Path to Manifolds from HS Algebra and Calculus?

Question

Is there a coherent path from high school algebra and beginning calculus to fully understanding the manifolds? In other words, can one self-study towards manifolds, only assuming a very modest mathematics background?

I understand that a lot of books on manifolds has been introduced, and although I had a course in topology (without any analysis), It's really hard for me to understand the subject.

So, I was just wondering if there could be a path from scratch that I can follow and also brush up on topology which I guess should be on the path.

Answer 1

In my opinion I don't think you would need commutative algebra or functional analysis to understand (at least the basics of) manifolds.

Maybe something like

1. Linear algebra (vector spaces and linear maps abstracly) - very important
2. Multivariable calculus/analysis - also very important
3. Real analysis & topology
4. Maybe a bit of abstract algebra helps (groups, rings, modules, etc.)

But I don't know what's a good order to do them. I had to keep going back to things I already read to understand them in a new way.

There are a lot of linear algebra books, it's important to learn about abstract vector spaces and linear maps. I heard good things about Axler - Linear algebra done right. I learned from Hoffman & Kunze.

Spivak - "Calculus on manifolds" is a very good multivariable calculus book that I learned from.

Rudin - "Principles of mathematical analysis", the first several chapters is just generally useful analysis and topology to have.

Then I first read Tu - "Intro to smooth manifolds", and that was good to learn from. Or Sean Carroll's GR book looks good.

Of course there are many alternative books, but I think these areas are the needed background.

Answer 2

I agree with the general path that Kreshav has laid out. Linear algebra is definitely the place to start, and it's a good topic in which to first get familiar with formal proofs in practice. I liked Serge Lang's Linear Algebra. It's then crucial to know multivariable calculus, preferably from an analysis perspective. You can pick up much of what you absolutely need about topology from a good intro analysis text (Rudin is popular). You'd then have the bare-bones tools necessary to approaching manifolds, and for doing so with an eye towards GR, the hands-down best text (in my opinion) is O'Neill's Semi-Riemannian Geometry. A text on Semi-Riemannian geometry is necessary so that one can separate results specific to Riemannian geometry (typically what is focused on in pure math contexts) from the Lorentzian case. With the mathematical footing solid, one might then look at a physics source like Wald to develop your understanding of the physical ideas, motivations, and history.

A lot of details fall by the wayside on that most direct path, in particular various results in point-set topology and a study of algebraic structures that familiarizes you with constructions like quotients and tensor products. What's more, O'Neill doesn't develop integration on manifolds (crucial for many physics discussions, such as an action approach), so one should look to an alternative text for that, perhaps Lee's smooth manifolds.

All that being said, many physicists get by without a single course in proof-based mathematics and manage to utilize GR sufficiently for their purposes all the same. With that in mind, you might do well enough by understanding multivariable calculus and vector spaces from a more computational perspective and starting with a text like Wald. Ultimately, the level of rigor you care to indulge in is up to you.