Customer Reviews

Differential Geometry (Dover Books on Mathematics)

5 star:		(8)
4 star:		(3)
3 star:		(4)
2 star:		(1)
1 star:		(0)

 

 

I had been seeking a book on differential geometry for self-study, as a preface to learning general relativity. A seasoned mathematics friend recommended Kreyszig.

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

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

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

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

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

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

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

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

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

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

I really like this book. I checked out Manfredo P. Do Carmo'sfrom the library and bought this one, and I prefer this one just oncontent. The concepts are explained in a very approachable style and in a nice order to give you an understanding of diff. geometry as well as what you might use it for.. This is not a math text for just math in my opinion. This is geared for you to use differential geometry. I thought most of the concepts are explained nicely but it doesn't hurt to read another book to get another point of view. One advantage this book has over a number of others is that every answer to the exercises is in the back of the book with a very nice solution.. If you're interested in the subject I think this book is a great deal.

Erwin Kreyszig, who was a teacher at various Universities in Germany and Canada, originally published this book as a free translation from the German textbook "Differentialgeometrie" in 1959. It is one of the few English books that are able to cover the material in a mainly geometric fashion thus following the German tradition of classical geometry. Erwin Kreyszig develops advanced geometrical concept without a huge mathematical background; however, this also has the effect the people that are used to a more mathematical approach might have a hard time to get used to the style of the book. In my opinion, this book is clearly a must-have for everybody interested in this area.

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.

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

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

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

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

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

Here's a breakdown of the contents:

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

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

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

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

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

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

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

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

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

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

Kreyszig conserves in this book the same style of simple explanation of his Advanced Mathematics for Engineering. Although he reserves the content for a treatment of the differential geometry in three dimensions, for that reason it doesn't exempt the generality of treating this topic in spaces of n dimensions. Excepting the Chapter 6 and 8, the rest of the book is an extension of that was missing on his Advanced Mathematics book: a tensors chapter, and of course, a more general treatment on surfaces theory. The first Chapter tries on lineal algebra preliminary questions, quite simple, but enough for the Chapter two that is about the theory of curves, Chapter also quite simple if we compare it with the general treatments that are usually shown the the classic books of calculus (Thomas, Leithold...). The chapters 3, 4 and 5 are the real center of the work of Kreyszig in the book. Quite clear, concise, where introduce to the reader to the riemanian geometry with such a harmony that the reader feels extremely attracted by his reading. The symbology is clear and the very coherent and easy presentation of continuing. The Chapter 7 are a small treaty on calculus differential absolute, topic well presented, with the same methodological lines that characterize to the works of Kreyszig, without complications, simple, soft, without minimizing the rigor, without seeking to end up embracing the content of the book of Levi-Civitta, but very enough and that leaves the reader prepared for future more technical readings.

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.

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)