Minimum reqs for differential geometry

Question

I want to study Differential Geometry for General Relativity. I find even the introductory books very tough. My background:

1. College calculus - a general course, not for mathematicians
2. Linear Algebra - introductory course
3. Some calculus I've managed to study myself - a bit of vector analysis, some diff. equations.

While reading Diff. Geometry books I feel that my math. background is probably lacking, but I don't know what areas exactly should I cover to start Diff. Geometry.

I've tried two books and I found both starting from math I'm not quite familiar with:

1. The Geometry of Physics by Frankel
2. Diff. geometry and Topology course by Fomenko and Mishenko

Do I really miss some necessary prerequisites, or should I just soldier on?

Advice is most welcome.

Answer 1

As Kobayashi and Nomizu said (or at least implied) in their classic book Foundations of Differential Geometry, you need at least three techniques of computation:

1. classical tensor calculus with indices;
2. exterior differential calculus of E. Cartan;
3. formalism of covariant differentiation $\nabla_XY$.

When I started self-studying differential geometry as a mechanical engineer, I approached the subject via Baby Rudin+Munkres -> Do Carmo -> Boothby+Marsden.

After all these years, I still think this is the most considerate, progressive approach, but not without problems: The notations are not entirely consistent; Rudin too concise, Munkres doesn't know how to be concise, Do Carmo tried too hard to hide the "manifold" concept, Marsden is obsessed with Banach space, etc etc...So after this period, I shifted to other texts, such as Helgason and Kobayashi & Nomizu, which I found to be the most complete one (and also a very physics related one for its treatment of connection theory using Ehresmann's principal bundle approach).

But could there be a better approach for me if I start all over again? I highly doubt it. One simply needs to be progressive. If I were to read Kobayashi & Nomizu at the very beginning, I may very well end up reading only 2 pages a day and understanding little to none.

My suggestion is: be progressive, go where necessary to fill in the gap, and I'm not saying you should only use the book I recommended.