**6. Affine Formulation**

In the usual formulation of General Relativity, one defines a covariant derivative ∇, which is a map among tensors. Then, one can define curvature and torsion and eventually get the field equations for ECSK theory or General Relativity by setting torsion to zero.

One may wonder if this latter formulation is equivalent to the one we have been implementing through principal bundles and principal connection.

The answer is positive and is given in the next two sections.

<sup>16</sup> Where we have introduced the notation Ω*<sup>k</sup>* (*M*, *<sup>P</sup>* <sup>×</sup>Ad <sup>g</sup>) :<sup>=</sup> <sup>Ω</sup>*<sup>k</sup>* (*M*, ad*P*).
