An affine connection $\nabla$ on a smooth manifold $M$ is a connection on the frame bundle $F M$ of $M$, i.e., the principal bundle of frames in the tangent bundle $T M$.
The components of the local Lie-algebra valued 1-form of an affine connection are called Christoffel symbols.
wikipedia affine connection
А.П. Норден, Пространства аффинной связности, 1976, djvu