Undergrad Spacetime Curvature via Triangle

[Narasoma]

I understand the mechanism of defining the curvature of a 2D manifold via triangle. But I don't understand how this works in 3D. Meanwhile, Lawrence Krauss mentioned in his book A Universe from Nothing it does.

How does this work in 3D?

 

[Orodruin]

The same way it works in 2D except in 3D there are more than one independemt component for the curvature tensor and which one you single out depends on the planar orientation of the triangle.

 

[anorlunda]

If you are looking for a visual analog to the triangle, here's one from Wikipedia.
It is animated, so watch it for a while until you see all the vectors.
https://en.wikipedia.org/wiki/Riemann_curvature_tensor

An illustration of the motivation of Riemann curvature on a sphere-like manifold. The fact that this transport may define two different vectors at the start point gives rise to Riemann curvature tensor. The right angle symbol denotes that the inner product (given by the metric tensor) between transported vectors (or tangent vectors of the curves) is 0.