Deriving curvature tensor from parallel transport

Question

I know that the curvature tensor $R$ of a Riemannian manifold can be derived, in a more intuitive way, by using the parallel transport. Can you give me a reference of this way to derive the curvature tensor?

Answer 1

On a 2D riemann manifold that lives in 3D space imagine that you have 2 vector fields on the manifold(v and u), then imagine one more vector field that directs to somewhere in the 3d space(w).Now imagine parallel transporting w along v and u vectors by order, you could do that by multiplying w with some matrices that transport w vector along the vector field(v and u). When you finish transporting you should have a vector that is in the same point with w at the start but if it is directing a different point in 3D space, difference between your first w vector and parallel transported version of it is what you will get when you use riemann curvature tensor.

Mathematical derivation comes from taking limits of v and u vectors and dividing the parallel transported version of w to vu, this makes the parallel transforming operation independent from the coordinate of the plane. So operation becomes the covariant derivative of w along u and v vectors.

Also there is a lie bracket term in riemann curvature tensor. I think it is there because it solves the problem of v and u vector fields are not intersecting at each point on the plane.