>>12004992Important: I'm a math major. I don't study physics but I have a lot of physics friends who are very advanced and I talk to them often
You'd definitely want to have a solid course on differential geometry. Before learning differential geometry, though, make sure you have a solid grounding on smooth manifolds (avoid books that treat manifolds as embedded in R^n). When I say differential geometry I'm not just talking about learning Riemannian Geometry, but also learning about principal bundles, connections on principal bundles, associated vector bundles, etc. Principal bundles serve as the foundation for Gauge Theory, which I've been told is important in particle physics, Yang-Mills theory, etc. For smooth manifolds, I like John Lee's "Introduction to Smooth Manifolds". A good book for Riemannian Geometry is "Riemannian Geometry" by Manfredo do Carmo. There's really no good book covering the relevant material on principal bundles, I personally like "Mathematical Gauge Theory: with applications to the standard model of particle physics" by Mark Hamilton.
A book that covers most (all?) of the stuff I mentioned above is Nakahara's "Geometry, Topology, and Physics". It's one of those books that you're supposed to use as a reference, rather than to learn material for the first time. Another book similar to Nakahara is "The Geometry of Physics" by Theodore Frankel. The difference between Nakahara and Frankel is that Nakahara is more rigorous.
