= Difference of affine connections is a tensor
{title2=$\Delta(X,Y)=\nabla_XY-\widetilde\nabla_XY$}
The difference of two <affine connections> is a smooth <tensor field> of type $(1,2)$. Linearity over <smooth functions> holds in the first slot by the connection axioms. In the second slot, the two extra terms $X(f)Y$ cancel, so $\Delta(X,fY)=f\Delta(X,Y)$. This <tensoriality> makes the value depend only on $X_p,Y_p$. In coordinates its components are the differences of the <Christoffel symbols>, although either collection of connection coefficients individually is not a tensor.
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