TITLE: What is torsion in differential geometry intuitively?
QUESTION [199 upvotes]: Hi,
given a connection on the tangent space of a manifold, one can define its torsion:
$$T(X,Y):=\triangledown_X Y - \triangledown_Y X - [X,Y]$$
What is the geometric picture behind this definition—what does torsion measure intuitively?

REPLY [3 votes]: Define a path as a set of instructions to move, as given in the reference frame of the starting point, i.e. one step forwards, one step to your right, one step back and one step left, all given with respect to the initial reference frame. Of course, in order to follow the instructions, we have to parallel transport the reference frame with the path as we move.
In a flat space, when we parallel transport our reference frame through a closed path (closed in the flat map we parallel transported), we end up at the initial point and the reference frame ends in its initial orientation. If we do this in a torsionless, curved space, we probably will end up somewhere else, unless the path is infinitesimal. Yet, in this case, the reference frame will probably be changed with respect to the initial frame. In Riemannian geometries, the reference frame will simply be rotated with respect to the original but relative angles  and the length of the basic vectors is left invariant. In non-Riemannian geometries relative angles and lengths of the basis vectors of the reference frame can vary.
What happens when you go through an infinitesimal closed path if torsion is not zero?
The set of instructions close, but we end up in a different point of the space. 
Torsion is the measure by which an infinitesimal closed path does not result in a closed loop.