I am a senior at a small liberal arts college, and this semester I will be taking Introduction to Differential Geometry at a big research university. I am very excited, but I have heard the course is quite difficult. The textbook is Differential Geometry of Curves & Surfaces by Do Carmo.
What material should I review? Which theorems from analysis, multivariate calculus, linear algebra etc. will make proofs more approachable? What are the most important proof techniques? What kind of computations should I be able to do without needing to double check a textbook?
Thanks a lot!
A differential geometry course could vary wildly, depending on who is lecturing it and what they might think is appropriate. If I were you, I would simply start working through Do Carmo. I can't think of much which could be more constructive preparation than reading the course textbook.
Having said that, I can give you advice based on what came up in the first differential geometry course which I took whilst I was an undergraduate. You should take this with a pinch of salt.
Calculus. Brush up on basically all calculus. This includes multivariate/vector calculus, if you've done it.
Linear algebra.
Any geometry you've already done.
Specific theorems you might want could be existence and uniqueness of solutions to ODEs, the Fundamental theorem of Calculus, and stuff like that.