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I have a hazy notion of some stuff in differential geometry and a better, but still not quite rigorous understanding of basics of differential topology.

I have decided to fix this lacuna once for all. Unfortunately I cannot attend a course right now. I must teach myself all the stuff by reading books.

Towards this purpose I want to know what are the most important basic theorems in differential geometry and differential topology. For a start, for differential topology, I think I must read Stokes' theorem and de Rham theorem with complete proofs.

Differential geometry is a bit more difficult. What is a connection? Which notion should I use? I want to know about parallel transport and holonomy. What are the most important and basic theorems here? Are there concise books which can teach me the stuff faster than the voluminous Spivak books?

Also finally I want to read into some algebraic geometry and Hodge/Kähler stuff.

Suggestions about important theorems and concepts to learn, and book references, will be most helpful.

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I enjoyed do Carmo's "Riemannian Geometry", which I found very readable. Of course there's much more to differential geometry than Riemannian geometry, but it's a start... –  Aaron Mazel-Gee Dec 9 '10 at 1:02
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This book is probably way too easy for you, but I learned differential geometry from Stoker and I really love this book even though most people seem to not know about it. I personally found de Carmo to be a nice text, but I found Stoker to be far easier to read. I think a lot of the important results are in this book, but you will have to look elsewhere for the most technical things. –  Matt Calhoun Dec 9 '10 at 1:10
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Again, possibly at too low a level, but everything I know about algebraic geometry I learned from working through Cox, Little, and O'Shea. This book is great for self study, in my opinion. I have tried to read the major algebraic geometry texts, but they are way over my head; this book on the other hand always makes complete sense to me. –  Matt Calhoun Dec 9 '10 at 1:20
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Also, Griffiths & Harris is a pretty standard "classical algebraic geometry" book. A word of advice: don't get caught up in chapter 0. It's about 100 pages of not-so-easy complex analysis review. (Or, do get caught up in it, if that's your thing.) –  Aaron Mazel-Gee Dec 9 '10 at 9:45
    
Are there any good courses videos of MIT/standford etc.? –  0x90 Feb 22 '13 at 18:20

8 Answers 8

ADDITION: I have compiled what I think is a definitive collection of listmanias at Amazon for a best selection of books an references, mostly in increasing order of difficulty, in almost any branch of geometry and topology. In particular the books I recommend below for differential topology and differential geometry; I hope to fill in commentaries for each title as I have the time in the future.

If you want to have an overall knowledge Physics-flavored the best books are Nakahara's "Geometry, Topology and Physics" and above all: Frankel's "The Geometry of Physics" (great book, but sometimes his notation can bug you a lot compared to standards).

If you want to learn Differential Topology study these in this order: Milnor's "Topology from a Differentiable Viewpoint", Jänich/Bröcker's "Introduction to Differential Topology" and Madsen's "From Calculus to Cohomology". Although it is always nice to have a working knowledge of general point set topology which you can quickly learn from Jänich's "Topology" and more rigorously with Runde's "A Taste of Topology".

To start Algebraic Topology these two are of great help: Croom's "Basic Concepts of Algebraic Topology" and Sato/Hudson "Algebraic Topology an intuitive approach". Graduate level standard references are Hatcher's "Algebraic Topology" and Bredon's "Topology and Geometry", tom Dieck's "Algebraic Topology" along with Bott/Tu "Differential Forms in Algebraic Topology."

To really understand the classic and intuitive motivations for modern differential geometry you should master curves and surfaces from books like Toponogov's "Differential Geometry of Curves and Surfaces" and make the transition with Kühnel's "Differential Geometry - Curves, Surfaces, Manifolds". Other nice classic texts are Kreyszig "Differential Geometry" and Struik's "Lectures on Classical Differential Geometry".

For modern differential geometry I cannot stress enough to study carefully the books of Jeffrey M. Lee "Manifolds and Differential Geometry" and Livio Nicolaescu's "Geometry of Manifolds". Both are deep, readable, thorough and cover a lot of topics with a very modern style and notation. In particular, Nicolaescu's is my favorite. For Riemannian Geometry I would recommend Jost's "Riemannian Geometry and Geometric Analysis" and Petersen's "Riemannian Geometry". A nice introduction for Symplectic Geometry is Cannas da Silva "Lectures on Symplectic Geometry" or Berndt's "An Introduction to Symplectic Geometry". If you need some Lie groups and algebras the book by Kirilov "An Introduction to Lie Groops and Lie Algebras" is nice; for applications to geometry the best is Helgason's "Differential Geometry - Lie Groups and Symmetric Spaces".

FOR TONS OF SOLVED PROBLEMS ON DIFFERENTIAL GEOMETRY the best book by far is the recent volume by Gadea/Muñoz - "Analysis and Algebra on Differentiable Manifolds: a workbook for students and teachers". From manifolds to riemannian geometry and bundles, along with amazing summary appendices for theory review and tables of useful formulas.

EDIT (ADDED): 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, the kind of algebraic formulation is not used (forget about pullbacks and functors, like Tu or Lee mention), that is why an old fashion geometrical treatment may be very helpful to complement modern titles. In the end, we must not forget that the old masters were much more visual an intuitive than the modern abstract approaches to geometry.

NEW!: the book by Mishchenko/Fomenko, along with its companion of problems and solutions, has been recently typeset and reprinted by Cambridge Scientific Publishers! The original Soviet editions can still be purchased by a much cheaper prize through URSS publishers (I got my copy that way as they have direct distribution in Spain).

If you are interested in learning Algebraic Geometry I recommend the books of my Amazon list. They are in recommended order to learn from the beginning by yourself. In particular, from that list, a quick path to understand basic Algebraic Geometry would be to read Bertrametti et al. "Lectures on Curves, Surfaces and Projective Varieties", Shafarevich's "Basic Algebraic Geometry" vol. 1, 2 and Perrin's "Algebraic Geometry an Introduction". But then you are entering the world of abstract algebra.

If you are interested in Complex Geometry (Kähler, Hodge...) I recommend Moroianu's "Lectures on Kähler Geometry", Ballmann's "Lectures on Kähler Manifolds" and Huybrechts' "Complex Geometry". To connect this with Analysis of Several Complex Variables I recommend trying Fritzsche/Grauert "From Holomorphic Functions to Complex Manifolds" and also Wells' "Differential Analysis on Complex Manifolds". Afterwards, to connect this with algebraic geometry, try, in this order, Miranda's "Algebraic Curves and Riemann Surfaces", Mumford's "Algebraic Geometry - Complex Projective Varieties", Voisin's "Hodge Theory and Complex Algebraic Geometry" vol. 1 and 2, and Griffiths/Harris "Principles of Algebraic Geometry".

You can see their table of contents at Amazon. Hope this helps... good luck!

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That's certainly a nice list! But your amazon link doesn't work. –  wildildildlife Feb 18 '11 at 11:53
    
I have changed the link to the Amazon list, hope now it works –  Javier Álvarez Feb 18 '11 at 12:54
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+1 for a great recommendation list. –  Mathemagician1234 Mar 6 '12 at 10:42
    
@JavierÁlvarez Sorry to bother you on this old post but the highest math class I have taken thus far is linear algebra. Would the list you recommend help me or should I start reading more basics books? I am going to take abstract algebra, complex analysis, and analysis 1 next semester. –  diimension Dec 13 '12 at 2:21
    
@diimension: there is no bother at all! this list, and my other Amazon listmanias, will be very useful to you AFTER your next semester when you get background on rigorous analysis and algebra. Then, books like Runde's and Munkres' on topology will be at your level and you should by all means try them. Pressley or Bär should be your start in differential geometry. Keep studying and everything will be at your reach! At your level right now you could start reading the basic book by "Jänich" on topology at the same time you study next semester courses. –  Javier Álvarez Dec 13 '12 at 8:02

I had seen a mention of this work on Differential Geometry by Theodore Shifrin at UGA giving it great comments mathoverflow.

It's currently a free and legal download. It's an entry level text and the prior responders have put a lot of effort into giving outstanding suggestions. But I thought it might be of interest.

You can download it with the link on Prof. Shifrin's home page:

http://www.math.uga.edu/~shifrin/

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+1. One of the best free sources on undergraduate differential geometry and it may be nearing completion for a publisher,so download it ASAP! –  Mathemagician1234 Oct 1 at 3:15

I would recommend Jost's book "Riemannian geometry and geometric analysis" as well as Sharpe's "Differential geometry".

The first book is pragmatically written and guides the reader to a lot of interesting stuff, like Hodge's theorem, Morse homology and harmonic maps.

The second book is mainly concerned with Cartan connection, but before that it has an excellent chapter on differential topology. Furthermore it treats Ehresmann connections in appendix A.

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I reviewed the fifth edition of Jost's book for MAA Reviews awhile back-while excellently organized and written, it's very condensed and terse.I think it's best suited for a second course in differential geometry after digesting a standard introductory treatment,like Petersen or DoCarmo.Sharpe is easier and beautifully written,but it has a rather unusual selection of topics-this also makes it better suited for a second course. –  Mathemagician1234 Oct 1 at 3:18

Like the other posters, I think Lee's books are fantastic. I'd start with his Introduction to Smooth Manifolds.

For differential geometry, I'd go on to his Riemannian Manifolds and then follow up with do Carmo's Riemannian Geometry. (That's what I did.)

For differential topology, after Lee's Smooth Manifolds, I'd suggest Differential Forms in Algebraic Topology by Bott, Tu and anything (and everything) by Milnor.

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I'm doing exactly the same thing as you right now. I'm self-learning differential topology and differential geometry. To those ends, I really cannot recommend John Lee's "Introduction to Smooth Manifolds" and "Riemannian Manifolds: An Introduction to Curvature" highly enough. "Smooth Manifolds" covers Stokes Theorem, the de Rham theorem and more, while "Riemnannian Manifolds" covers connections, metrics, etc.

The attention to detail that Lee writes with is so fantastic. When reading his texts that you know you're learning things the standard way with no omissions. And of course, the same goes for his proofs.

Plus, the two books are the second and third in a triology (the first being his "Introduction to Topological Manifolds"), so they were really meant to be read in this order.

Of course, I also agree that Guilleman and Pollack, Hirsch, and Milnor are great supplements, and will probably emphasize some of the topological aspects that Lee doesn't go into.

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+1. Lee's "trilogy" is probably the best "one author"source on graduate topology and geometry that currently exists, but it's sheer length is really going to make it a tough choice for coursework. They're really best suited for a self-studying student working through them at his or her own pace. For that, the books are matchless. –  Mathemagician1234 Oct 1 at 3:22

I would like to recommend Modern Differential Geometry of curves and surfaces with Mathematica, by Alfred Gray, Elsa Abbena, and Simon Salamon. You can look at it on Google books to decide if it fits your style. If you are a Mathematica user, I think this is a wonderful avenue for self-study, for you can see and manipulate all the central constructions yourself. I use Gray's code frequently; I was a fan.

PS. Here is how he died: "of a heart attack which occurred while working with students in a computer lab at 4 a.m."!

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Alternatively, if you're a Maple guy, there's Oprea's [ Differential Geometry and Its Applications ](books.google.com/books?id=xb48zk0wJfIC). –  J. M. Dec 9 '10 at 2:25
    
@J.M. Is the Maple-based book comparable in quality to the Mathematica-based book, or should I skip it for the sake of another book? My library doesn't have access to the Mathematica-based book, hence my question. –  user89 Jan 29 at 1:28
    
Joseph, have you had a chance to look at Frankel's book "Geometry of Physics"? If you have, I would be curious to know your opinion of it. (P.S. mining your MO/M.SE answers for resource recommendations has been an absolute godsend for me -- sincerely: thanks!) –  user89 Jan 29 at 1:35
    
@twirlobite: I own the book but haven't looked at it for some time. I alway have found the lack of perspective on the front cover a bit jarring :-) cover here –  Joseph O'Rourke Jan 29 at 1:53
    
@JosephO'Rourke Ah! I have tried reading through it a bit, but I am not sure it's a text that I enjoy. What about Oprea's book? Is it comparable in quality to the one you mentioned? –  user89 Jan 29 at 2:14

For differential topology, I would add Poincare duality to something you may want to know. A good textbook is Madsen and Tornehave's From Calculus to Cohomology. Another nice book is John Lee's Introduction to Smooth Manifolds.

For differential geometry, I don't really know any good texts. Besides the standard Spivak, the other canonical choice would be Kobayashi-Nomizu's Foundations of Differential Geometry, which is by no means easy going. There is a new book by Jeffrey Lee called Manifolds and Differential Geometry in the AMS Graduate Studies series. I have not looked at it personally in depth, but it has some decent reviews. It covers a large swath of the differential topology, and also the basic theory of connections. (As a side remark, if you like doing computations, Kobayashi's original paper "Theory of connections" is not very hard to read, and may be a good starting place before you jump into some of the more special-topic/advanced texts like Kolar, Slovak, and Michor's Natural operations in differential geometry.)

A book I've enjoyed and found useful (though not so much as a textbook) is Morita's Geometry of differential forms.

I can't help you with algebraic geometry.

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@Willie: As you seem to know a bit (a lot) about this, could you suggest what would be a nice book to start with if someone is interested in harmonic analysis and PDEs and wants to know how to do this kind of stuff on non-Euclidean spaces (I guess that is what Diff Geom is about?)? Also, do you have a reference where there things are applicable in PDE (or harmonic analysis)? –  Jonas Teuwen Dec 8 '10 at 23:19
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@Jonas: I don't actually know much about harmonic analysis on non-Euclidean spaces. AFAIK most of the introductory material in that direction is in the context of symmetric spaces, and a standard reference for that is Helgason's book "Differential Geometry, Lie Groups, and Symmetric Spaces". Then you may want to look at Joseph Wolf's "Harmonic analysis on commutative spaces". In a slightly different direction, you can also look at Eli Stein's "Topics in harmonic analysis related to the Littlewood Paley theory". –  Willie Wong Dec 8 '10 at 23:34
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For PDEs, the information in most advanced texts are perfectly applicable to the case of manifolds (at least in regard to scalar functions; sections of vector bundles can get a bit trickier). So much of Hormander's "Analysis of Linear Partial Differential Operator" is applicable and Taylor's "Partial Differential Equation" also (the latter also explicitly formulate the discussion on manifolds, though the text in general is very dense). You may also want to look at Jost's "Riemannian Geometry and Geometric Analysis". There are in fact lots of words written about PDEs on manifolds... –  Willie Wong Dec 8 '10 at 23:41
    
... so it is hard to give a concrete recommendation. Another problem is that each branch of PDEs has its own trove of literature: from exterior differentiation systems (Cartan and others), to elliptic geometric PDEs (Einstein manifolds, conformal geometry, harmonic maps etc), to parabolic theory (Ricci flow, mean curvature flow), and to hyperbolic theory (wave maps, general relativity), it is hard to give the reference without knowing what your goals are. –  Willie Wong Dec 8 '10 at 23:46
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@Jonas: Then let me give a quick description of differences on the manifold setting. Simple Fourier analysis does not carry well directly to the manifold setting, since Fourier analysis requires some symmetries. You can do it by looking at coordinate patches, but the (pseudo)differential operators you define will depend on the coordinate chart you chose (though usually the principal part is invariant under coordinate change). In the absence of symmetries which allows you to define the Fourier transform group theoretically, you can otherwise do frequency decomposition using spectral theory... –  Willie Wong Dec 9 '10 at 1:42

Guillemin and Pollack's "Differential Topology" is about the friendliest introduction to the subject you could hope for. It's an excellent non-course book. Good supplementary books would be Milnor's "Topology from a differentiable viewpoint" (much more terse), and Hirsch's "Differential Topology" (much more elaborate, focusing on the key analytical theorems).

For differential geometry it's much more of a mixed bag as it really depends on where you want to go. I've always viewed Ehresmann connections as the fundamental notion of connection. But it suits my tastes. But I don't know much in the way of great self-learning differential geometry texts, they're all rather quirky special-interest textbooks or undergraduate-level grab-bags of light topics. I haven't spent any serious amount of time with the Spivak books so I don't feel comfortable giving any advice on them.

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+1 for Guillemin/Pollack-one of the great classic textbooks on any subject by 2 masters. –  Mathemagician1234 May 5 '12 at 18:41

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