Editing Connection form (section)

====Torsion====
The Levi-Civita connection is characterized as the unique [[metric connection]] in the tangent bundle with zero torsion.  To describe the torsion, note that the vector bundle ''E'' is the tangent bundle.  This carries a canonical solder form (sometimes called the [[canonical one-form]], especially in the context of [[classical mechanics]]) that is the section ''θ'' of {{nowrap|1=Hom(T''M'', T''M'') = T<sup>∗</sup>''M'' ⊗ T''M''}} corresponding to the identity endomorphism of the tangent spaces.  In the frame '''e''', the solder form is {{nowrap|1=''θ'' = Σ<sub>''i''</sub> ''e''<sub>''i''</sub> ⊗ ''θ''<sup>''i''</sup>}}, where again ''θ''<sup>''i''</sup> is the dual basis.

The torsion of the connection is given by {{nowrap|1=Θ = ''Dθ''}}, or in terms of the frame components of the solder form by
:<math>\Theta^i(\mathbf e) = d\theta^i+\sum_j\omega^i_j(\mathbf e)\wedge\theta^j.</math>
Assuming again for simplicity that '''e''' is holonomic, this expression reduces to
:<math>\Theta^i = \Gamma^i{}_{kj} \theta^k\wedge\theta^j</math>,
which vanishes if and only if Γ<sup>''i''</sup><sub>''kj''</sub> is symmetric on its lower indices.

Given a metric connection with torsion, one can always find a single, unique connection that is torsion-free, this is the Levi-Civita connection. The difference between a Riemannian connection and its associated Levi-Civita connection is the [[contorsion tensor]].