[Math] frustrating experience about differential geometry

Question

I am felling rather frustrated now, after taking a long time to study differential geometry, but with little progress…

Indeed my major is mainly numerical analysis. I am studying modern geometry, because I am interested in some points of geometric mechanics, like in Arnold and Jerry Marsden's beautiful books. And on the other hand, I am preparing for some researches related to some "geometric" topics, like general relativity.

I began to study differential manifold etc. from undergraduate, mostly by myself. I began by S.S.Chern's book. I thought it was a mistake, because that book is very formal. Indeed I read that for several times, but when I close the book, I did not know what I have read…

Now the situation gets better, but still difficult… I try to drive the formulations by myself, try to write them down. But the progress is still quite slow.

When I deal with PDEs, I can see clearly the essential points in estimates. But when I turn to Riemannian geometry, Lie groups etc., I get lost in the confusing notions, and have no idea to start MY OWN proof. For example, I used much time to understand what a "pull back" is. It seems no two books use the same notation!

So could you give me some suggestions? Is it because these stuffs are indeed much harder then analysis, or I haven't found the right way?

Thanks a lot!

Answer 1

The basics of differential geometry is secretly just multivariable calculus; in my opinion, the main, new ideas it introduces are:

• the use of scalar fields where one is used to talking about variables (e.g. using $y$ to mean the function on $\mathbb{R}^3$ that sends a point to its middle coordinate)
• paying attention to what should be a vector (in coordinates, that means column vector) and what should be a covector (in coordinates, that means row vector). e.g. if $f$ is a scalar field, $\nabla f$ should be a covector, not a vector
• differentials (e.g. things like "$\mathrm{d}x$") really are mathematical objects, not merely a bit of notation used when writing an integral
• being able to study geometry using only as many dimensions as needed -- e.g. you're used to writing tangent vectors to the sphere as vectors in three-space. How would you capture the essential properties of a tangent vector to the sphere if you only allow yourself to work with two dimensions?
• new notation to simplify complicated calculations involving many variables

It may be helpful to do something like review doing calculus on a sphere -- e.g. scalar and tangent vector fields on a sphere, path integrals along a curve on the sphere, surface integrals over the sphere -- and see how things line up with the concepts you're learning in differential geometry.