Book Description

This book is meant for a one year course in Riemannian Geometry. The approach the author has taken deviates in some ways from the standard path. Instead of discussing variational calculus, the author introduces a more elementary approach which simply uses standard calculus together with some techniques from differential equations. This book will appeal to readers with a knowledge of standard manifold theory, including such topics as tensors and Stoke's theorem. Scattered throughout the text is a variety of exercises which will help the readers to deepen their understanding of the subject. ... Read more

Book Description

This is a book on topology and geometry and, like any books on subjects as vast as these, it has a point-of-view that guided the selection of topics. Naber takes the view that the rekindled interest that mathematics and physics have shown in each other of late should be fostered and that this is best accomplished by allowing them to cohabit. The book weaves together rudimentary notions from the classical gauge theory of physics with the topological and geometrical concepts that became the mathemtical models of these notions. We ask the reader to come to us with some vague notion of what an electromagnetic field might be, a willingness to accept a few of the more elementary pronouncements of quantum mechanics, a solid background in real analysis and linear algebra and some of the vocabulary of modern algebra. To such a reader we offer an excursion that begins with the definition of a topological space and finds its way eventually to the moduli space of anti-self-dual SU(2) connections on S4 with instanton number -1. ... Read more

Reviews (2)

MATH AND TOPOLOGY
Topology is very important scince in the fields of mathematics. And it using in many of another sinceis.

required reading for a topologist interested in physics
As a mathematician turned physics grad student, it is often difficult to read "Math for Physicists" books simply because of the focus on making "numbers churn out;" which, at least for me personally, more difficult to get a handle on the subject and then, in turn, use it fruitfully.

This book on the other hand, is exemplary of why I got into physics in the first place. The first chapter (Physical motivations) and the last chapter (Gauge Fields and Instantons) can be read by any one with undergraduate topology under their belt and come away with a more powerful understanding of gauge theory than, in my opinion, can be found in other introductory gauge theory texts I've been directed to.

Of course I'll read all those said texts as well, but I'm thankful that I found this one. ... Read more

Book Description

This is the first volume of a three-volume introduction to modern geometry, with emphasis on applications to other areas of mathematics and theoretical physics. Topics covered include tensors and their differential calculus, the calculus of variations in one and several dimensions, and geometric field theory. This material is explained in as simple and concrete a language as possible, in a terminology acceptable to physicists. The text for the second edition has been substantially revised. ... Read more

Reviews (1)

Required background reading...
..if you want to understand the much of Arnol'd's book on classical mechanics. Written for physicists in language that physicists can follow, the book starts with advanced calculus (geometry of surfaces and curves in 2D and 3D) and provides a readable and informative introduction to Riemannian geometry, including connections defined by structure coefficients of a Lie algebra, all the way through gauge theories. However, the books by Schutz and by Nakahara cover interesting topics not included here, so see them as well. ... Read more

Book Description

The theory for frames and bases has developed rapidly in recent years because of its role as a mathematical tool in signal and image processing. In this self-contained work, frames and Riesz bases are presented from a functional analytic point of view, emphasizing their mathematical properties. This is the first comprehensive book to focus on the general properties and interplay of frames and Riesz bases, and thus fills a gap in the literature.

Key features:

* Basic results presented in an accessible way for both pure and applied mathematicians

* Extensive exercises make the work suitable as a textbook for use in graduate courses

* Full proofs included in introductory chapters; only basic knowledge of functional analysis required

* Explicit constructions of frames with applications and connections to time-frequency analysis, wavelets, and nonharmonic Fourier series

* Selected research topics presented with recommendations for more advanced topics and further reading

* Open problems to stimulate further research

"An Introduction to Frames and Riesz Bases" will be of interest to graduate students and researchers working in pure and applied mathematics, mathematical physics, and engineering. Professionals working in digital signal processing who wish to understand the theory behind many modern signal processing tools may also find this book a useful self-study reference. ... Read more

Reviews (1)

A new tool!
The subject of the book is a new tool in math, with a host of exciting applications. Of course, the subject has roots in classical ideas from harmonic analysis. But the book covers an explosive and exciting variety of developments since roughly 1990, and it is presented in the form of a graduate text. The basic idea begins with linear algebra, and progresses to expansions in function spaces, and multiresolutions. It will be useful to anyone who wants to learn from scratch about the underlying principles, the new results, and the applications. It is well written. A student of mine picked it up accidentally from my desk, and couldn't put it down. After awhile, he had completely forgotten what he came to see me about. It could have been some of the applications, such as antenna theory, wavelets, time-frequency analysis, uses in Radar, speech processing... ... Read more

Book Description

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

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

Reviews (2)

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

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

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

Book Description

In recent years the methods of modern differential geometry have become of considerable importance in theoretical physics and have found application in relativity and cosmology, high-energy physics and field theory, thermodynamics, fluid dynamics and mechanics. This textbook provides an introduction to these methods - in particular Lie derivatives, Lie groups and differential forms - and covers their extensive applications to theoretical physics. The reader is assumed to have some familiarity with advanced calculus, linear algebra and a little elementary operator theory. The advanced physics undergraduate should therefore find the presentation quite accessible. This account will prove valuable for those with backgrounds in physics and applied mathematics who desire an introduction to the subject. Having studied the book, the reader will be able to comprehend research papers that use this mathematics and follow more advanced pure-mathematical expositions. ... Read more

Reviews (6)

Integrability conditions discussed
Written in a attractive and even seductive way, relying more on Lie algebraic language than is typical, this book is probably as stimulating an intro. to modern geometry as you can find, within certain limits. The section on noncoordinate bases might have been more clearly written, however. Frobenius's theorm is discussed, something that Fomenko et al should have covered, and the section on connections can be worked throuigh independently of the heavy machinery of exterior differential forms, which is attractive for physics students.

Not as good as "a first course in general relativity"
I had read first the "first course in general relativity"and was exited,so i fygured out that this book from the same author would reach the same standards,but it didnt.If Ihadnt read the first book from Schutz this book would be incomprenheceble.The greatest problem i think is the lack of exercices.Without them you cant really go anywhere.Another problem ,i believe,is the short space given to analyzeeach topic.Eventhough i understand tensor calculus very well I just cant get anywhere with the differential forms.
Eventhough its not the worst book out there its not the best either.My advise,buy a better book.

A Very Accessible Book ! Buy It !
This is a very enjoyable and clearly written book. From a physics point of view the approach is rather abstract, so although differential geometry is developed from 'scratch', it is probably better to have studied a more elementary text on the theory of 2-surfaces in 3-space first (eg Faber's book Differential Geometry and Relativity Theory ). The first chapter sets the mathematical background expected of the reader. The rudiments of analysis, topology, calculus of many variables and basic linear algebra is reviewed.The ensuing chapters cover differential geometry from a 'modern' viewpoint but the style is quite relaxed and the links to 'co-ordinate approach' are well explained. The exercises concentrate on the abstract approach. Throughout the book the underlying structure of manifolds is concentrated upon. No extra 'structure' eg connections and 'distance' concepts are added until the final chapter on Riemannian spaces. For example the metric tensor throughout the body of the book is merely used as a map between a tangent space and its dual space. It is only used as a 'distance' operator in the final chapter.For the purposes of independent study this is a sound book, there are hints and partial solutions for many of the exercises, which is always a welcome feature for those studying entirely on their own.

A Great Introduction to Diff. Geometry
This book presents the basic concepts of differential geometry in a clear, concise manner using modern notation. Schutz's writing style is very readable and there is a considerable breadth of coverage. In areas where one might wish for greater depth, Schutz provides excellent references. My only regret is that the physical applications chapters weren't longer. An excellent starter book and a good quick reference if you continue in differential geometry, GR or field theory.

a little time and no love
I am very angry with paying this much money for a book which obviously dosen't live up to it's name. Think I that Schuz's writing "exposition" is no more than symbolic buffoonery. If you want a book that will knack ya sax off than get a copy of Schutz's "A First Course in General Relativity". ... Read more

Book Description

This book begins with a new approach to the geometry of pseudo-Finsler manifolds. It also discusses the geometry of pseudo-Finsler manifolds and presents a comparison between the induced and the intrinsic Finsler connections. The Cartan, Berwald, and Rund connections are all investigated. Included also is the study of totally geodesic and other special submanifolds such as curves, surfaces, and hypersurfaces.Audience: The book will be of interest to researchers working on pseudo-Finsler geometry in general, and on pseudo-Finsler submanifolds in particular. ... Read more

Book Description

Besides being a problems/solutions addition to the well established textbook by Cheng & Li, this book introduces several new topics. It provides the reader with a self-contained approach to the subject, suitable even for those not familiar with the textbook. All problems have been given a descriptive title, enabling the reader to select according to his preferences. ... Read more

Book Description

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

Reviews (3)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Book Description

With special emphasis on new techniques based on the holonomy of the normal connection, this book provides an up-to-date, complete introduction to submanifold geometry. It offers a thorough survey of these techniques and their applications and presents a framework for various recent results to date found only in scattered research papers. The treatment introduces all the basics of the subject, and along with some classical results and hard-to-find proofs, presents new proofs of several recent results. Appendices furnish the necessary background material, exercises give readers practice in using the techniques, and discussion of open problems will help stimulate interest in the field. ... Read more

Reviews (1)

A fine text for research needs
This is a comprehensive book on, well, the modern aspects of submanifolds.In particular, my interest in this text was for the large section devoted to isoparametric submanifolds (and hypersurfaces) first introduced by Cartan.There is a set of appendices *refreshing* the basics of differential geometry and group theory in the context of the book's main topics. ... Read more

Book Description

In Riemannian geometry, measurements are made with both yardsticks and protractors. These tools are represented by a family of inner-products. In Riemann-Finsler geometry (or Finsler geometry for short), one is in principle equipped with only a family of Minkowski norms. So ardsticks are assigned but protractors are not. With such a limited tool kit, it is natural to wonder just how much geometry one can uncover and describe?It now appears that there is a reasonable answer. Finsler geometry encompasses a solid repertoire of rigidity and comparison theorems, most of them founded upon a fruitful analogue of the sectional curvature. There is also a bewildering array of explicit examples, illustrating many phenomena which admit only Finslerian interpretations. This book focuses on the elementary but essential items among these results. Much thought has gone into making the account a teachable one. ... Read more

Reviews (1)

Mainly for experts in Finsler geometry
The authors claim to turn the subject of Finsler geometry with this book
into a more teachable one and to have a candid style of writing.
This is definitly true for the first 50 pages, where the concepts of
Finsler geometry are very well explained and the exercises
are manageable and perfectly interrelated with the text.

Then the Chern connection and the curvature tensor of Finsler geometry
drop out of the heaven without any explanation of the ideas leading to these
constructions. So one has to derive them alone. In doing so older
texts on Finsler geometry like the Grundlehren text of Rund are more
helpful than this volume.

But the book was apparently prepared with great care. The layout must be called
beautyful and it really facilitates reading this book. Many references
to the literature and classical papers of the subject are included. So beginners,
which want to get a first aquaintance in Finsler geometry, find at least some help.

... Read more

Book Description

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

Reviews (3)

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

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

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

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

Book Description

Fibre bundles, now an integral part of differential geometry, are also of great importance in modern physics--such as in gauge theory.This book, a succinct introduction to the subject by renown mathematician Norman Steenrod, was the first to present the subject systematically.

It begins with a general introduction to bundles, including such topics as differentiable manifolds and covering spaces.The author then provides brief surveys of advanced topics, such as homotopy theory and cohomology theory, before using them to study further properties of fibre bundles. The result is a classic and timeless work of great utility that will appeal to serious mathematicians and theoretical physicists alike. ... Read more

Reviews (3)

This is where it started
For those individuals who want an in-depth, insightful, and solid understanding of fiber bundles this book must be read. In spite of its date of publication, it still is of considerable value in this regard. Modern treatments of fiber bundles are very formal and the underlying motivation gets swept away in the thirst for rigor. Fiber bundles are now ubiquitous in differential topology, algebraic topology, differential geometry, and algebraic geometry, and have also found a place in theoretical physics, thanks to the success of gauge field theories. Therefore a mastery of fiber bundles is essential for entering any of these fields. But fiber bundles are fascinating objects in and of themselves, and studying them for their own sake needs no apology.

The author does use some antiquated notation, but that is not really a hindrance to the study of the book. The reader will no doubt have some background in differential geometry and topology before attempting this book, so the appropriate translation to more modern notation should be straightforward. Once started, and with a little thought adjustment to the idiosyncracies of the author's writing style, the reader will find a plethora of neat examples and insights into the subject. In particular, part 3 on the cohomology theory of bundles is exceptionally valuable in that it gives the reader a detailed overview of the origin of what are not called Stiefel-Whitney classes. The theory of characteristic classes has of course advanced and matured extensively since this book first appeared, but all of the modern treatments are lacking in that they do not give the reader an appreciation of the fundamentals of the subject. Indeed, the construction of the obstruction to the construction of a cross-section to a bundle is the starting point for many of the ideas in obstruction theory that one finds in differential topology. And yes, the procedures the author uses can be "cleaned-up" and made more concise, but the price one pays in such an endeavor is the loss of an appreciation of the concepts behind the scene.

Since the book is a monograph, there are no exercises, and this is probably the only minus to the book. Also, some knowledge of the German language would be useful to a reader who has it, since the author makes references to papers written in German and much of the terminology in the book shows its roots in the German language. One good example of this is the Reidemeister theory of cohomology groups based on a bundle of coefficients, called Uberdeckung by Reidemeister.

There is no question as to why this book remains in print, and it will no doubt continue to be well into the 21st century. IT is a good example of the idea that something new may not be something better. After finishing it, the reader will be amply prepared to enter into the continually-evolving theory of fiber bundles and their applications, all of which are interesting and important.

Excellent introduction to fiber bundles
This book supplies a lot of intuition and background that more modern texts seem to assume of the reader. Steenrod's writing is meticulous and extremely clear. My opinion is that one can learn just as much out of this seemingly outdated text and probably even more than from the modern texts.

... True, more slick machinery has been developed since Steenrod's time, but those big machines are hardly transparent. Steenrod assumes very little of the reader; he even has a quick course in homotopy groups, although he assumes the reader knows the basics of homology/cohomology. Perhaps most importantly, since many of the ideas in the book were new at the time, he doesn't assume that the reader is already comfortable with those ideas. All together this makes a very accessible book indeed.

Still attractive.
A nostalgic but still attractive book on (homotoy theory of) fiber bundles. This book is not very accessible as it predates the development of modern machinery of algebraic topology, but is worth reading. ... Read more

Book Description

This first edition of this book quickly became an established text in this fast-developing branch of mathematics. This second edition has been significantly revised and expanded. It includes a section on new developments and an expanded discussion of Taubes' and Donaldson's recent results. ... Read more

Reviews (1)

A must for researchers new to the field
An authoritative and comprehensive reference...McDuff and Salamon have done an enormous service to the symplectic community: their book greatly enhances the accessibility of the subject to students and researchers alike.

The discussion begins with classic topology and cover a variety of final year undergraduate topics such as complex manifolds and inverse differential techniques before moving into the vastly complex world of Symplectic Topology.

A must for researchers new to the field ... Read more

Reviews (2)

A nice easy going book.
Can't agree with my friend from Ann Arbor. This is the most accesible book on Riemannian Geometry. Or to be precise, this is not even a book on Riemannian Geometry, It's only a guide! The author doesn't take the universal intrinsic approach to the subject, but only look at everything as a subspace of Eucleadean space, and see how those apparently extrinsically defined concepts indeed have intrinsic meannings. And that gives you a taste of the real part, which you must turn to some other books.

Riemannian Geometry - A Beginner's Guide
Well I am only on page 10 and am considering quiting this book. I have a BSEE with quite a lot of mathematics history, but I know nothing about Riemann geometry. I got bogged down right from the beginning with this book. It's not that the material - so far - is that demanding, but the explanations are terse at best and some formulas seem to pop onto the page from hyperspace. Some variables are undefined, unfamiliar nomenclature is used without explanation and there is no exposition to show from where some rather complicated formula appear. Too much is assumed by the author for a book that claims to be a "beginners guide". You can waste a lot of time trying to guess what is on the authors mind.

I get the impression that by adding another 10 pages or so of elucidating math and text this might be a nice little survey of the subject matter, but as is I can't recommend it. ... Read more

Book 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, and with trying to answer them using calculus techniques. It is a subject that contains some of the most beautiful and profound results in mathematics, yet many of them are accessible to higher level undergraduates. Elementary Differential Geometry presents the main results in the differential geometry of curves and surfaces while keeping the prerequisites to an absolute minimum. Nothing more than first courses in linear algebra and multivariate calculus are required, and the most direct and straightforward approach is used at all times. Numerous diagrams illustrate both the ideas in the text and the examples of curves and surfaces discussed there. The book will prove an invaluable resource to all those taking a first course in differential geometry, for their lecturers, and for all others interested in the subject. ... Read more

Reviews (2)

(Most) College students will love this book
I am using this book for a 2 semester Differential Geometry course at my university. The school used to use Do Carmo, but apparently the book was too advanced for the undergraduate level, so this semester they decided to switch over to give this one a test and see how it worked out. This book is not bad. It is basically Do Carmo rehashed for the not so mathematically mature. In all seriousness, the book even follows almost the exact same flow as Do Carmo, the topics are just presented with less rigor. The exercises are rather tedious at the end of each chapter, and in my opinion they don't really help to enhance the subject matter.

On the other hand, if you fall in the category that most of the math majors at my university fall in (i.e. the category of people who really don't care, they just want to get an A and graduate, and don't care about mathematics), then you'll love this book. Why? Because the solution to every single problem is at the end of the book. In my opinion this is a huge flaw. It would be great if everyone were honest and everybody was genuinely interested in the learning Differential Geometry, but that isn't the case. So 90% of my class simply copies the answers out of the back of the book and hands it in to get a 100 on the homework assignments. Pretty sad if you ask me. The book is almost there. Without full solutions to every problem, this book would get 5 stars. But those students who simply turn to the back of the book 15 seconds after looking at the problem statement will learn nothing from this book, so I have to knock it down 2 stars. After all, what good is a book if it doesn't serve it's intended purpose. Perhaps some people would rate a book by "how easy is it to get an A in the class if this is the textbook", in which case they would probably rate this book 5 stars.

Differential Geometry is a hard subject. It's _supposed_ to be hard. We're not talking about taking the reciprocal of a fraction here, it's Differential Geometry. You're _supposed_ to think about these problems for a long time. So if you're a professor considering this book for a course I would recommend against it. The text is good, but the students won't learn anything from it. I've suggested to my professor that perhaps it would be good to not assign problems from the text, but rather get problems from other textbooks where students can't look at the answers.

In my opinion that is the only flaw with this book. Otherwise I think it's a great introduction, and about as elementary as you can really make the subject. If another book was too hard, then this is the one for you.

Also, if you're interested in this book for self study it's a good choice since obviously you're genuinely interested in the subject matter and won't be tempted to look at the answer at the first opportunity.

A quick introduction to differential geometry
Pressley's gives you a very comprehensible and down to earth introduction to differential geometry.
By avoiding the more modern and abstract generalizations of differential geometry to more than three dimensions you really feel that you grasp what the theorems and methods are about. In this way you are able to work your way quickly through the book and avoid getting stuck and loosing interest. Another plus is that the book contains lots of examples and fully worked answers to all exercises, which makes it perfect for self-study.
The downside is that the book is not as exhuastive as you perhaps would like, when you have looked at books like O'Neills and DoCarmos ... on the other hand, you only need to spend a fraction of the time to get through Pressleys.
I definitely think that this is a much better book that Struiks classical work (by being more structured and goal oriented) and an excellent introduction to further studies in differential and Riemann geometry. ... Read more

Book Description

This edition of the invaluable text Modern Differential Geometry for Physicists contains an additional chapter that introduces some of the basic ideas of general topology needed in differential geometry. A number of small corrections and additions have also been made.

These lecture notes are the content of an introductory course on modern, coordinate-free differential geometry which is taken by first-year theoretical physics PhD students, or by students attending the one-year MSc course "Fundamental Fields and Forces" at Imperial College. The book is concerned entirely with mathematics proper, although the emphasis and detailed topics have been chosen bearing in mind the way in which differential geometry is applied these days to modern theoretical physics.This includes not only the traditional area of general relativity but also the theory of Yang-Mills fields, nonlinear sigma models and other types of nonlinear field systems that feature in modern quantum field theory.

The volume is divided into four parts: (i) introduction to general topology; (ii) introductory coordinate-free differential geometry; (iii) geometrical aspects of the theory of Lie groups and Lie group actions on manifolds; (iv) introduction to the theory of fibre bundles. In the introduction to differential geometry the author lays considerable stress on the basic ideas of "tangent space structure", which he develops from several different points of view - some geometrical, others more algebraic. This is done with awareness of the difficulty which physics graduate students often experience when being exposed for the first time to the rather abstract ideas of differential geometry. ... Read more

Reviews (1)

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

Book Description

This is the third version of a book on Differential Manifolds; in this latest expansion three chapters have been added on Riemannian and pseudo-Riemannian geometry, and the section on sprays and Stokes' theorem have been rewritten. This text provides an introduction to basic concepts in differential topology, differential geometry and differential equations. In differential topology one studies classes of maps and the possibility of finding differentiable maps in them, and one uses differentiable structures on manifolds to determine their topological structure. In differential geometry one adds structures to the manifold (vector fields, sprays, a metric, and so forth) and studies their properties. In differential equations one studies vector fields and their integral curves, singular points, stable and unstable manifolds, and the like. ... Read more

Reviews (3)

Modern, but....
This book is a proper subset of Lang's later book "Fundamentals of Differential Geometry (Graduate Texts in Mathematics, 191)".

Not a "first book", ok as reference
Lang's book is definitely not useful as textbook for classes or for self-guided study (learnt this the hard way). He is rather abstract and provides zero motivation for the theory. The book is obviously made for people who learnt diff. geometry elsewhere but want to read a cleaner and more modern treatment. To this end, Lang's book is useful. The best part is that manifolds are infinite-dimensional right away. This is probably the only reason for buying Lang instead of/in addition to Dieudonne as a reference. Otherwise, the book is a little too terse; fiber bundles are merely hinted at. Moreover, I think some of the proofs are unnecessarily complicated, such as the one for Frobenius theorem.

Maybe Lang's best book.
Well, we have here another book on differential manifolds, and another book by Serge Lang. Lang is well-known by writing (lots of) books on different topics in analysis and algebra, all of them in a quite "Bourbaki-like" style: attaining maximum generality, with less motivation than most students would like. This is no surprise, because Lang himself is a Bourbakist.

So, what's interesting about D&RM? It's a book very much like Lang's other books, only that here the Bourbakist's approach is quite happy: it's one of the very few books on his subject to present most of his results in infinite-dimensional (Banach) version, a must if you are interested in nonlinear functional analysis or dynamical systems. The exposition is very clean and clear: Lang uses categories all the way to estabilish the main relations between the different differential-topological structures and tools, and he does not hesitate in stating and using tools from analysis, such as Lebesgue measure and functional analysis' main theorems. The proofs are very polished and, in a certain sense, beautiful, a philosophy that permeates most of the book. As if it weren't enough, the book still contains an appendix with a Von Neumann's seminar about the spectral theorem.

All things considered, it's a quite "state-of-the-art" book about the basics of differential manifolds, from an analyst's perspective. This perspective provides differential topology with a lot of additional clarity and power. I don't know if most physicists would like this book, because its motivations, if any, are sparse and sometimes quite obscure, as long as physical applications are concerned. For a mathematician, however, this book is a gem: it's Lang at its best, and the perfect opening door to global analysis (the nonlinear analysis on infinite dimensional manifolds, a vast field of mathematics that encompasses dynamical systems and nonlinear functional analysis). Despite all that, I would also recommend to physicists to at least tackle this book, as an antidote to all the crap that the so-called "differential topology for physicists" books put on their heads, because I don't know a cleaner and more precise presentation of differential manifolds so far. ... Read more