differential Geometry and its Applications (Classroom Resource Materials) (Mathematical Association of America Textbooks). Other problem books on DG are rare. I will mentionFedenko's (Mir-Moscow) (similar to M. Lipschutz's)and Mishchenko-Solovyev-Fomenko (Problems in DG and Topology, Mir- Moscow).

Slightly Subpar for Mathematical Topics in the Schaum Outline Series
While the few solved problems have been carefully selected, and the topics covered continue to reflect Martin Lipschultz normal high standards of exposition, overall this volume is a sub par effort for topics in this series.

The problem lies with the progression of topics, and the erratic treatment -- both of which seem to lack rhyme or reason and leaves the reader with no sense of continuity or cohesion to the substance: Why not, for instance, have "vectors" and "vector functions of a real variable," followed by "vector functions of a vector variable?" And why throw topology right into the middle of this mix? Was it only to get to the idea of Homeomorphisms? If so, should this not have been done much earlier on in the book, maybe even as early as the very first chapter, providing a smoother transition to vector functions of higher mathematical forms?Or better yet, perhaps the author should have merely mentioned the importance of elementary topology, in passing, and then referred the reader to an introductory topology textbook, or as a last resort, he could have added topology as an appendix? But not just toss it in the middle unexpectedly without explanation in an almost completely disconnected fashion. This smattering of topology just seemed so much out of place here. And in any case, it surely was insufficient to tie down the concepts needed to build the necessary bridge between topology and differential geometry. Yes, it did help in understanding the parametric representations of surfaces, but the reader still "was on his own" and had to hustle mightily to make the intended connections.

As well, throughout the book, the lurching back and forth leaves the reader without any sense of coherence on which to build confidence in either the theory of these many complex topics, or problem-solving in the field of differential geometry, more generally. Thus I would argue at the very least that this volume should be relabeled "Selected Topics in Differential Geometry," or better yet "Eclectic Topics in Differential Geometry.'

Its real merit is as a supplement only: neither as a text, nor as a robust basis for developing skills beyond the basics for solving problems in Differential Geometry.

Still, since there is so little basic material available in the field, this remains a useful, even if not an entirely valuable, resource. Three stars.

Differential Geometry review
I have found this to be an excellent addition to my library.

Good as a basic textbook and a source of solve problems
This book is intended to assist upper level undergraduate and graduate students in their understanding of differential geometry, which is the study of geometry using calculus.Usually students study differential geometry in reference to its use in relativity. I personally have a rather oddball application for the subject - modeling of curved geometry for computer graphics applications. The fundamental concepts are presented for curves and surfaces in three-dimensional Euclidean space to add to the intuitive nature of the material.
The book presumes very little in the way of background and thus starts out with the basic theory of vectors and vector calculus of a single variable in the first two chapters. The following three chapters discuss the concept and theory of curves in three dimensions including selected topics in the theory of contact.
Great care is given to the definition of a surface so that the reader has a firm foundation in preparation for further study in modern differential geometry. Thus, there is some background material in analysis and in point set topology in Euclidean spaces presented in chapters 6 and 7. The definition of a surface is detailed in chapter eight. Chapters 9 and 10 are devoted to the theory of the non-intrinsic geometry of a surface. This includes an introduction to tensor methods and selected topics in the global geometry of surfaces. The last chapter of the outline presents the basic theory of the intrinsic geometry of surfaces in three-dimensional Euclidean space.
Exercises are primarily in the form of proofs, and there are plenty of worked examples. Since the examples are kept to no more than three dimensions, the outline contains plenty of good instructive diagrams that illustrate key concepts. This Schaum's outline has quite a bit of instruction in it past the bare required minimum, but you might still want a good primary textbook. My personal favorite is Pressley's "Elementary Differential Geometry". Overall I find this to be a very good outline and source of solved problems on the subject and I highly recommend it. ... Read more

Customer Reviews (1)

correction
This is actually in response to the other review. Do Carmo has written two books on differential geometry and the previous commentator refers to only the undergraduate version. Riemannian geometry is the title of the other.
as for these, I personally find them too long to use except as a reference or when i'm confused. A real mathematician will get the detail of the proof and other technicalities on his or her own, so even if there are faults I find that any book with text of explaination and examples is worth it's weight. this is what makes it a five volume (as far as I know) set. i like it as a reference. ... Read more

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

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

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

Customer Reviews (3)

Direct explanation
This books gives a very direct explanation of the main concepts in differential forms. I would recommend it for anyone wanting to get to the main concepts quickly and cleanly.

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

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

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

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

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

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

There are plenty of exercises to boot.

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

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This monograph presents the basic theorems of differential geometry in three-dimensional space, including a thorough coverage of surface theory. By means of a series of carefully selected and representative mathematical models this monograph also explains at length how these theorems are used in three-dimensional elasticity and in shell theory. The presentation is essentially selfcontained, with a great emphasis on pedagogy. In particular, no "a priori" knowledge of differential geometry or of elasticity theory is assumed, the only requirements are a reasonable knowledge of basic analysis, functional analysis, and some acquaintance with ordinary and partial differential equations.

Customer Reviews (1)

A nice little book on the subject
This book was required for a class, so I'm very glad I found it.It turns out to be quite a nice little book on the subject.It is not as in-depth as some texts (thereby sacrificing the 5th star), but for an introductory book, it's very good.I particularly like the way it separates out "Exercises" (with answers) and "Theorems and Proofs" (which you have to do without knowing the answers) giving practice both in deriving results known in advance and problems which you have to solve without knowing the answer.It comes in a nice little red leather binding. ... Read more

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Here is a genuine introduction to the differential geometry of plane curves for undergraduates in mathematics, or postgraduates and researchers in the engineering and physical sciences. This well-illustrated text contains several hundred worked examples and exercises, making it suitable for adoption as a course text. Key concepts are illustrated by named curves, of historical and scientific significance, leading to the central idea of curvature. The author introduces the core material of classical kinematics, developing the geometry of trajectories via the ideas of roulettes and centrodes, and culminating in the inflexion circle and cubic of stationary curvature. ... Read more

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Product Description

The geometry which is the topic of this book is that determined by a map of one space N onto another, M, mapping a diffusion process, or operator, on N to one on M.

Filtering theory is the science of obtaining or estimating information about a system from partial and possibly flawed observations of it. The system itself may be random, and the flaws in the observations can be caused by additional noise. In this volume the randomness and noises will be of Gaussian white noise type so that the system can be modelled by a diffusion process; that is it evolves continuously in time in a Markovian way, the future evolution depending only on the present situation.

This is the standard situation of systems governed by Ito type stochastic differential equations. The state space will be the smooth manifold, N, possibly infinite dimensional, and the "observations" will be obtained by a smooth map onto another manifold, N, say. We emphasise that the geometry is important even when both manifolds are Euclidean spaces. This can also be viewed from a purely partial differential equations viewpoint as one smooth second order elliptic partial differential operator lying above another, both with no zero order term.

We consider the geometry of this situation with special emphasis on situations of geometric, stochastic analytic, or filtering interest. The most well studied case is of one Brownian motion being mapped to another with a consequent skew product decomposition (or equivalently the case of Riemannian submersions). This sort of decomposition is generalised and a key to the rest of the book. It is used to study in particular, classical filtering, (semi-)connections determined by stochastic flows, and generalised Weitzenbock formulae.

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Conformal invariants (conformally invariant tensors, conformally covariant differential operators, conformal holonomy groups etc.) are of central significance in differential geometry and physics. Well-known examples of such operators are the Yamabe-, the Paneitz-, the Dirac- and the twistor operator. The aim of the seminar was to present the basic ideas and some of the recent developments around Q-curvature and conformal holonomy. The part on Q-curvature discusses its origin, its relevance in geometry, spectral theory and physics. Here the influence of ideas which have their origin in the AdS/CFT-correspondence becomes visible.

The part on conformal holonomy describes recent classification results, its relation to Einstein metrics and to conformal Killing spinors, and related special geometries.

Product Description
One of two volumes which lay the foundations for understanding differential geometry. This work familiarizes readers with various techniques of computation. ... Read more

Customer Reviews (3)

I would feel taken
I badly want a copy of this book.But the currently advertised cost ($123 for a paperback) is just too high.

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

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

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

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

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

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

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Differential geometry and topology have become essential tools for many theoretical physicists. In particular, they are indispensable in theoretical studies of condensed matter physics, gravity, and particle physics. Geometry, Topology and Physics, Second Edition introduces the ideas and techniques of differential geometry and topology at a level suitable for postgraduate students and researchers in these fields.

The second edition of this popular and established text incorporates a number of changes designed to meet the needs of the reader and reflect the development of the subject. The book features a considerably expanded first chapter, reviewing aspects of path integral quantization and gauge theories. Chapter 2 introduces the mathematical concepts of maps, vector spaces, and topology. The following chapters focus on more elaborate concepts in geometry and topology and discuss the application of these concepts to liquid crystals, superfluid helium, general relativity, and bosonic string theory. Later chapters unify geometry and topology, exploring fiber bundles, characteristic classes, and index theorems. New to this second edition is the proof of the index theorem in terms of supersymmetric quantum mechanics. The final two chapters are devoted to the most fascinating applications of geometry and topology in contemporary physics, namely the study of anomalies in gauge field theories and the analysis of Polakov's bosonic string theory from the geometrical point of view.

Geometry, Topology and Physics, Second Edition is an ideal introduction to differential geometry and topology for postgraduate students and researchers in theoretical and mathematical physics. ... Read more

Customer Reviews (13)

Excellent review of math for (particle) physicists
I bought this book to supplement my knowledge of mathematics which frequently is involved in understanding Particle physics concepts. The book is terse, but peppered with examples and insights about the definitions, and so far it is really fun to read. Seems like a good investment.

Too many errors to be useful for study
Reading all the glowing reviews of this book, I wonder whether the reviewers actually tried to use the book to understand the material, or just checked the table of contents. There are so many misprints, throughout, that one wonders if the book was proofread at all. Some of the mistakes will be obvious to every physicist - for example, one of the Maxwell equations on page 56 is wrong - others are subtle, and will confuse the reader. The careful reader, who wants to really understand the material and tries to fill in the details of some of the derivations, will waste a lot of time trying to derive results that have misprints from intermediate steps which have different misprints! Some chapters are worse than others, but the average density of misprints seems to be more than one per page.
The book might be useful as a list of topics and a "road map" to the literature prior to 2003, but that hardly justifies the cost (or the paper) of a whole book.

Geometry Topology and Physics: A condesed view
This book provide a complete and useful review of geometrical instuments of mathematical physics from the beginnig to the most advanced topics of interest. It can be used by students at the beginnig of thei studies in this topics, and it's found to be a useful gallery for higher level students (or scholar).

An excellent book
This is the best book of its type, that is, a book that contains almost all if not all the advance mathematics a theoretical physicist should know. I have studied chapters 2-9 and it has the perfect balance between rigorous presentation of topics and practical uses with examples. The level is for advance graduate students. The range of topics covered is wide including Topology topics like Homotopy, Homology, Cohomology theory and others like Manifolds, Riemannian Geometry, Complex Manifolds, Fibre Bundles and Characteristics Classes. I believe this book gives you a solid base in the modern mathematics that are being used among the physicists and mathematicians that you certainly may need to know and from where you will be in a position to further extent (if you wish) into more technical advanced mathematical books on specific topics, also it is self contained but the only shortcoming is that it brings not many exercises but still my advice, get it is a superb book!

A great reference book.
No doubt, the interplay of topology and physics has stimulated phenomenal research and breakthroughs in mathematics and physics alike.

Unfortunately, there is so much mathematics to master that the average graduate physics student is left bewildered.....until now.

The text is an excellent reference book. I emphasize reference. The book presupposes an acquaintance with basic undergraduate mathematics including linear algebra and vector analysis.

The author covers a wide range of topics from tensor analysis on manifolds to topology, fundamental groups, complex manifolds, differential geometry, fibre bundles etc.

The exposition in necessarily brief but the main theorems and IDEAS of each topic are presented with specific applications to physics. For example the use of differential geometry in general relativity and the use of principal bundles in gauge theories, etc.

Unfortunately, there are very few exercises necessitating the use of supplementary texts. However, to the author's credit appropriate supplementary texts are provided. The author goes to great lengths to show which texts inspired the chapters and follows the same line of presentation.

Perhaps the greatest attribute of the text is to take disparate branches of mathematics and coallate them under one text with applications to physics. In doing so one gains a better grasp of how the fields of mathematics interact in the domain of physics. ... Read more

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Customer Reviews (4)

Peculiar
This book contains material about differential geometry that is very hard to find in any other book, if possible at all. However for people who feel uncomfortable with different approaches of what they already know, a word of warning is in order: the book builds everything on 5 axioms about parallel structures in bundles. All other approaches (frame bundles, connections e.t.c.) are deduced later in the book. Five axioms might take a hard-swallowing and the trade-off is that they mimic the intuition for Euclidean spaces. Of course any serious reader will not expect to learn differential geometry from one book, so overall it is a useful addition to your collection. Finally, looking at the size of the book and the material it covers you can expect the text to be pretty dense and this is actually the case. Several books on smooth manifolds are suggested in the beginning as companions and Warner's book, which is cited consistently for background results, is a prerequisite for this book.

not for engineers
this book is not for engineers, there is no introduction to those
topics mentioned, if you do not have some mathematical background on manifolds etc, the book will not help you. i m giving 4 stars
just to warn the engineers like me trying to get into differential geometry.

Finally in print again
I learned a great deal from this book
in my second year of grad school and
have recommended it to dozens of people
since then.It is wonderful to see it
back in print.A fantastic introduction
to differential geometry.

the best intro to diff. geom. ever -- period
I have been recommending this book to my colleagues and students since 1981. Finally, they can get a copy easily.

Prequisites are modest, and should be part of the standard math graduate curriculum anyway: the equivalent of Chapters 1 --3 of Warner (differential manifolds, tensors and forms, and a minimal introduction to Lie groups).

Given these, it is simply the best introduction ever written. ... Read more