Editing Affine connection (section)

==Formal definition on the frame bundle==
{{see also|Connection (principal bundle)}}

An affine connection may also be defined as a [[connection (principal bundle)|principal {{math|GL(''n'')}} connection]] {{mvar|ω}} on the [[frame bundle]] {{math|F''M''}} or {{math|GL(''M'')}} of a manifold {{mvar|M}}. In more detail, {{mvar|ω}} is a smooth map from the tangent bundle {{math|T(F''M'')}} of the frame bundle to the space of {{math|''n'' × ''n''}} matrices (which is the [[Lie algebra]] {{math|'''gl'''(''n'')}} of the [[Lie group]] {{math|GL(''n'')}} of invertible {{math|''n'' × ''n''}} matrices) satisfying two properties:
# {{mvar|ω}} is [[equivariant]] with respect to the action of {{math|GL(''n'')}} on {{math|T(F''M'')}} and {{math|'''gl'''(''n'')}};
# {{math|''ω''(''X<sub>ξ</sub>'') {{=}} ''ξ''}} for any {{mvar|ξ}} in {{math|'''gl'''(''n'')}}, where {{mvar|X<sub>ξ</sub>}} is the vector field on {{math|F''M''}} corresponding to {{mvar|ξ}}.

Such a connection {{mvar|ω}} immediately defines a [[covariant derivative]] not only on the tangent bundle, but on [[vector bundle]]s [[associated bundle|associated]] to any [[group representation]] of {{math|GL(''n'')}}, including bundles of [[tensor]]s and [[tensor density|tensor densities]]. Conversely, an affine connection on the tangent bundle determines an affine connection on the frame bundle, for instance, by requiring that {{mvar|ω}} vanishes on tangent vectors to the lifts of curves to the frame bundle defined by parallel transport.

The frame bundle also comes equipped with a [[frame bundle#Solder form|solder form]] {{math|''θ'' : T(F''M'') → '''R'''<sup>''n''</sup>}} which is '''horizontal''' in the sense that it vanishes on [[vertical bundle|vertical vectors]] such as the point values of the vector fields {{mvar|X<sub>ξ</sub>}}: Indeed {{mvar|θ}} is defined first by projecting a tangent vector (to {{math|F''M''}} at a frame {{mvar|f}}) to {{mvar|M}}, then by taking the components of this tangent vector on {{mvar|M}} with respect to the frame {{mvar|f}}. Note that {{mvar|θ}} is also {{math|GL(''n'')}}-equivariant (where {{math|GL(''n'')}} acts on {{math|'''R'''<sup>''n''</sup>}} by matrix multiplication).

The pair {{math|(''θ'', ''ω'')}} defines a [[bundle isomorphism]] of {{math|T(F''M'')}} with the trivial bundle {{math|F''M'' × '''aff'''(''n'')}}, where {{math|'''aff'''(''n'')}} is the [[Cartesian product]] of {{math|'''R'''<sup>''n''</sup>}} and {{math|'''gl'''(''n'')}} (viewed as the Lie algebra of the affine group, which is actually a [[semidirect product]] – see below).