Understanding the Riemann curvature tensor is like doing a spinal tap
Back in the day when I was doing spinal taps, I spent far more time setting them up (positioning the patient so that the long axis of the spinal column was parallel to the floor and the vertical axis of the recumbent patient was perpendicular to the floor) than actually doing the tap. Why? because then, all I had to do was have the needle parallel to the floor, with no guessing about how to angle it when the patient had rolled (usually forward into the less than firm mattress of the hospital bed).
So it is with finally seeing what the Riemann curvature tensor actually is, why it is the way it is, and why the notation describing it is such a mess. Finally on p. 290 of Needham’s marvelous book “Visual Differential Geometry and Forms” the Riemann curvature tensor is revealed in all its glory. Understanding it takes far less time than understanding the mathematical (and geometric) scaffolding required to describe it, a la spinal taps.
Over the years while studying relativity, I’ve seen it in one form or other (always algebraic) without understanding what the algebra was really describing.
Needham will get you there, but you have to understand a lot of other things first. Fortunately almost all of them are described visually, so you see what the algebra is trying to describe. Along the way you will see what the Lie bracket of two vector fields is all about along with holonomy. And you will really understand what curvature is. And Needham will give you 3 ways to understand parallel transport (which underlies everything — thanks Ashutosh)
Needham starts off with Gauss’s definition of curvature of a surface — the angular excess of a triangle, divided by its area.
Here is why this definition is enough to show you why the surface of a sphere is curved. Go to the equator. Mark point one, then point two 1/4 of the way around the sphere. Form longitudes (perpendiculars) there and extend them as great circles toward the North pole. You now have a triangle containing 3 right angles, (clearly an angular excess from Euclid who states that the sum the angles of a triangle is two right angles). The reason, of course, is because the sphere is curved.
Ever since I met a classmate 12 years ago at a college reunion who was a relativist working with Hawking, I decided to try to learn relativity so I’d have something intelligent to say to him if we ever met again (COVID19 stopped all that although we’re still both alive).
Now that I understand what the math of relativity is trying to describe, I may be able to understand general relativity.
Be prepared for a lot of work, but do start with Needham’s book. Here are some links to other things I’ve written about it. It will take some getting used to as it is very different from any math book you’ve ever read (except Needham’s other book).
7 Mar 22 — https://luysii.wordpress.com/2022/03/07/visual-differential-geometry-and-forms-q-take-3/
27 June 22 — https://luysii.wordpress.com/2022/06/27/the-chinese-room-argument-understanding-math-and-the-imposter-syndrome/
17 July 22 — https://luysii.wordpress.com/2022/07/17/a-visual-proof-of-the-the-theorem-egregium-of-gauss/