Which course should I choose before tensor calc?

Question

I'm doing tensor calc at the beginning of next year, and I have a choice of elective next semester. I'm trying to decide between abstract algebra and differential geometry. I will do both eventually, but I will do one this year second semester, and one next year second semester (so one before and one after tensor calc). Should I take abstract algebra or differential geometry before tensors? Which would be more useful?

Importantly, the second half of the tensor calc is an intro to GR, and is pretty physics-based, whereas the first half is really just focusing on math and proofs of tensors, manifolds, etc.

Thanks :)

Answer 1

Here are the classes I propose you to consider taking:

1. Intro To Differential Geometry. This is a bit tricky business because most books about this topic is not really introductory at all and it assumes the readers strong background in Advanced Multivariable Calculus. Thus you really want to take:

2. Advanced Calculus. And there are several good books out there. Any of them would do a good job to help you gain the necessary backgrounds to understand proofs, writing proofs about theory of curves in Geometry. The "differential " part of the Geometry is the hard part, and the advanced calculus background can help you overcome this obstacle to survive the main course.

3. Intro To Topology. Both Point Set and Algebraic. As the Geometry uses the concepts of "open" and "closed" often in its proofs and properties of curves such as: connectedness, hausdoff, Tychonoff, etc..So you really want to be prepared and ready to deal with this notions and to use them in your own proofs.

   • After reading or taking those courses above, go straight to the classic: Differential Geometry by M. Spivak. This is a popular series of books that are for Differential Geometers.