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.
A coordinate-free treatment first appeared in
Harley Flanders?, Development of an extended exterior differential calculus. Transactions of the American Mathematical Society 75:2 (1953), 311–311. doi.
wikipedia affine connection
А.П. Норден, Пространства аффинной связности, 1976, djvu
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