Editing Connection form (section)

===Soldering and torsion===
Suppose that the fibre dimension ''k'' of ''E'' is equal to the dimension of the manifold ''M''.  In this case, the vector bundle ''E'' is sometimes equipped with an additional piece of data besides its connection: a [[solder form]].  A '''solder form''' is a globally defined [[vector-valued form|vector-valued one-form]] θ ∈ Ω<sup>1</sup>(''M'',''E'') such that the mapping
:<math>\theta_x : T_xM \rightarrow E_x</math>
is a linear isomorphism for all ''x'' ∈ ''M''.  If a solder form is given, then it is possible to define the '''[[torsion (differential geometry)|torsion]]''' of the connection (in terms of the exterior connection) as
:<math>\Theta = D\theta.\, </math>
The torsion Θ is an ''E''-valued 2-form on ''M''.

A solder form and the associated torsion may both be described in terms of a local frame '''e''' of ''E''.  If θ is a solder form, then it decomposes into the frame components
:<math>\theta = \sum_i \theta^i(\mathbf e) e_i.</math>
The components of the torsion are then
:<math>\Theta^i(\mathbf e) = d\theta^i(\mathbf e) + \sum_j \omega_j^i(\mathbf e)\wedge \theta^j(\mathbf e).</math>
Much like the curvature, it can be shown that Θ behaves as a [[Covariance and contravariance of vectors|contravariant tensor]] under a change in frame:
:<math>\Theta^i(\mathbf e\, g)=\sum_j g_j^i \Theta^j(\mathbf e).</math>

The frame-independent torsion may also be recovered from the frame components:
:<math>\Theta = \sum_i e_i \Theta^i(\mathbf e).</math>