Sooner or later, I’m going to want to mention the Levi-Cevita connection in my course, so I want to understand the meaning of the fact that it has no torsion. I think I have enough understanding for teaching purposes. See these Mathoverflow threads 1 2 and these John Baez essays 1 2 3 (scroll down) for some of the ideas I might present.
However, I am left with a question. I haven’t found any integral characterizations of zero-torsion, only infinitesimal ones. Let me explain what I mean by some examples:
Let be a smooth manifold.
Let be a differential -form on . Then is zero if and only if, for all contractible -spheres in , we have .
Let be a vector bundle on , and let be a connection on . Then the curvature of vanishes if and only if, for every contractible loop in , the holonomy around is trivial.
Both of these characterizations involve quantifying over contractible spheres, and say that some quantity is identically zero.
In contrast, let be a connection on . Every characterization of torsion vanishing which I can find involves looking at sufficiently small discs/geodesics/etcetera and involves the order of vanishing of some quantity being small (usually instead of ), rather than being identically zero.
I find this aesthetically unpleasing. Any ideas?