the index on torsion tensor on the section above, one can perform a similar operation with the solder form, and construct a tensor Here is the scalar product. This tensor can be expressed as The quantity is the contorsion form and is exactly what is needed to add to an arbitrary connection to get the torsion-free Levi-Civita connection. That is, given an Ehresmann connection , there is another connection that is torsion-free. The vanishing of the torsion is then equivalent to having or This can be viewed as a field equation relating the dynamics of the connection to that of the contorsion tensor. Derivation One way to quickly derive a metric compatible affine connection is to repeat the sum-sum difference idea used in the derivation of the Levi–Civita connection but not take torsion to be zero. Below is a derivation. Convention for derivation (Choose to define connection coefficients this way. The motivation is that of connection-one forms in gauge theory): We begin with the Metric Compatible condition: Now we use sum-sum difference (Cycle the indices on the condition): We now use the below torsion tensor definition (for a holonomic frame) to rewrite the connection: Note that this definition of torsion has the opposite sign as the usual definition when using the above convention for the lower index ordering of the connection coefficients, i.e. it has the opposite sign as the coordinate-free definition in the below section on geometry. Rectifying this inconsistency (which seems to be common in the literature) would result in a contorsion tensor with the opposite sign. Substitute the torsion tensor definition into what we have: Clean it up and combine like terms The torsion terms combine to make an object that transforms tensorially. Since these terms combine together in a metric compatible fashion, they are given a name, the Contorsion tensor, which determines the skew-symmetric part of a metric compatible affine connection. We will define it here with the motivation that it match the indices of the left hand side of the equation above. Cleaning by using the anti-symmetry of the torsion tensor yields what we will define to be the contorsion tensor: Subbing this back into our expression, we have: Now isolate the connection coefficients, and group the torsion terms together: Recall that the first term with the partial derivatives is the Levi-Civita connection expression used often by relativists. Following suit, define the following to be