What books or aid can I use to learn differential geometry

[Mathmanman]

I am very curious with what differential geometry is.
Can you send me links, books, and etc? I want to learn it.
Thank you in advance

 

[Simon Bridge]

It's actually the same answer as to your other thread:
https://www.physicsforums.com/showthread.php?t=755164
... even though the question is different.

 

[Mathmanman]

Ok, but I need some resource that also gives me problems to solve...

 

[micromass]

What is your current knowledge? Do you know calculus? Multivariable calculus? Linear Algebra? Topology? Real Analysis? Etc.

Differential Geometry is essentially split into two parts. The first part is classical differential geometry and deals with curves and surfaces embedded in Euclidean space. The second part abstracts this theory and does away with the underlying Euclidean space. It is the theory of manifolds.

I highly suggest to learn the classical case first. It is also very beautiful. Things you should learn are the Theorema Egregium and the Gauss-Bonnet theorem.

The classical book to consider is Do Carmo: https://www.amazon.com/dp/0132125897/?tag=pfamazon01-20 The exercises are often not easy.

There is also Pressley: https://www.amazon.com/dp/184882890X/?tag=pfamazon01-20 This is more elementary

One of my favorites is Bar: https://www.amazon.com/dp/B00AKE1X8E/?tag=pfamazon01-20 But this book suffers from a real lack of exercises

Also very good is Millman and Parker: https://www.amazon.com/dp/0132641437/?tag=pfamazon01-20 This one has very good exercises which aren't too difficult. But the book is quite old and feels quite old.

Finally, there is O' Neill https://www.amazon.com/dp/0120887355/?tag=pfamazon01-20 This book does everything with the modern language of forms. This might be weird to people.

 

[Mathmanman]

Yes I learned integral and differential calculus and linear algebra.

 

[micromass]

Yes I learned integral and differential calculus and linear algebra.

Then any of the books I listed should be fine.

 

[Daverz]

Also very good is Millman and Parker: https://www.amazon.com/dp/0132641437/?tag=pfamazon01-20 This one has very good exercises which aren't too difficult. But the book is quite old and feels quite old.

This amuses me because I can remember buying my copy probably not too long after it came out.

I think it's still an excellent, readable text.

There's also this very polished online text:

http://www.math.uga.edu/~shifrin/ShifrinDiffGeo.pdf

To prepare for a presentation using differential forms, the book by Bachmann is a gentle -- though very brief -- introduction:

https://www.amazon.com/dp/B001RGEHWI/?tag=pfamazon01-20

 

[finnk]

For a more modern introduction to differential geometry start with The Shape of Space by Jeffrey Weeks then read John M Lee Trilogy (Topological Manifolds - Smooth Manifolds - Riemannian Manifolds).

About classical differential geometry, i love differential geometry by JJ Stoker, not sure why it's not mentioned often. the other good one is geometry from differentiable viewpoint by John McCleary.

 

[NumericalFEA]

I am very curious with what differential geometry is.
Can you send me links, books, and etc? I want to learn it.
Thank you in advance

Differential geometry has a variety of applications. For example, mechanics of shells is one of such areas, because it profoundly deals with surfaces in 3D space. If you want a book of real value, you need something with computer source codes, implementing various differential geometry algorithms, so you'd have some really working stuff "to play with". For example, if you are interested to learn about such thing as lines of principal curvature on surfaces, including related numerical methods and software codes, try this (Chapter 5 is devoted specifically to that subject):

https://www.amazon.com/dp/0646594044/?tag=pfamazon01-20

 

[whyevengothere]

Marcel Berger's books(geometry revealed,differential geometry and a panoramic view of riemannian geometry) are apparently masterful,but I think they're research-level,am I wrong?