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Connection (principal bundle)
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==Connections on frame bundles and torsion== If the principal bundle ''P'' is the [[frame bundle]], or (more generally) if it has a [[solder form]], then the connection is an example of an [[affine connection]], and the curvature is not the only invariant, since the additional structure of the solder form ''ΞΈ'', which is an equivariant '''R'''<sup>''n''</sup>-valued 1-form on ''P'', should be taken into account. In particular, the [[Torsion (differential geometry)|torsion form]] on ''P'', is an '''R'''<sup>''n''</sup>-valued 2-form Ξ defined by :<math> \Theta=\mathrm d\theta+\omega\wedge\theta. </math> Ξ is ''G''-equivariant and horizontal, and so it descends to a tangent-valued 2-form on ''M'', called the ''torsion''. This equation is sometimes called the ''(Cartan's) first structure equation''.
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