17
votes
4answers
1k views

Why is it important that partial derivatives commute?

I am asking this in the context of differential geometry (specifically Riemannian). When the Levi-Civita Connection is defined, we require that the torsion tensor is 0, which in local coordinates ...
3
votes
2answers
800 views

Interpretation of Curvature and Torsion

Dear all, When dealing with General Relativity one uses the Levi-Civita connection with is torsion-free. Thus the commutator of the covariant derivatives yields $[\nabla_\mu,\nabla_\nu]V^\rho = ...
18
votes
4answers
2k views

Rolling without slipping interpretation of torsion

This is, in a sense, a follow up to this question. Hehl and Obukhov try to give an intuitive description of torsion. I am confused about their description. I am looking at the following paragraph on ...