Where to start learning Differential Geometry/Differential Topology?

Solution 1:

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.

Solution 2:

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.

Solution 3:

I highly recommend Topology from the Differentiable Viewpoint by Milnor.

Solution 4:

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.