Introductory Differential Geometry Book With Lots of Intuition

[Poopsilon]

So I took an analysis class which covered chapters 9 and 10 of Rudin's PMA, for those of you who don't know that's multivariable analysis and differential forms, and I have taken a course in vector calculus but never a proper course on differential geometry. Thus my introduction to the subject has been a bit backwards and short on both geometry and intuition.

Thus I was hopping you fine fellows could recommend me a good introductory book on differential geometry which is rigorous, but that will first discuss the geometry of curves and surfaces in R3 in terms of vector analysis, before moving on to differential forms, so that I can properly appreciate the motivation for such abstractions.

 

[mathwonk]

I highly recommend the free differential geometry notes by Ted Shifrin.

http://www.math.uga.edu/~shifrin/The book by David Bachman of Pitzer College on the geometry of differential forms, read here as a community project some years ago, are excellent for grasping the meaning of this tool.

https://www.amazon.com/s/ref=nb_sb_...words=david+bACHman&x=0&y=0&tag=pfamazon01-20I annoyed some people at the time by pointing out tiny mathematical errors in his exposition, but the book does a great job of what it intends to do, explain the geometry behind differential forms, as well as how to calculate with them.

 

[Poopsilon]

You're right Mathwonk those notes by Ted Shifrin do look excellent. And while I have your attention, I also took an introductory class on algebraic geometry recently which used Ideals, Varieties and Algorithms, which was a bit too computational for my tastes, and so now I'm looking for something that takes a cleaner more geometric approach to the subject. I was thinking of getting the Red Book by Mumford, what do you think?

 

[mathwonk]

that red book is a great book by a fields medalist, and it is superb. having said that, although necessary, it is not sufficient for most of us, who need more examples, and for that i recommend shafarevich, BAG.

 

[Poopsilon]

Excellent, thanks =].

 

[Obis]

I also recommend Shifrin's differential geometry. It's short, interestingly and cleverly written.