# Where to start learning Differential Geometry/Differential Topology?

I realize that this may be a very general question, perhaps even an unclear one (if it is I apologize), but as someone looking for the best way to start learning about these topics, I find that there is no clear path to learning Differential Geometry / Differential Topology, as there is with Analysis or General Topology, or even Abstract Algebra

For example in Analysis, most agree that Principles of Mathematical Analysis by Walter Rudin is the place to begin, for Topology, Munkres book is the standard reference, and for Algebra, most tend to use either Dummit and Foote, Artin, Fraleigh or Lang.

For Differential Geometry/Differential Topology, I find that there are no standard texts, the only one I know of is Lee's Introduction to Smooth Manifolds, however I feel I currently lack the prerequisites to tackle that book properly.

Now I understand that to recommend a book to someone, you would need some gauge of their mathematical ability/maturity, but it is next to impossible to demonstrate that, so instead I can give a list of books that I'm currently reading through, and plan to read through in the next 3-6 months.

• Principles by Mathematical Analysis (Baby Rudin)
• Linear Algebra Done Right (by Sheldon Axler)
• Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach (by Hubbard and Hubbard)

## What I plan on reading soon

• Calculus on Manifolds by Spivak
• Topology by Munkres
• Complex Analysis by Alfhors
• Abstract Algebra by Dummit and Foote

But after that I'm lost as to where to go further. I'm lost between Analysis on Manifolds by Munkres, A Comprehensive Introduction to Differential Geometry by Spivak, and do Carmo's Differential Geometry of Curves and Surfaces.

Or should I just skip all those intermediate books and go straight to Lee's Introduction to Smooth Manifolds?

A Side note I find that the more challenging a book I read is, and the more I struggle through a book, I develop a deeper understanding of the topics in the book, and a greater appreciation of the subject I'm studying as a whole. Based on the books I've read/plan to read, please recommend books that are not easy, but difficult and challenging.

• Another note, I do not intend on becoming a Differential Geometer, but I do want to learn Differential Geometry/Differential Topology quite deeply – Perturbative Oct 28 '16 at 7:19
• For differential topology (which I find quite different from differential geometry in spirit) I would advise you to take a look at the beautiful short book Differential Topology by Milnor. – Daniel Robert-Nicoud Oct 28 '16 at 7:41
• I have looked at Spivak's book (the comprehensive one) briefly while I was an undergrad and I think you will like it for your purposes (I might get myself a copy someday for similar reasons). The book appeared to be very thorough in its coverage of the material and it starts with the basics of the subject. I've done some differential geometry in a topics class where we studied Lie Theory, and I really wise I understood the foundational aspects of manifolds better than I do currently (atlases and charts and all that). – Justin Benfield Oct 28 '16 at 7:54
• Why not study from multiple sources at once? It is often helpful to have multiple expositions and sources of problems; if you find one unhelpful, you can switch to another. – Will R Oct 28 '16 at 10:25
• What do you guys think of "Topics in Differential Geometry" by Michor? See amazon.co.uk/Differential-Geometry-Graduate-Studies-Mathematics/…. – Trajan Dec 23 '18 at 20:20

Differential Geometry by Barrett O'Neil and Introduction to Manifolds by Tu. The second is my all time favorite. It filled so many gaps for me.

• just wanted to thank you for making me aware of Introduction to Manifolds by Tu, skimming through the book, it's the best book I've found on the topic so far for an introduction – Perturbative Nov 8 '16 at 9:21
• No problem. Yes, it is a wonderful book. – Faraad Armwood Nov 8 '16 at 16:32
• Loring Tu's An Introduction to Manifolds (be sure to use the Second Edition) is not only the best and easiest introductory book to differential topology, but it is one of the best undergraduate books in mathematics, period. It is in my opinion at the quality level of Atiyah-MacDonald's Commutative Algebra, and this is the highest compliment I can think of (although the comparison between books on such completely different subjects is of course very subjective). – Georges Elencwajg Oct 1 '17 at 9:16
• Well said @GeorgesElencwajg. – Faraad Armwood Oct 1 '17 at 15:54

It's been around 11 months since I first asked this question. I thought I would share my path to learning Differential Topology and Differential Geometry. Hopefully this will be of some help to others who are also hoping to learn Differential Topology and Differential Geometry.

Firstly I read most of the contents of the General Topology part of Munkres. I tried afterwards to go through Calculus on Manifolds by Spivak, but I got bored really quickly, and as a book I didn't particularly enjoy reading or working out of it that much.

So I jumped straight ahead to reading Topology from the Differentiable Viewpoint by Milnor, this quickly became one of my favourite books I've ever read. There was a saying I read somewhere on MathOverflow which said

Run don't walk your way to Milnor's Topology from the Differentiable Viewpoint

That couldn't have been more true. (If as a reader to this answer, this is the most important thing to take away) You just need a bit of General Topology and the basics of multivariable calculus and linear algebra to tackle it. In it's short 50 pages, it takes you deep into Differential Topology. I'm planning on rereading it again.

I'm currently reading Differential Topology by Guillemin and Pollack which is a superb supplement to Milnor's book.

The only drawback (although not a bad one) is with Milnor's and Guillemin and Pollack's books, all smooth manifolds are embedded in some euclidean space $\mathbb{R}^n$, and aren't abstract, though due to Whitney's Embedding Theorem this isn't too much of an issue.

I am also currently reading Introduction to Smooth Manifolds by John Lee which is an incredibly well written book, it's clear, filled with tons of examples and exercises. I've also browsed through Introduction to Manifolds by Tu but compared to Lee's book I don't use it as much.

Finally, I think a book that is worth mentioning is Introduction to Topological Manifolds also by John Lee which acts as a great first encounter to topological manifolds.

I highly recommend Topology from the Differentiable Viewpoint by Milnor.

You mentioned do Carmo's Differential Geometry of Curves and Surfaces but if you want to study modern differential geometry it may be more appropriate to focus on his excellent text Riemannian geometry, published a decade later. It combines geometric clarity with a teaching experience of decades (do Carmo's, that is). I personally used it in teaching a course in Riemannian geometry and warmly recommend it. All that is required is a solid basis in advanced calculus. Do Carmo's textbook is certainly not exhaustive in any sense but it gives you a pleasant point of entry which you can use as a springboard for further studies in differential geometry.

• I love Do Carmo's books, and highly recommend them! Riemannian Geometry by Do Carmo is a beautiful textbook. It does not cover everything, but it covers a nice selection of topics, up to a proof of the topological sphere theorem, if I remember well, in the last chapter, with the famous quarter-pinched condition. And it also covers many more basic concepts: geodesics, exponential maps, curvature, completeness, with related theorems. – Malkoun Oct 28 '16 at 10:41
• I still think it's preferable to preface a graduate course in Riemannian geometry with some concrete, hands-on experience with curves and surfaces. But I realize that lots of people disagree with me. – Ted Shifrin Oct 29 '16 at 23:11
• @TedShifrin, I agree with you, and moreover that's precisely what I do when I teach differential geometry. The OP seems already to be familiar with do Carmo's undergraduate book. – Mikhail Katz Oct 30 '16 at 7:52

Use Guillemin and Pollack's for Differential Topology, it is a jewel.