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

$[\nabla_\mu,\nabla_\nu]V^\rho = R_{\mu\nu}{}^\rho{}_\lambda V^\lambda.$

Usually, it is said that curvature is the responsible of the change of the direction of the vector under parallel transport through the two different paths.

However, in general there are three different effects in the transportation:

- Change of direction of the vector.
- Non-closure of the path (say, if one moves 1meter along each direction).
- Rotation about its own axis.

whilst the general commutator is

$[\nabla_\mu,\nabla_\nu]V^\rho = R_{\mu\nu}{}^\rho{}_\lambda V^\lambda- T^\lambda{}_{\mu\nu}\nabla_\lambda V^\rho.$

Is it possible to give a meaning to the curvature and torsion in term of these intuitive geometry or is not possible in general?

Thank you!