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I am an engineering major and looking for a straightforward, easy to understand basic book on differential geometry to get started. At starting point, I am not looking for a comprehensive book (may be Spivak's Comprehensive Introduction to Differential Geometry series). I have background of linear algebra and advanced calculus. I am looking to learn topics such as Lie derivative, covarient-contravarient derivatives, pushforward pullback operations, Riemannian manifolds, moving frames, etc. As I am not from math major, I am confused over many previous questions asking suggestions for differential geometry, such as this, this, this, and many other answers on the similar questions.

From search on good books on the topic, I found out O'neill's Elementary Differential Geometry is meant for first course on the differential geometry. However, as I have very limited knowledge about the differential geometry, I am not sure whether this would be good starting point.

Can someone suggest me a good reference (and prerequisites, if necessary,for studying above topics)? Thanks in Advance.

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343_458
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  • Tu's Introduction to Manifolds is good. I do not recommend O'Neills book. Frankly, I think it's a pretty bad book -- it's all about annoying calculations, it only talks about surfaces (and curves), and it has the very nasty habit of making many references to formulas / definitions in previous chapters without repeating them, making you spend long periods of times looking back to find things, which is mildly infuriating. – Pedro Dec 13 '15 at 02:42

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I highly recommend Introduction to Manifolds by Loring Tu. The reason is that he specifically delays the need for topology so as to get into the meat of the subject quicker. This means that he develops the material on $\mathbb{R}^n$ first, and then uses these results plus minimal topology to generalize to manifolds. Tu's book really is the best to get into the subject quickly and concisely, particularly someone without a course in topology. I learned the material from a course that used Lee but used Tu as a reference for qualifying exams and also as a reference since then.

Introduction to Smooth Manifolds by John M. Lee is good, but it is more verbose, less focused, and requires a lot more topology to get through. Tu's book covers the same core material.

Given your background, you have the minimal prequisities for Tu, but it might be a little tough at first recalling a lot of your vector calculus and linear algebra. You will have to pick up pieces of algebra, analysis, and topology along the way, but I don't see a reason why you need to learn all of that before trying out Tu. Just go through it and read up on subjects when you need to.

If you truly find yourself stuck on Tu, then I recommend the following two books to help out:

  • Advanced Calculus: A Geometric View by James Callahan
  • Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach by Hubbard and Hubbard

Both of these books should get you back on track regarding any missing material from your background. They are both excellent.

Other books to be aware of regarding supplemental material are:

  • The Geometry of Physics by Theodore Frankel - Really fantastic book, but I haven't gone through all of it as I have Tu and Lee. It is quite comprehensive, and there isn't a heavy emphasis (from what I have read) on proofs. Thus there are a ton of examples and applications.
  • Applied Differential Geometry by William Burke - Lots of great pictures and insight. It's too quirky to use as your main learning source though, but it's great for supplemental insight.
  • Geometrical Vectors by Gabriel Weinreich - This is a very short but insightful read that would help put vectors, covectors, etc. into context.

Three other books on the subject that I am aware of but have no experience with:

  • First Steps in Differential Geometry by Andrew McInerney
  • Manifolds and Differential Geometry by Jeffrey M. Lee
  • Differential Geometry: Curves - Surfaces - Manifolds by Wolfgang Kuhnel
nikofeyn
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My 3 favourites are:

  • Introduction to Smooth Manifolds - John M. Lee
  • Introduction to Manifolds - Loring Wu. Tu
  • Analysis on Manifolds - James R. Munkres - expensive and hard to get. I'd recommend for a physicist.

However I am loving:

  • Smooth Manifolds - Rajnikant Sinha

And I think that may be better for a physicist. However I've only scanned a Library e-book, my copy will arrive on Monday hopefully.

A great topology book to complement ITSM by JML is:

  • Introduction to Topological Manifolds - John M. Lee
  • Topology - James R. Munkres
  • Introduction to Topology - Gamelin and Greene (nice cheap Dover book, VERY good)
  • Introduction to Topology - Bert Mendelson (nice cheap Dover book, VERY good, not as broad as the others but very good none the less)
Alec Teal
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  • Any questions about specifics please do ask. – Alec Teal Dec 13 '15 at 03:00
  • It seems that to read Lee's book or Tu's book, one should have good background in linear or abstract algebra to understand things such as homomorphism, dual vector space, field, etc. Do you suggest any book for that? – 343_458 Dec 27 '15 at 19:54