Differential geometry recommendations

[Terilien]

what is a good book in differential geometry. I currently know calculus, a bit about differential equations, a bit of linear algebra and a bit about tensors. I also know some variational calculus.

Of course what I know won't really help. I've skimmed through some physics sources and mathematical sources and have decided exactly what I want. I currently don't have any.

I'm looking for something clear and intuitive. something that doesn't throw formal definitions in your face without explaning what motivates them. I would like something that explains what lead to certain concepts and theorems. That's what tunred me away from most mathematics books. i don't think it helps a beginner to start a book that opens with "Let a be (insert formal terms here thing)".

I would however like the book to have mathematical rigour. Things, even if somewhat obvious, should be proven and not taken for granted. NOTHING should be taken for granted. This feature turned me away from a great deal of physics texts.

so I would like something that is informal and intuitive but still rigorous. a kind of medium.

PS.: I will admit that I'm a very impatient person. I am working on it though.

Some chapters introducing topology would also be nice.

I personally think that formality these days is killing mathematics. I admit that it is important, but it is also important to see how definitions arise.

 

[garrett]

I really like this book:

https://www.amazon.com/dp/0521539277/?tag=pfamazon01-20

And if you want to know how the basic elements of differential geometry all weave together, informally, you might have fun poking around my wiki:

http://deferentialgeometry.org/#Welcome manifold 1-form"

 

[AlphaNumeric]

Nakahara - Geometry, Topology and Physics is an excellent book, though it is aimed at graduate students so chunks of it are perhaps a bit more advanced than you're after but it does cover topology and differential geometry (you'd never have guessed from the name ) from fairly basic things as well as put them in a physical context so you can see what applications such things have outside of "It's an interesting bit of maths". Applications like general relativity, quantum mechanics, super fluids, string theory, gauge theories, supersymmetry etc. I never got the notion of topologies until I sat down and read this book, and I'm postgrad!

I'd advise flicking through it before buying if you're a 1st or 2nd year student, but if you're a 3rd year I very much recommend it.