Undergraduate differential geometry texts 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.)
 A: 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.
A: 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
A: There is our book What is differential geometry: curves and surfaces.
It is written for those who either plan to work in differential geometry, or at least want to have a good reason not to do it.
(Sorry for self-advertisement.)
A: 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. 
A: 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...)   
A: 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.
A: 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.
A: 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.
A: 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. 
A: 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:
A: 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" 
quite readable. 
(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.) 
If this is too advanced for your purpose, I would recommend
"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
A: There's also Modern Differential Geometry of Curves and Surfaces with Mathematica by Alfred Gray.
A: 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!
A: 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).
A: 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.
A: I enjoyed teaching "Curves and Surfaces" with notes of Theodore Shifrin, which are here: http://alpha.math.uga.edu/~shifrin/ShifrinDiffGeo.pdf 
A: 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.  
A: 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.
