Differential Geometry

No one cares about that autistic shit.

Hilbert wasn't a physicist.

Who is that supposed to be? Newton?

no shit, that's why modern math sucks so much fucking dick and is ridden with redundant pseudo-intellectual jargoneering. physicishits are so fucking shallow and retarded they think they are reinventing math. stick to pre-19th century number theory and statistics if you want "real" math.

Know topology first, say Munkres. And basic linear algebra and calculus.

Lee's books on Topological Manifolds and Differential Manifolds are good. There's an incomplete manuscript by Robbin and Salamon I found online which is very similar to Lee, but I found slightly more clear for the tensor algebra/cohomology part. There's also a Dover book "Tensor Analysis on Manifolds". Both of these could be used concurrently with Lee for the algebra part. After those, find any handout on De Rham cohomology online, and then pick up a book on Riemannian Manifolds (say the springer one).

looks like that preprint got more developed; might be worth seeing if you can learn from it entirely, as the part I read was very clear.

if you want a self-contained and very precise, but dry treatment, get Lee's trilogy
if you're in for a fun ride, get Spivak's Comprehensive Introduction to Differential Geometry

interesting take; I actually found Lee to be pretty conversational in style. Even moreso than how people talk about Hatcher. He does a good job at telling you what things are like in practice vs general theory, and qualitatively why such and such constraints are imposed.

I'm recommending "Semi-Riemannian Geometry With Applications to Relativity" by Barret O'Neill for diffgeo. It's for some reason not well known, but it's a great textbook.

Choosing my first "final year" math subject for my math major.

Can Topology be applied for jobs?

If I dont like trig will i not like topology?