TITLE: Geometric meaning of symmetric connection
QUESTION [40 upvotes]: If $(M, g)$ is Riemannian manifold, there is unique connection $\nabla$, called Levi-Civita connection, satisfying the following conditions:
1) Compatibility with Riemannian metric, i.e. $\nabla(g)$=0
2) Symmetricity, i.e. $\nabla_X(Y)-\nabla_Y(X)=[X, Y]$
While the former seems quite natural (equally, parallel transform of a vector along a curve does not change its length), what geometric intuition is hidden behind the later?
The only idea I have, except for the fact it makes a compatible connection unique, is that it is in some way similar to the equality of partial derivates along commuting fields.

REPLY [5 votes]: The presence of torsion in a connection allows you to extend parallel transport in a manifold from being a linear mapping to a more general affine mapping. The torsion then generates the translation part of the affine mapping. But parallel transport is defined between tangent spaces which are of course linear spaces and these do not admit affine mappings in a natural way. So instead we think of each tangent space to be the associated vector space of an underlying affine space and the affine parallel transport is between these tangent affine spaces.
These tangent affine spaces capture perhaps in some sense the classical notion of infinitesimal close points. The points of a tangent affine space are all infinitesimally close to each other as well as being infinitesimally close to the manifold point which the affine space is the fiber of.
The points of each affine space are on equal footing, there is no distinguished point among them. We may ask if it is possible to choose an origin in each tangent affine space which could be identified with the base manifold point, thus "attaching" the tangent affine spaces to the manifold. This can be carried out by choosing some initial point in one tangent affine space and then trying to parallel transport this point throughout the whole manifold. If there is no torsion this succeeds, the manifold is generated as a submanifold in the tangent affine space bundle and the bundle becomes isomorphic to the usual tangent bundle. But if there is torsion this is not possible, the tangent affine spaces "slip", they are not "attachable".
This can be expressed by
$D^2P=T(X,Y)$
where $D$ is the exterior covariant derivative, $P$ is a point function mapping a manifold point to a point in its tangent affine space and $T(X,Y)$ is the torsion two-form.
Thus a non-vanishing torsion two-form means that $DP$ is not closed and there is not a well defined point function $P$ allowing us to consistently choose origins in the tangent affine spaces.