# Introductory texts on manifolds

I was studying some hyperbolic geometry previously and realised that I needed to understand things in a more general setting in terms of a "manifold" which I don't yet know of.

I was wondering if someone can recommend to me some introductory texts on manifolds, suitable for those that have some background on analysis and several variable calculus. A lecturer recommended to me "Analysis on Real and Complex Manifolds" by R. Narasimhan, but it is too advanced.

I had a look at Loring W. Tu's text on manifolds and it seemed accessible.

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A related question: books on Calculus on Manifolds math.stackexchange.com/q/14475/823 –  Baudrillard Jun 20 '11 at 15:46
If you're interested in things mostly centred around 2-dimensional hyperbolic geometry, Singer and Thorpe's "Elementary Topology and Geometry" is quite nice. But without more specifics from you it's not so clear what to recommend. –  Ryan Budney Jun 20 '11 at 18:11

(Another interesting answers to a similar question are in Teaching myself differential topology and differential geometry. You may find interesting other books which are recommended there).

Just as you mention it, I strongly recommmend the new edition of Tu - "An Introduction to Manifolds" since it is accessible but also very well-organized and motivated and basically starts up from multivariable calculus and ends up with cohomology of manifolds (it is very useful for example to get the needed background to follow his other more advanced and topologically focused text Bott/Tu - "Differential Forms in Algebraic Topology"). Moreover it includes hints and solutions to many problems!.

A little bit more advanced and dealing extensively with differential geometry of manifolds is the book by Jeffrey Lee - "Manifolds and Differential Geometry" (do not confuse it with the other books by John M. Lee which are also nice but too many and too long to cover the same material for my tastes). You can use it as a complement to Tu's or as a second reading. It is much more complete since it deals with all the stuff in Tu's but includes a lot more like vector bundles and connections, Riemannian geometry, etc.

In the same spirit of the previous book but a little better in my opinion, and even more complete, is the title by Nicolaescu - "Lectures on the Geometry of Manifolds". Its table of contents is amazing in scope dealing with some advanced topics most other introductory books avoid like classical integral geometry, characteristic classes and pseudodifferential operators. It supposedly builds everything up just from a background in linear algebra and advanced multivariable calculus. It may seem a little bit advanced at first, but it is the best book to read with/after Tu's. Its exercises are quite solvable and I learned a lot from it.

In the end, my advise is to get Tu's and if you feel comfortable after a while with it and want to learn more on the geometry of manifolds, get Nicolaescu's (or Lee's).

Besides this, I strongly recommend you get the incredible book by Gadea/Muñoz - "Analysis & Algebra on Differentiable Manifolds: A Workbook for Students and Teachers". This title is quite overlooked outside of Spain I believe, but it is a very insightful and detailed treatise of solved problems about almost every introductory topic of the differential geometry of manifolds.

If you look for an alternative to Tu's I believe the best one is John M. Lee - "Introduction to Smooth Manifolds"; it is a well-written book with a slow pace covering every elementary construction on manifolds and its table of contents is very similar to Tu's. Other alternative maybe Boothby - "Introduction to Differentiable Manifolds and Riemannian Geometry" since it also builds everything up starting from multivariable analysis. If you prefer a transition from differential curves and surfaces focusing on riemannian geometry you have Kühnel - "Differential Geometry: Curves, Surfaces, Manifolds".

However, I would argue that one of the best introductions to manifolds is the old soviet book published by MIR, Mishchenko/Fomenko - "A Course of Differential Geometry and Topology". It develops everything up from $\mathbb{R}^n$, curves and surfaces to arrive at smooth manifolds and LOTS of examples (Lie groups, classification of surfaces, etc). It is also filled with LOTS of figures and classic drawings of every construction giving a very visual and geometric motivation. It even develops Riemannian geometry, de Rham cohomology and variational calculus on manifolds very easily and their explanations are very down to Earth. If you can get a copy of this title for a cheap price (the link above sends you to Amazon marketplace and there are cheap "like new" copies) I think it is worth it. Nevertheless, since its treatment is a bit dated, it lacks the kind of hard abstract algebraic formulation used nowadays (forget about functors or exact sequences, like Tu or Lee mention), that is why I believe an old fashion geometrical treatment may be very helpful to complement modern titles for a person entering the subject needing a good geometrical foundation. In the end, we must not forget that the old masters that founded the subject were much more visual an intuitive than the modern abstract approaches to geometry, and that motivation was what culminated in the unified abstract approach of nowadays.

Since this last book is out of print and the publisher does not longer exist, you may be very interested in an online "low-quality" copy which can be downloaded here (the 3 files linked in rapidshare).

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Thanks for your suggestion in fact thanks to everyone here who is more experienced than me for giving their suggestions. I had a look at the John Lee book and it starts off with topological manifolds which is different from Tu's book that starts off with differentiable functions. –  user38268 Jun 21 '11 at 12:46
Do not confuse the two books by John Lee. There is a first volume on "topological manifolds" and a second volume on "smooth manifolds" (and even a third one on "Riemannian geometry"). It depends on what you are interested in. In my opinion "topological manifolds" is just a book about topology, most titles when considering manifolds mean "smooth" ones since differential geometry works mainly in that category. One usually has already taken a course in topology when getting into manifolds. Lee's books are great anyway so you must pick whatever you are interested in. –  Javier Álvarez Jun 21 '11 at 14:38
+1 for a very nice list of recommendations indeed which I mostly agree with. Sadly,the HUGE problem with Gadea/Muñoz is that the solutions are given CONCURRENT with the problems. They would have had a far more useful and versatile book if they had separated the problems and their full solutions into 2 different sections of the book. If they had done that,the book would probably have been a huge success as a necessary supplement to some of the great exercise-less lecture notes on the subject-such as S.S. Chern's classic notes (in whatever incarnation you have) or Taubes. –  Mathemagician1234 Oct 25 '12 at 7:37
Thank you, Javier, for a very nice list of books. Following it, I have finished Tu's textbook, and started Jeffrey Lee's "Manifolds and Differential geometry". I want to ask what you think about the book of S. Ramanan "Global Calculus". Don't you think that at some point one should learn ringed space approach to manifolds using sheaves? –  Bananeen Jul 27 '14 at 9:15
@Bananeen: thank you. Yes, I do think that one should also learn at least the basics of the sheaf approach for manifolds via ringed spaces, and Ramanan is an excellent introduction to that. This is fundamental if one wishes to understand differential geometry in a similar language to modern algebraic geometry, although this approach is usually not required or even explained in most university courses. If you like algebraic-flavored treatments (but without sheaves), don't forget the advanced and formal book by Michor -"Topics in Differential Geometry". –  Javier Álvarez Jul 27 '14 at 11:09

Introduction to Smooth Manifolds by John M. Lee is a great text on the subject. It covers similar material to Loring W. Tu's text. Lee's book is big (~650 pages) but the exposition is clear and the book is filled with understandable examples. You will be able to find course notes that follow this book, and it's always nice to see the same things in different perspectives.

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I am immensely fond of Shastri's new book Elements of Differential Topology, mainly because it is concise and covers a lot.

It is only 310 pages long, but the font is extremely small, so there are a lot of things in there.

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I haven't had a chance to really look at it yet,but what I HAVE seen of it looks pretty good! –  Mathemagician1234 Oct 25 '12 at 7:39

Another suggestion:

1) Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach by John Hubbard: (text's website)

From what I've read of the excerpts, reviews, and table of contents, it looks to be a great book. Hubbard covers all the necessary linear algebra and presents to you calculus on manifolds, while integrating it into vector calculus. I look forward to going through this book. He also has some very nice physical applications, which includes Maxwell's equations.

2) An Introduction to Manifolds by Loring Tu (As others have suggested!)

The more abstract and general than Hubbard, but it is entirely accessible to upper-level undergraduates. This book gives differential forms based upon their general definition, which requires the development of multi-linear and tensor algebra. Highly recommended, esp. new edition.

3) Manifolds and differential geometry, by Jeffrey Marc Lee (Google Books preview)

4) Also, I just recently recommended this site in answer to another post; the site is from Stanford University; it offers a vast menu of detailed handouts used as the text for a class there on Differential Geometry, each handout accessible/downloadable as a pdf. They were found to be quite helpful to the user, and I've bookmarked the page, myself, for future reference.

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I enjoyed Fecko's Differential Geometry and Lie Groups for Physicists. It doesn't contain complete bottom-up theory building and omits hard proofs but it is a very neat general introduction to the basics of manifolds; it explains very well why the stuff should work the way it does and also provides very nice (usually physical) applications. In addition, it contains a big amount of interesting exercises.

In the latter chapters it also briefly scratches interesting topics from algebraic topology and group theory. I think an informal and high-level book like this is useful addition to the rigorous texts. Especially for beginners.

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Luckily there are lots of good books on manifolds. Lee's 'Introduction to Smooth Manifolds' seems to have become the standard, and I agree it is very clear, albeit a bit long-winded and talky. Warner's Foundations of Differentiable Manifolds is an 'older' classic.

Javier already mentioned Jeffrey Lee's 'Manifolds and Differential Geometry' and Nicolaescu's very beautiful book. I'd like to add:

Conlon - Differentiable Manifolds

Isham - Modern Differential Geometry for Physicists

Morita - Geometry of Differential Forms

Michor - Topics in Differential Geometry

Contrary to what you might suspect from the title, Isham's text is very mathematical; basically there is no physics at all. It is just a very clear introduction to manifolds (with a 50 page introduction to topology) covering vector fields, differential forms, Lie groups, Fibre bundles, and connections.

Morita has a way of explaining some quite advanced topic in a very understandable manner. Also he is not afraid to view a concept of different angles, first in an elementary way and later in more advanced terms (e.g. the notion of a 'connection' in a vector bundle, then in a general fibre bundle, and then looking back at the vector bundle notion with the from the more general perspective).

Michor's text might be considered as a 'second' textbook, at least if you look at the topics he covers. He has an extensive chapter about Lie groups. He covers differential forms and De Rham Cohomology (which is where most other books mentioned in this thread stop), and then talks about cohomology with compact support, Poincaré Duality, and cohomology of compact connected Lie Groups. Next bundles and connections, Riemannian Manifolds, Isometric Group Actions, Symplectic and Poisson Geometry are treated. So yeah, it's quite heavy and probably not an introduction, although I've found it useful at times when I learned this stuff for the first time (a year ago).

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Tu's book is definitely a great book to read for someone who doesn't know the first thing about manifolds. I have sampled many books on manifold theory and Tu's seems the friendliest. The most illuminating aspect of it, for me at least, is the fact that it presents the basics of differential and integral calculus on $\mathbb{R}^n$ in a coordinate-free fashion before even mentioning what a manifold is. Some might consider this boring, but I found it extremely helpful when similar concepts were introduced for abstract smooth manifolds. Since I was already familiar with the concept in the case of $\mathbb{R}^n$, I had some nice geometric intuition at my disposal (for instance, I find it impossible to understand what a 1-form is (a cross-section of the cotangent bundle, yeah...) without considering first what a 1-form on $\mathbb{R}^n$ is!). In addition, this approach teaches you to "think in a coordinate-free way", but in the familiar Euclidean space most students already feel comfortable with.

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Bishop and Goldberg, Tensor Analysis on Manifolds.

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+1. A very good and underrated book-and available very cheap from Dover! –  Mathemagician1234 Oct 25 '12 at 7:40

I'm way late to the party, but for an example requiring very very few prerequisites, Reyer Sjamaar's notes on Manifolds and Differential Forms are very well-organized and accessible. They can be accessed for free here on his website. Perhaps too elementary, but I'm not entirely sure of your background.

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