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.

Customer Reviews (3)

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.

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 hasdone an excellent job of communicating the essential aspects ofdifferential geometry to the reader. The book assumes a fairly low level ofmathematical ability having calculus as the primary prerequisite. From thishumble beginning, Dr. Lipschultz takes the reader through the necessarydiscussions of vector functions, curvature, fundamental forms, and tensoranalysis. Given the theoretical nature of the subject,Dr. Lipschultz hasincluded most of the theorems and associated proofs necessary for a generalunderstanding of the subject. However, this book is not a substitute for aserious study of differential geometry. In addition most of the problemsare limited to two dimensional surfaces and this reader would have enjoyeda more adventurous investigation of higher dimensional spaces. Like allSchaum's series, the text is chock full of problems and their solution. Irecommend this book for anyone interested in quickly gaining a workingknowledge of the subject. ... Read more

Book Description

Customer Reviews (14)

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

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

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

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

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

Good book - rather short
It's a good book, but not for an interested reader of popularized science. For the scientist or student it is rather short. Still a good book to have though.

Very bad book
If you want to study differential geometry for the first time, I would recommend the book "Differential geometry of curves and surfaces", by M. do Carmo, instead of this one. I bought Kreyzig's book for self-study but gave up after a few weeks. I then bought do Carmo's book and felt relieved that it was totally different from Kreyzig's.

Kreyzig's book has several issues, especially when compared to do Carmo's book: it is too small for the number of topics it deals with, it has few exercises and the notation is cumbersome. Part of these issues is probably due to the fact that this book was first published in 1959, and though I have seen older math books which haven't fallen behind, this one seems to have suffered from this. It took me a lot of time to understand some of the sections because of the bad notation, and the lack of some good exercises contributed to this problem. The book does cover the standard material that is expected of it and has a reasonable number of exercises, but, still, the exposition is very bad, and you get the feeling that the book could have been much better.

Despite the (important) fact that it costs about 10 times less than do Carmo's book, I strongly recommend that you save your time and enhance your learning by buying do Carmo's book instead. ... 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.

Customer Reviews (6)

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

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

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

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

(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. ... Read more

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

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

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

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

Customer Reviews (7)

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

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

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

I digress.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[Review updated in May 2006]

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 ondifferential forms, for example. ... Read more

Product Description
Differential geometry has a long, wonderful history.It has found relevance in areas ranging from machinery design to the classification of four-manifolds to the creation of theories of nature's fundamental forces to the study of DNA. This book studies the differential geometry of surfaces with the goal of helping students make the transition from the compartmentalized courses in a standard university curriculum to a type of mathematics that is a unified whole. It mixes together geometry, calculus, linear algebra, differential equations, complex variables, the calculus of variations, and notions from the sciences. Differential geometry is not just for mathematics majors.It is also for students in engineering and the sciences.The mix ofideas offer students theopportunity to visualize concepts through the use of computer algebra systems such as Maple. The book emphasizes that this visualization goes hand-in-hand with the understanding of the mathematics behind the computer construction. Students will not onlyseegeodesics on surfaces, but they will also observe the effect that an abstract result such as the Clairaut relation can have on geodesics. Furthermore, the book shows how the equations of motion of particles constrained to surfaces are actually types of geodesics.The book is rich in results and exercises that form a continuous spectrum, from those that depend on calculation to proofs that are quite abstract. ... Read more

Customer Reviews (3)

clearest undergrad differential geometry text around
This is a very well-written text on modern differential geometry for undergraduates. The content of the book is similar to O'Neill's "Elementary Differential Geometry" (e.g. covariant derivatives, shape operators), but it's easier to read. There are many undergrad texts around -- O'Neill, do Carmo, Pressley -- but this one is the most lucidly written one hands-down.

Afer going through Oprea, one might like to tackle O'Neill's "Elementary Differential Geometry" and Vols 2-4 of Spivak's "Comprehensive Introduction to D.G."

Like O'Neill, Oprea develops surface theory using the shape operator. But Oprea takes shortcuts and doesn't develop the theory in quite the same generality as O'Neill does. For example, Oprea doesn't introduce differential forms and the exterior calculus. As a consequence, Oprea restricts himself to the Serret-Frenet equations whereas O'Neill introduces Cartan's structural equations -- of which Serret-Frenet is simply a special case -- as the method of moving frames in full generality. The structural equations are then used (by O'Neill) in both curve and surface theory.

Nice introduction and applications of differential geometry
I found this book to be a fine introduction to this subject. I was particularly pleased with the practical examples outlined in the book. Even though I am not extremely proficient with Maple, I found the exercises using this software provided important illustrations of applications.

Not a text for a rigorous mathematics course
This book is not to be used as a rigorous introduction to differential geometry.There are some definitions and theorems that are casuallydescribed, and the motive behind particular definitions are vague. Thosenot interested in MAPLE might find constant instructions for MAPLEannoying. Not to be completely negative, there are some good excercizes inthe text that I especially enjoyed. ... Read more

Book Description

Customer Reviews (4)

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

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

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

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

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

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

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

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

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

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

The price of the book makes it an attractive purchase.

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

Customer Reviews (16)

The book is Fine but not detailed enough
To be honest, the topics contained in this book are interesting and useful. But, I believe that a beginner in learning Differential Geometry should at least understand some basic principles before reading this book. It will help you to extend your view only if you are well prepared.

Very Well-Suited to Terrestrial Navigation and Geodesy
I work as a navigation engineer and once took a course in differential geometry which used DoCarmo's textbook.The formulas for terrestrial navigation are best understood and most easily derived using differential geometry.But the method of presentation in DoCarmo's book and the subjects covered provide the best starting point for development of the theory of terrestrial navigation.Moreover, it is one of the few books that discuss the Darboux frame, which turned out to be very useful for understanding the kinematics of rotations.I would recommend this book for the serious engineer in the navigation field.

Good book.
Most popular textbook on Differential geometry. Feynman once said the most popular one may not be the best, which is not completely true here. If you want another view of differential geometry, Su Buqing's Lectures on Differential Geometry is a great little book.

Best DG book out there
This book is rather expensive, but when compared to the other books available, it is not a waste of your money. It has plenty of exercises, many of them with answer or hints in the back of the book, and its exposition is broad, very clear and concise.

It is hard to tell being a math student, but I think anyone with a solid knowledge in multi-variable calculus (Apostol's book would be perfect) or, better yet, who has taken multi-variable analysis course would find this book accessible. One of the advantages of this book is that it is self-contained, so even though it uses, for example, the inverse function theorem (which is something unavoidable for a DG book), it has an appendix on differentiability and continuity which covers this.

The exercises range from easy to very hard, but because of the exposition and of the way the exercises are stated (the tougher ones are many times itemized so that they drive you to the answer) it is rare to find a problem that the reader will not be able to solve upon a little thinking.

The greatest advantage of this book is its clear and well-written exposition. It has very few errors, mostly typographical. It covers a lot of topics and its notation is extremely simple and suggestive, which, believe me, is of great help in a DG book. In short, if you want or have to learn differential geometry, save your time and get this book. As another reader very intelligently put it, there is a reason why this is a classic.

There is a reason why it is a classic.
Before talking about the book itself, let me tell you that I am a mathematician, and when I took a differential geometry course and used do Carmo's book, I already knew I wanted to be a mathematician. So, is this a book for mathematicians? Well, yes, but not exclusively. It is certainly written from a mathematician's point-of-view, and it assumes some maturity on the part of the reader. For instance, the exercises contain very little in the way of drill, and are used to enhance the theory (as pointed out by another reviewer). It seems to me that the author believes that mature readers can provide their own `drill' exercises. So, you won't find many exercises asking for you to find pricipal curvatures for this or that surface, and that other as well; exercises in this book have a theoretical flavor to them. This, of course, makes for some hard exercises, and I do remember spending a lot of time over them, often working together with other students taking the same course. The upside is that we learned the material, and thoroughly. Also, the author provided plenty, plenty of examples. The figures are very well drawn and really allow you to see what is going on - even though these days, with powerful computer packages like Maple, Mathematica, Matlab, and others, any student can provide his/her own pictures. But just because now we can use computers, I wouldn't say the text shows signs of age. It is jus as clear now in its exposition of topics and concepts as it was many years ago. So, even though there are many good alternatives in the market, if I had to teach a course now on this subject, or even better, if I were a student now taking this subject, I would certainly have this book at the top of my list of possible textbooks. ... Read more

Customer Reviews (5)

understandable, clear differential geometry book
There are many differential geometry books out there. Some are very rigorous others not. This book walks the road in the middle. Intuition is developed in the first few chapters by discussing familiar surfaces in R^n, and then a discussion on more abstract manifolds follow.

The book requires some very basic knowledge of linear algebra and some multivariate calculus knowledge. So basically every undergrad in the sciences should find this book easy to understand, and a good introduction to differential geometry.

Took the class and the book
I had the class from Prof. Parker ~20 years ago. (BS Mathematics 83 from SIU)It was a wonderful class and this is a wonderful book.I still have my signed! copy.I am now a professor of EE and a large research university and this is still a subject that I love.Credit that to the Book and Prof. Parker.

A solid introduction
It is hard to disagree with the idea that one must pursue the learning of mathematics in way that might be at odds with its axiomatic structure. One can pursue the study of differentiable manifolds without ever looking at a book on classical differential geometry, but it is doubtful that one could appreciate the underlying ideas if such a strategy were taken. Some background in linear algebra, topology, and vector calculus would allow one to understand the abstract definition of a differentiable manifold. However, to push forward the frontiers of the subject, or to apply it, one must have a solid understanding of its underlying intuition.

Thus a study of classical differential geometry is warranted for someone who wants to do original research in the area as well as use it in applications, which are very extensive. Differential geometry is pervasive in physics and engineering, and has made its presence known in areas such as computer graphics and robotics. In this regard, the authors of this book have given students a fine book, and they emphasize right at the beginning that an undergraduate introduction to differential geometry is necessary in today's curriculum, and that such a course can be given for students with a background in calculus and linear algebra. They also do not hesitate to use diagrams, without sacrificing mathematical rigour. Too often books in differential geometry omit the use of diagrams, holding to the opinion that to do so would be a detriment to mathematical rigour. Much is to be gained by the reading and studying of this book, and after finishing it one will be on the right track to begin a study of modern differential geometry.

A Perfect Introduction
A Must !!!After reviewing a few dozen books in the subject, this is without any doubt one of the best. It it written with rare clarity, and givesenough motivation and examples to understand the more abstract and difficult aspects of the field. The book is intended for advanced undergraduate (with good understanding of linear algebra and calculus III) and should be read prior to an abstract course in differential geometry (such as is covered in the books of Warner and Hicks).

Another Differential Geometry Book - So So
This book I also purchased as a resource for studying differential geometry. It's a little bit better than the one by Thorpe, but not by much. The text is dedicated to the 'hard-core' mathematical, and even they would have to have some experience/guidance in this subject.I'm a self-learning type of guy, with an MS in physics. Too many questions arise to justify this book for the self-learner. There are problems, and a FEW examples. ... Read more

Customer Reviews (9)

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

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

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

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

Volume 1: A nice study of de Rham Cohomology
This book is the first volume of the 3rd edition in a five volume series on differential geometry. The emphasis on this first volume is the study of differential forms and de Rham Cohomology Theory. Spivak also considers two 'bonus' topics: integral manifolds & foliations and Lie groups.

You'll need some prerequisites to get started. For the differential topology material (including Sard's Theorem and Whitney's 2n+1 Embedding Theorem), I recommend Hirsch's "Differential Topology". For results on determinants and symmetric groups, I use Hungerford's "Algebra", now in its 12th printing. For the general topology material (Hausdorff spaces, Urysohn metrization, etc.), I recommend Munkres "Topology", 2nd edition.

Spivak begins this volume with a review of topological manifolds in Chapter 1. The author provides the basic definitions and gives lots of examples of surfaces and other manifolds. The discussion of manifolds and surfaces continues in the Chapter 1 Exercises. (The author routinely used the exercise set to continue the thread of discussion.) Quick mention of the surface classification theorem is made, although for the proof of this, you'll need to look in Hirsch or Munkres. The reader gets to have fun gluing topological handles onto and cutting disks out of the 2-sphere.

Chapter 2 reviews some of the basic concepts from differential topology, including the fundamental Whitney Embedding Theorem and Sard Critical Point Theorem. Basic properties of smooth maps are also studied.

Chapter 3 studies the general vector bundle and specializes to the tangent bundle of a smooth manifold. The author is keen on the idea that the reader 'grok' (i.e. understand intuitively) the tangent bundle and the associated induced maps and commutative diagrams. The notion of orientability is also introduced.

Multilinear forms and their tensor product are studied in Chapter 4. This is a key building block in the construction of de Rham cohomology. The author gets side tracked a bit with a discussion of differences in classical/modern notion.

Chapter 5 is a very nice chapter on vector fields. Instead of just appealing to results from differential equations (as is usually done) to build integral curves and the flow of a vector field, Spivak establishes these needed results from differential equations using a very accessible integral equations/fixed point argument. Once the flow of a vector field is show to exist (locally), Lie derivatives and Lie brackets are then studied.

Following the integral curves & vector fields material in the previous chapter, the author detours a bit and studies the problem of integral manifolds of dimensions other than 1 along with applications to foliations in Chapter 6. Spivak establishes a basic version of the Frobenius Integrability Theorem and uses examples to motivate the result before diving into the proof.

The basics of de Rham cohomology are established in Chapter 7 and Chapter 8. Alternating and skew-symmetric forms are discussed, although is may be easiest to establish some of the needed results on the symmetric group of permutations after reviewing Hungerford's algebra text. Differential forms and their wedge product are defined, and Frobenius' Theorem can now be restated in terms of differential forms. Two versions of Stokes Theorem are established and this result is applied to integrating forms on manifolds and studying properties of the degree of a proper map of between manifolds. The formal definition of the de Rham cohomology groups is given and some basic calculations are carried out.

The author does something curious with one of the main results of de Rham cohomology, namely the homotopy-invariance property. He starts this with a discussion section in Chapter 7 (not a called out theorem) in which contractible manifolds are show to have zero cohomology in all dimension by an explicit calculation showing all closed k-forms are exact. The results that the author establishes in Chapter 7 for this `one-off' calculation are precisely what are needed to show the more general result that homotopic maps induce equivalent homomorphisms of de Rham cohomology later in Chapter 8.

Chapter 9 is a very nice chapter covering several foundational topics of Riemannian geometry; include the Riemannian metric, geodesics, the exponential map, geodesic completeness and tubular neighborhoods.

Chapter 10 is a short chapter on Lie groups and is something of a detour from the main thread. The author uses the material as a source of application of the material from the first nine chapters.

Returning to de Rham cohomology in Chapter 11, more foundational results from algebraic topology are studied, including exact sequences, Poincare Duality, the Thom class and the index of a vector field.

The book contains many wonderful geometric diagrams which help motivate the material. In most cases, the author is very careful to highlight theorems, propositions and lemmas. Occasionally key results will be 'buried' in a series of discussion paragraphs, which makes referring to these results later on somewhat difficult. The author never, ever calls out or highlights any of his definitions. This can be somewhat frustrating, especially when trying to track down one of these definitions. Fortunately the index to the book is reasonably good.

The Great American Differential Geometry Book
Michael Spivak begins these five volumes stating his modest aim to write the "Great American Differential Geometry book." He surely has.Instead of listing the numerous subjects Spivak treats clearly and beautifully in these volumes, I'd like to call out the delightful travelogue style in which they are written, using history, anecdotes, and opinion to explain, illuminate, and, when possible, motivate the gleaming modern edifice. Spivak's opinions are sprinkled lightly here and there like easter eggs. How could you not love a math book that uses the subtitle "The Debauch of Indices," or dismisses Eric Temple Bell's history as "supercilious remarks of questionable taste"? Also, don't miss the annotated bibliography in volume 5. The fact that legions of professionals refer to these books in their original *typewritten* format [1st & 2d editions] is a further testament to their quality. The third edition is typeset using TeX and, though beautiful, still manages to retain a little of the quirky typewritten appearance. One quibble: I was disappointed to see that this edition did not use Richard Bassein's bizarre artwork [think 70s psychedelic] for the covers; I admit that this stuff weirded me out originally, but have grown to love it -- where else could I see fuzzy trolls in crowns made from Enneper's minimal surface?

Let Spivak take you "All the Way With Gauss-Bonnet." ... Read more

Book Description

Customer Reviews (3)

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

Book Description
This text has been adopted at: University of Pennsylvania, Philadelphia University of Connecticut, Storrs Duke University, Durham, NC California Institute of Technology, Pasadena University of Washington, Seattle Swarthmore College, Swarthmore, PAUniversity of Chicago, IL University of Michigan, Ann Arbor "In the reviewer's opinion, this is a superb book which makes learning a real pleasure."ne de Mathematiques Pures et Appliquees"This main-stream presentation of differential geometry serves well for a course on Riemannian geometry, and it is complemented by many annotated exercises."F. Mathematik "This is one of the best (if even not just the best) book for those who want to get a good, smooth and quick, but yet thorough introduction to modern Riemannian geometry."es MathematicaeContents: Differential Manifolds * Riemannian Metrics * Affine Connections; Riemannian Connections * Geodesics; Convex Neighborhoods * Curvature * Jacobi Fields * Isometric Immersions * Complete Manifolds; Hopf-Rinow and Hadamard Theorems * Spaces of Constant Curvature * Variations of Energy * The Rauch Comparison Theorem * The Morse Index Theorem * The Fundamental Group of Manifolds of Negative Curvature * The Sphere Theorem * IndexSeries: Mathematics: Theory and Applications ... Read more

Customer Reviews (8)

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

The best introduction to Riemannian Geometry I have read
Most books about Riemannian Geometry can be quite difficult to read and understand if you are not lucky to have a good teacher that explains you the contents in a more intuitive way. This is not the case of the present book. Of course good teachers will not do any harm, but I would dare to say that this book can be read through and understood without their help. The exposition and the proofs are elegant and concise, but one still gets all the essential intuitive hints to grasp the fundamental notions in RG.

This has been my experience with RG: I did not have good teachers (of RG), and the first 4 or 5 books I tried to study discouraged me and made me think I could never understand RG. Until I found Do Carmo's book. Then I found out that RG was intelligible.

I consider this book the best available introducion to RG in less than 300 pages, and I would definitely use it if I had to teach RG. Of course some important topics must be missing in a relatively short introduction, and if one wants to continue studying RG other books (such as Sakai's one) should be studied as well, but for quickly acquiring a sound basis of RG this is the best I know of.

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

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

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

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

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

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

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

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

Book Description

Discrete differential geometry is an active mathematical terrain where differential geometry and discrete geometry meet and interact. It provides discrete equivalents of the geometric notions and methods of differential geometry, such as notions of curvature and integrability for polyhedral surfaces. Current progress in this field is to a large extent stimulated by its relevance for computer graphics and mathematical physics. This collection of essays, which documents the main lectures of the 2004 Oberwolfach Seminar on the topic, as well as a number of additional contributions by key participants, gives a lively, multi-facetted introduction to this emerging field.

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

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

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

Book Description
AN INTRODUCTION TO DIFFERENTIAL GEOMETRY WITH USE OF THE TENSOR CALCULUS By LUTHER PFAHLER EISENHART.Preface Since 1909, when my Differential Geometry of Curves and Surfaces was published, the tensor calculus, which had previously been invented by Ricci, was adopted by Einstein in his General Theory of Relativity, and has been developed further in the study of Riemannian Geometry and various generalizations of the latter. In the present book the tensor calculus of cuclidean 3-space is developed and then generalized so as to apply to a Riemannian space of any number of dimensions. The tensor calculus as here developed is applied in Chapters III and IV to the study of differential geometry of surfaces in 3-space, the material treated being equivalent to what appears in general in the first eight chapters of my former book with such additions as follow from the introduction of the concept of parallelism of Levi-Civita and the content of the tensor calculus. Of the many exercises in the book some involve merely direct appli cation of the text, but most of them constitute an extension of it. In the writing of the book I have received valuable assistance and criticism from Professor H. P. Robertson and from my students, Messrs. Isaac Battin, Albert J. Coleman, Douglas R. Crosby, John Giese, Donald C. May, and in particular, Wayne Johnson. The excellent line drawings and half-tone illustrations were conceived and executed by Mr. John H. Lewis. Princeton, September 27, 1940 LUTHER PFAHLER EISENHART. In this edition a number of errors have been corrected in the text. On page 298 there are notes dealing with revisions not incorporated in the text. Princeton, April 9, 1947 LUTHER PFAHLER EISENHART. Contents CHAPTER I CURVES IN SPACE SECTION PAGE 1. Curves ami surfaces. The summation convention 1 2. Length of a curve. Linear element , 8 3. Tangent to a curve. Order of contact. Osculating plane 11 4. Curvature. Principal normal. Circle of curvature 16 5. TBi normal. Torsion 19 6r The Frenet Formulas. The form of a curve in the neighborhood of a point 25 7. Intrinsic equations of a curve 31 8. Involutes and evolutes of a curve 34 9. The tangent surface of a curve. The polar surface. Osculating sphere. . 38 10. Parametric equations of a surface. Coordinates and coordinate curves trT a surface 44 11. 1 Tangent plane to a surface 50 tSffDovelopable surfaces. Envelope of a one-parameter family of surfaces. . 53 CHAPTER II TRANSFORMATION OF COORDINATES. TENSOR CALCULUS 13. Transformation of coordinates. Curvilinear coordinates 63 14. The fundamental quadratic form of space 70 15. Contravariant vectors. Scalars 74 16. Length of a contravariant vector. Angle between two vectors 80 17. Covariant vectors. Contravariant and covariant components of a vector 83 18. Tensors. Symmetric and skew symmetric tensors 89 19. Addition, subtraction and multiplication of tensors. Contraction.... 94 20. The Christoffel symbols. The Riemann tensor 98 21. The Frenet formulas in general coordinates 103 22. Covariant differentiation 107 23. Systems of partial differential equations of the first order. Mixed systems 114 CHAPTER III INTRINSIC GEOMETRY OF A SURFACE 24. Linear element of a surface. First fundamental quadratic form of a surface. Vectors in a surface 123 25. Angle of two intersecting curves in a surface. Element of area 129 26. Families of curves in a surface. Principal directions 138 27. The intrinsic geometry of a surface. Isometric surfaces 146 28. The Christoffel symbols for a surface. The Riemannian curvature tensor. The Gaussian curvature of a surface 149 X CONTENTS 29. Differential parameters 155 30. Isometric orthogonal nets. Isometric coordinates 161 31... ... Read more

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

Customer Reviews (4)

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

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

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

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