Here's an interpretation I have held for while (since I never had the chance to sort it out myself... but this might be my opportunity!)... if it's wrong, please correct me.

For simplicity, take an affine space ("a vector space, but forget the origin").

Given a point x on that space, its location is represented by a tuple of numbers that depends on the choice of origin.

Given two points x and y on that space, the displacement from x to y (i.e., y-x) is a vector, independent of any choice of origin.

Is something like this happening with the Christoffel symbols and the tensor C

^{c}_{ab}?

(If so, is this why the Christoffel symbols are sometimes called

**affine connections**? Or does "affine" refer to the "affine parameter" in the geodesic equation? Based on

http://planetmath.org/encyclopedia/C...n.html#foot196 , I might have to go find the references to Cartan.)