Variation of Levi-Civita connection ($\delta {\Gamma }^a{}_{bc}=\frac{1}{2}{g}^{ad}({\nabla }_b\delta g_{cd}+{\nabla }_c\delta g_{bd}-{\nabla }_d\delta g_{bc})$)

For a family of Levi-Civita connections, the difference of affine connections is a tensor. Dividing by the family parameter and taking the limit shows that its variation is tensorial. Vary the metric compatibility equation and retain the symmetry of the two lower indices. Adding the two cyclic identities and subtracting the third gives the displayed result. In a normal coordinate frame it is the direct derivative of the Christoffel formula; its tensorial form then holds in every frame.