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I am working through Visser's notes http://msor.victoria.ac.nz/twiki/pub/Courses/MATH465_2012T1/WebHome/notes-464-2011.pdf section 3.5 onward. I am trying to differentiate between the torsion and the Riemann curvature tensor in a heuristic manner.

It appears from "Geometric interpretation 1" (page 73) that the torsion tensor is related to going along both paths of a parallelogram to get to opposite corners, and then finding the difference...how the infinitesimal parallelogram does not close.

The Riemann tensor is introduced simply through the noncommutativity of the the covariant derivative, though looking around on the internet, I have also found references to infinitesimal parallelograms. Could someone explain, in the same sense as above, what motivates the calculation of the Riemann tensor (i.e. commutator of covariant derivatives)?

I have heard references to the "twist" of frames along a geodesic for the torsion as opposed to the "roll" for the Riemann curvature --- if you could explain these, that would also be appreciated.

Cheers