Editing Connection form (section)

===Example: the Levi-Civita connection===
As an example, suppose that ''M'' carries a [[Riemannian metric]]. If one has a [[vector bundle]] ''E'' over ''M'', then the metric can be extended to the entire vector bundle, as the [[bundle metric]]. One may then define a connection that is compatible with this bundle metric, this is the [[metric connection]].  For the special case of ''E'' being the [[tangent bundle]] ''TM'', the metric connection is called the [[Riemannian connection]]. Given a Riemannian connection, one can always find a unique, equivalent connection that is [[torsion tensor|torsion-free]]. This is the [[Levi-Civita connection]] on the tangent bundle ''TM'' of ''M''.<ref>See {{harvtxt|Jost|2011}}, chapter 4, for a complete account of the Levi-Civita connection from this point of view.</ref><ref>See {{harvtxt|Spivak|1999a}}, II.7 for a complete account of the Levi-Civita connection from this point of view.</ref>

A local frame on the tangent bundle is an ordered list of vector fields {{nowrap|1='''e''' = (''e''<sub>''i''</sub> {{!}} ''i'' = 1, 2, ..., ''n'')}}, where {{nowrap|1=''n'' = dim ''M''}}, defined on an open subset of ''M'' that are linearly independent at every point of their domain.  The [[Christoffel symbols]] define the Levi-Civita connection by
:<math>\nabla_{e_i}e_j = \sum_{k=1}^n\Gamma_{ij}^k(\mathbf e)e_k.</math>
If ''θ'' = {{mset|1=''θ''<sup>''i''</sup> {{!}} ''i'' = 1, 2, ..., ''n''}}, denotes the [[dual basis]] of the [[cotangent bundle]], such that ''θ''<sup>''i''</sup>(''e''<sub>''j''</sub>) = ''δ''<sup>''i''</sup><sub>''j''</sub> (the [[Kronecker delta]]), then the connection form is
:<math>\omega_i^j(\mathbf e) = \sum_k \Gamma^j{}_{ki}(\mathbf e)\theta^k.</math>

In terms of the connection form, the exterior connection on a vector field {{nowrap|1=''v'' = Σ<sub>''i''</sub>''e''<sub>''i''</sub>''v''<sup>''i''</sup>}} is given by
:<math> Dv=\sum_k e_k\otimes(dv^k) + \sum_{j,k}e_k\otimes\omega^k_j(\mathbf e)v^j.</math>
One can recover the Levi-Civita connection, in the usual sense, from this by contracting with ''e''<sub>i</sub>:
:<math> \nabla_{e_i} v = \langle Dv, e_i\rangle = \sum_k e_k \left(\nabla_{e_i} v^k + \sum_j\Gamma^k_{ij}(\mathbf e)v^j\right)</math>

====Curvature====
The curvature 2-form of the Levi-Civita connection is the matrix (Ω<sub>''i''</sub><sup>''j''</sup>) given by
:<math>
\Omega_i{}^j(\mathbf e) = d\omega_i{}^j(\mathbf e)+\sum_k\omega_k{}^j(\mathbf e)\wedge\omega_i{}^k(\mathbf e).
</math>
For simplicity, suppose that the frame '''e''' is [[Holonomic basis|holonomic]], so that {{nowrap|1=''dθ''<sup>''i''</sup> = 0}}.<ref>In a non-holonomic frame, the expression of curvature is further complicated by the fact that the derivatives dθ<sup>i</sup> must be taken into account.</ref>  Then, employing now the [[summation convention]] on repeated indices,
:<math>\begin{array}{ll}
\Omega_i{}^j &= d(\Gamma^j{}_{qi}\theta^q) + (\Gamma^j{}_{pk}\theta^p)\wedge(\Gamma^k{}_{qi}\theta^q)\\
&\\
&=\theta^p\wedge\theta^q\left(\partial_p\Gamma^j{}_{qi}+\Gamma^j{}_{pk}\Gamma^k{}_{qi})\right)\\
&\\
&=\tfrac12\theta^p\wedge\theta^q R_{pqi}{}^j
\end{array}
</math>
where ''R'' is the [[Riemann curvature tensor]].

====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]].