My post yesterday on the torsion tensor and symmetry had a serious error. For some reason I thought that connections can be pulled back. I am correcting the latter part of that post (where I used that erroneous claim) here. I decided not to repeat the (as far as I know) correct earlier part.
Proposition 1 Let be a surface in , and let be a symmetric connection on . Then
Assume first is an immersion. Then at some fixed
if is a vector field on some neighborhood of which is -related to , and similarly for . Similarly,
The difference between these two quantities is , because by a general theorem about -relatedness preserving the Lie bracket for a morphism. As before, we can approximate by an immersion at , and we get the general case.
That was much easier than what I was trying to do earlier. Blogging is a learning experience.