An affine connection on a smooth manifold is a connection on the frame bundle of , i.e., the principal bundle of frames in the tangent bundle . The components of the local Lie-algebra valued 1-form of an affine connection are called Christoffel symbols. A coordinate-free treatment first appeared in: See also: Wikipedia: Affine connection Formulation in synthetic differential geometry: Wolfgang Bertram; section 10–11 in: Differential Geometry, Lie Groups and Symmetric Spaces over General Base...