How to learn differential geometry?

[SMHPhysics]

Hello everyone!

I just wanted to ask a question about how I should study for differential geometry. Now, as I have it, I've got a few suggestions for books, of which two stand out prominently:

1. John Lee's Introduction to smooth manifolds

2. De Carmo

Which one would be best for self study, bearing in mind that I'm at high school, although I do have familiarity with real analysis to the level of 'Calculus' by Spivak, linear algebra to the level of Georgi Shilov, and multivariable calculus as well, but little familiarity with sets and groups. So I suppose I am somewhat familiar with writing out proofs. By the way, I expect that both are mathematically rigorous, but also I want to see differential geometry in general relativity as well. If you have any other recommendations for books, by all means let me know.

Thank you!ALSO: PLEASE DO NOT HIJACK THIS THREAD!

 

[lavinia]

I personally believe that it is wise to start with classical differential geometry. This because the examples are easy to visualize and the mathematical structures are less dense. You can learn using multivariate calculus only and do not need to know the theory of differentiable manifolds as a prerequisite.

There are beginning books which link classical geometry to the modern formulations like the wonderful book by Singer and Thorpe.

Personally I would read Struiks' book first then Singer and Thorpe. Read them together.

 

[quasar987]

A widely used book is Elementary Differential geometry by Pressley.

It studies the riemannian geometry of curves and surfaces, which are the 1 and 2-dimensional manifolds respectively and are used to build intuition for the general theory which you can find in Lee's book(s).

 

[micromass]

Lee's smooth manifold books is really a wonderful book. But I'm afraid that you miss the prereqs and the mathematical maturity to tackle the book.

I agree with the suggestions of lavinia and quasar, they are very good. Another book that I really like is: https://www.amazon.com/dp/0132641437/?tag=pfamazon01-20
I personally like how millman explains things.

 

[WannabeNewton]

You can do Lee's book, technically, without a pre requisite knowledge of classical geometry of surfaces but he does assume you have thorough understanding of smooth and topological manifolds. Do Carmo is essentially the opposite. Ideally, it would prove good to have knowledge of both. Do Carmo's book on differential geometry of curves and surfaces would lead naturally to his book on riemannian geometry and, unlike some other diff geo of surfaces books, Do Carmo's problems contain very little computations and mostly proofs which is a huge plus. Lee's books on topological and smooth manifolds would also naturally lead to his book on curvature. Lee's book covers less topics than Do Carmo's book but Lee is an amazing teacher so you can't go wrong with his text (maybe you could get both; they are pretty cheap even if new). If you are interested in general relativity in particular and want a book that utilizes differential geometry at a level comparable to that of Do Carmo \ Lee (comparable for a physics textbook at least) then the unequivocal choice would be Wald's General Relativity; Spacetime and Geometry by Sean Carroll would be a great close - second but it is at a slightly lower level. However, like micromass said you need to have the necessary pre - requisites down otherwise it won't really be a pragmatic venture.