Can anyone suggest any basic undergraduate differential geometry texts on the same level as Manfredo do Carmo's Differential Geometry of Curves and Surfaces other than that particular one?

(I know a similar question was asked earlier, but most of the responses were geared towards Riemannian geometry, or some other text which defined the concept of "smooth manifold" very early on. I am looking for something even more basic than that.)

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The answer depends on what kind of stuff you want the book to cover. Would you like it to get to Gauss-Bonnet? Would you like it to get to Stokes' theorem? Do you want to talk about applications to, say, topology? Also this should probably be community wiki. –  Kevin H. Lin Dec 5 '09 at 21:12
Just curious - why don't you like Do Carmo? I'm taking a course that uses this book. –  David Corwin Sep 21 '10 at 1:59
Well, I never said I didn't like do Carmo (although I must admit, it's not among my favorites) -- it's simply that at the time of this posting, I wasn't really aware of any alternatives. Basically, I just wanted to explore other treatments of the subject. When learning a subject, I almost always use at least two texts simultaneously. –  Jesse Madnick Sep 21 '10 at 4:31

Nobody has mentioned Spivak's five book series. Here's the first book: http://www.amazon.com/Comprehensive-Introduction-Differential-Geometry-Vol/dp/0914098705 .

It assumes some knowledge of differential topology, and of course some standard results from linear algebra and topology, but we used this book in my undergraduate differential geometry class.

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The first two chapters of Vol. 2 require pretty much no prerequisites; they're on curves and surfaces. –  Akhil Mathew Dec 5 '09 at 14:07
Spivak also wrote a book called "Calculus on Manifolds". Depending on the undergraduate, this might be a nice place to start. –  Spice the Bird Sep 14 '11 at 4:23
In vol. 1 he repeatedly refers to 'Calculus on Manifolds' (in particular any actual calculus required), so it might be prudent to consider that book vol. 0 of his magnum opus. It is probably a bit hard to start on vol. 1 without any previous knowledge. –  Ketil Tveiten Sep 14 '11 at 8:42

I enjoyed teaching "Curves and Surfaces" with notes of Theodore Shifrin, which are here: http://alpha.math.uga.edu/~shifrin/ShifrinDiffGeo.pdf

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I haven't read them, but if Ted wrote them, they must be good! –  Deane Yang Dec 6 '09 at 22:50
a quick peer reveales a great mastery, but I am enemy of the ambiguously use rows and columns for vectors... anyway a vote +1 –  janmarqz Dec 7 '09 at 1:07
An OUTSTANDING set of notes by Shifrin and when it finally becomes a book,it may very well become the benchmark for undergraduate DG courses. –  The Mathemagician Jul 9 '10 at 3:01
Just like calculus on manifolds should be read before Spivak 5 volume, perhaps Ted's multivariable mathematics text serves as a great prerequisite builder for his differential geometry text. –  John Jiang May 11 '12 at 5:57

I particularly like Wolfgang Kuhnel's "Differential Geometry: Curves - Surfaces - Manifolds".
The autor goes from curves to surfaces and from surfaces de Riemannian geometry in a very nice way. Even if you're not intersted in the manifold part it is a good book about curves and surfaces. The language is modern and the exposition of the subject very clear. It is better than Manfredo's book in my opinion.

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I really like Barrett O'Niell's \textit{Elementary Differential Geometry, Revised 2nd Edition}. Here is the amazon link: http://www.amazon.com/Elementary-Differential-Geometry-Revised-Second/dp/0120887355/ref=sr_1_1?ie=UTF8&s=books&qid=1260890327&sr=8-1

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Used this for my undergrad diff geom course (along with Montiel & Ros) and it was good. –  Andrew Sep 13 '11 at 20:50

Did someone already mention Geometry of differential forms by Do Carmo?. It is the 2-dimensional version of Riemannian Geometry by the same author. Quite nice since one can see how differential forms work in a riemannian geometry point of view. Here the author works out everything in 2 dimensional manifolds by using definitions that latter on He is going to generalize for high dimensions.

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It's actually called "Differential Forms and Applications", and I wouldn't exactly call it the 2-dimensional version of Riemmanian Geometry. It's quite different, and very good for a undergraduate differential geometry course. –  Spiro Karigiannis Feb 7 '11 at 14:56

If you are looking for text that is good for an undergraduate course in differential geometry, I would suggest Differential Geometry of Curves and Surfaces by Banchoff and Lovett. See http://www.amazon.com/Differential-Geometry-Curves-Surfaces-Banchoff/dp/1568814569/ref=sr_1_3?ie=UTF8&qid=1317835776&sr=8-3 . It was published in 2010 so did not show up on this earlier.

The book comes with online computer graphics that help develop an intuition for the topics in the book.

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As an undergraduate I used Elements of Differential Geometry by Millman and Parker. The prerequisites are solid multi-variable calculus and linear algebra. It works through basic material on curves and surfaces in the plane and three space, and then transitions to studying basic material on manifolds defined intrinsically. I recommend it for an undergraduate course for serious students with minimal background.

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I used to teach out of this one. Lots of great exercises. It is especially good for space curves. It sort of falls down in explaining the Theorema Egregium. They fail to get across the notions of intrinsic and extrinsic clearly. –  Charlie Frohman Mar 22 '10 at 4:18

I don't know if it can be considered as an undergraduate book, but I really liked The Geometry of Physics: An Introduction

It is covering a lot of different topics and found it was a fascinating introduction.

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Curves and surfaces by Montiel and Ros. A modern approach to the contents of Do Carmo's, but focusing on developing and using analytical methods, particularly integration. This book is actually used for an introductory course on the geometry of curves and surfaces at my home university (Granada).

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I used this along with O'Neill's text. What's great about this Montiel & Ros book is that Spain really has become a great place for finding geometers. –  Andrew Sep 13 '11 at 20:52

I learned from this set of lecture notes, and I've never come across anything better. This was in the Budapest Semesters in Math program, and the instructor (who also wrote the notes) had the clearest presentation I've ever seen:

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missing link?... –  Thierry Zell Sep 13 '11 at 20:29
Thanks, I had copied and pasted a link, but somehow it failed to point to a viable URL. I fixed it. –  David White Sep 13 '11 at 22:58

I've reviewed a few books online for the MAA. When I learned undergraduate differential geometry with John Terrilla, we used O'Neill and Do Carmo and both are very good indeed. O'Neill is a bit more complete, but be warned - the use of differential forms can be a little unnerving to undergraduates. That being said, there's probably no gentler place to learn about them. I do think it's important to study a modern version of classical DG first (i.e. curves and surfaces in R3, emphazing vector space properties) before going anywhere near forms or manifolds - linear algebra should be automatic for any student learning differential geometry at any level.

Of the textbooks mentioned here:

• I love Millman and Parker as well, although it's not as complete as one would like. I'd love to see Dover put out a nice cheap paperback of it. Thorpe is OK, but doesn't excite me; his notation gets unnecessarily dense. That being said, he does emphasize linear algebra aspects and covers quite a few topics not found in the other texts.

• Gray's mammoth tome is probably the single most complete source on classical DG: everything is very clearly done with lots of fascinating computer drawn images and historical asides. But the incomprehensibly inserted program code is really distracting and breaks the flow and organization of the text - it should be relegated to software or online. For that reason, I can't really recommend it as a class text, but it definitely should be kept on reserve when teaching such a course.

• Spivak and Frankel, although both wonderful texts, are really graduate level.

Lastly, there are lots of free online resources for students now - the aforementioned lecture notes by Shifrin are outstanding, and we should enjoy them as long he makes them freely available before converting them to a real book. (Really looking forward to the finished product in a few years,though...)

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(This was originally an answer by Andrew L; since the question has returned to the front page, I took the liberty of reformatting the text slightly, to try and improve readability.) –  Yemon Choi Sep 14 '11 at 0:00

I had an undergraduate course out of Elementary Topics in Differential Geometry by John Thorpe and thought it was a good book. It gets to some advanced material (e.g. the Gauss-Bonnet theorem) without a huge amount of technical preliminaries by sacrificing some generality.

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I would like to add that Thorpe focuses on $n$-dimensional surfaces in $\mathbb{R}^{n+1}$, right from the outset. This gives the book a distinct point of view, which I find attractive. –  Joseph O'Rourke Sep 14 '11 at 1:02

I'm not sure whether the following is too advanced, but I found
"Introduction to Topological Manifolds (Graduate Texts in Mathematics) (Paperback) by John Lee"
(Edit: As Ho Chung Siu pointed out, Lee's Intorduction to Topological Manifolds is written in a different spirit to Do Carmo. I'm sorry, I totally misread that the questioner is searching a kind of "Do Carmo text" (so elementary texts in curves and surfaces are searched, right?). Perhaps Lee's Introduction to Smooth Manifolds is more appropriate, but I think it's also too advanced, but anyway my suggestions below should be adequate.)

"Elementary Differential Geometry by Christian Bär (see for exmaple here)"

Furthermore I would warmly recommend Nigel Hitchin's lecture notes "Geometry of surfaces" : http://people.maths.ox.ac.uk/~hitchin/hitchinnotes/hitchinnotes.html

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Lee's book is definitely different from Do Carmo's, isn't it? I remember that one is mainly on basic algebraic topology: classification of surface and fundamental groups. –  Ho Chung Siu Dec 5 '09 at 14:24
You're right, it is different from Do Carmo's. But I think is an introductory text to differential geometry on the same (skill-) level. But as you said, if you are interested in differential goemetry of curves and surfaces the book by Bär or Hitchin's Lecture notes are more appropriate. –  Spinorbundle Dec 5 '09 at 16:27
I'd love it if our undergraduate were prepared for Hitchin's lecture notes, but they don't appear to have any problems, nor many undergraduate-level detailed examples, and it freely uses terminology that most undergraduates never see -- things like differential forms. –  Ryan Budney Dec 5 '09 at 20:01
We teach differential forms (in R^n) to our undergraduates in Edinburgh, actually. They see this in their first (and only :() differential geometry course, which is all about surfaces (in three-dimensional euclidean space). This is a third-year (out of 4) course and we do coordinate-independent calculus on R^n at the very start of the course. From past experience teaching the course, I would say that most students can cope. –  José Figueroa-O'Farrill Dec 5 '09 at 22:53
I don't doubt it. But I think most institutions in North America don't have the kind of infrastructure you have. We will soon have a multi-variable calculus course where students can learn about differential forms, but at present there's no way we can make it a prerequisite to a differential geometry course. –  Ryan Budney Dec 6 '09 at 2:56

There's also Modern Differential Geometry of Curves and Surfaces with Mathematica by Alfred Gray.

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I heard a rumor that when he test taught out of that book, the second semester of the class got cancelled because no one signed up for it. –  Charlie Frohman Mar 22 '10 at 4:21
@Charlie If the draft version looked anything like the finished product,I can see why.Let's be honest-it's scary to students,a book that size!! –  The Mathemagician Jul 9 '10 at 3:03

commentaries like without much formalism like those to the Thorpe's book, I think, are really discouraging for everyone, then students don't want to do complex calculations 'cuz they are ugly... Nah!

I believe that the std covariant derivative of $\mathbb{R}^3$ and the induced connection to a surface, via the Gauss equation (to quick deduce a formula for the gaussian curvature), paves the way to grasp better thing like the geodesic curvature and the Gauss-Bonnet must-do's, for: O'Neil is suitable!

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Not just that they are ugly, but that they're not geometric. I really think it is harder to gain insight by seeing the coordinate definition of a connection as opposed to the invariant one(s). –  Akhil Mathew Dec 6 '09 at 3:09
Look, I wouldn't use the old coordinated version for the covariant derivative (std and induced) instead better I like $D_XY=(JY)X$ and $\nabla_XY=D_XY-\langle D_XY,n\rangle n$, after in the working examples anyone'll be very pleased to see the glow in the eyes of the students who see how the curvature of a 2-sphere of radius $R$ is $R^{-2}$, of a plane zero, ect... modern geometry isn't just geometry alone, it is important to see it thru algebraic procedures, what for? the multidimensional assault –  janmarqz Dec 6 '09 at 3:43

Pressley's Elementary Differential Geometry isn't so bad. It's similar to Do Carmo in many ways. It's part of Springer's UTM series.

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Other Do Carmo like books include Millman and Parker's "elements of differential geometry". I like it more than Do Carmo as it has a more organic feel. I haven't read Pressley's book. –  Ryan Budney Dec 5 '09 at 4:51
And what about O'Neill's book, Elementary Differential Geometry, which I like better than do Carmo's book, because it uses moving frames rather than local co-ordinates. –  Deane Yang Dec 5 '09 at 4:57
I've heard that the second edition of O'Neill's book is riddled with typos. But I haven't seen it myself. –  Mark Meckes Dec 5 '09 at 13:58
The problem with Pressley is that it does not presuppose any linear algebra, which most students taking such a course do know. As a result, it makes many simple results look unbelievably complicated. –  Spiro Karigiannis Feb 7 '11 at 14:57

Look at the excellent book of Toponogov and Rovenski, Differential Geometry of Curves and Surfaces: A Concise Guide. It is very clear and straightforward. I taught from Pressley's book and from O'Neill's book, and they are also very good.

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