Parallel Transport along a sphere

Question

I was watching this video on general relativity (https://youtu.be/D3GVVkPb3OI?t=7m) and i'm confused by the definition of curvature, more specifically what you mean by parallel transporting.

Assumption 1:A cylinder is flat.

Assumption 2:A sphere is curved.

So if were were to parallel transport along the circumference of a cylinder, then it should be parallel as it is flat, correct?

If i were to put a sphere of the same radius inside this cylinder such that its great circle (equator) touches the cylinder, then shouldn't the parallel transport also stay parallel?

I've seen this diagram as an explanation (https://i.stack.imgur.com/FNcBH.jpg) but by that logic wouldn't a cylinder like the one i described above also behave the same?

p.s. I do not understand any of the math, with its matrices and so on, i just know very basic 3D-geometry. I'm out of my depth and gasping for breath, but if anyone could tell me 1)where i'm wrong and 2)where do i start to understand all this, i would greatly appreciate it.

Answer 1

In my experience parallel transport is very complicated to understand if you don't have a good grasp of the basics of differential geometry.

Assumption 1:A cylinder is flat.

Assumption 2:A sphere is curved.

There are two problems with your assumptions, specifically your definition of "flat" and "curved".

A cylinder has Gaussian curvature equal to $0$ while a sphere doesn't, but both are definitely curved in the usual sense of the word.

The fact that a cylinder is "flat" in the way you use the word means that there a direction through which it can be cut and "unrolled" into a plane, so if you parallel transport a vector in a direction that is orthogonal to it, you will obtain what is essentially a parallel movement also on the plane.

On a sphere however there is no such way of cutting, so you cannot simply project a vector onto the plane maintaining all of its properties.

So if were were to parallel transport along the circumference of a cylinder, then it should be parallel as it is flat, correct?

If i were to put a sphere of the same radius inside this cylinder such that it great circle (equator) touches the cylinder, then shouldn't the parallel transport also stay parallel?

Parallel transport is by definition parallel. You should imagine living on the surface and seeing a vector being transported, what you see as parallel is not the same as what you would see as parallel if you were to look at it abstracted from the surface in a purely euclidean world.

Answer 2

This part is a bit hard to understand

"If i were to put a sphere of the same radius inside this cylinder such that its great circle (equator) touches the cylinder, then shouldn't the parallel transport also stay parallel?"

You can parallelly transport a tangent vector in a sphere along the great circle, there is no problem with that. You can parallelly transport a vector along any curve.

However, if you consider a closed loop and transport your vector along the curve, then your vector will remain unchanged (after traveling along the loop) in the cylinder.

In a sphere, this is still true for great circles, but is not true if you consider a loop that travels 90% along the equator, then goes up to the nort-pole, and then returns back to the starting position. Your vector will completely change after this trip.