Editing Affine connection (section)

====Definition via absolute parallelism====

Let {{mvar|M}} be a manifold, and {{mvar|P}} a principal {{math|GL(''n'')}}-bundle over {{mvar|M}}. Then an '''affine connection''' is a 1-form {{mvar|η}} on {{mvar|P}} with values in {{math|'''aff'''(''n'')}} satisfying the following properties
# {{mvar|η}} is equivariant with respect to the action of {{math|GL(''n'')}} on {{mvar|P}} and {{math|'''aff'''(''n'')}};
# {{math|''η''(''X<sub>ξ</sub>'') {{=}} ''ξ''}} for all {{mvar|ξ}} in the Lie algebra {{math|'''gl'''(''n'')}} of all {{math|''n'' × ''n''}} matrices;
# {{mvar|η}} is a linear isomorphism of each tangent space of {{mvar|P}} with {{math|'''aff'''(''n'')}}.
The last condition means that {{mvar|η}} is an '''[[absolute parallelism]]''' on {{mvar|P}}, i.e., it identifies the tangent bundle of {{mvar|P}} with a trivial bundle (in this case {{math|''P'' × '''aff'''(''n'')}}). The pair {{math|(''P'', ''η'')}} defines the structure of an '''affine geometry''' on {{mvar|M}}, making it into an '''affine manifold'''.

The affine Lie algebra {{math|'''aff'''(''n'')}} splits as a semidirect product of {{math|'''R'''<sup>''n''</sup>}} and {{math|'''gl'''(''n'')}} and so {{mvar|η}} may be written as a pair {{math|(''θ'', ''ω'')}} where {{mvar|θ}} takes values in {{math|'''R'''<sup>''n''</sup>}} and {{mvar|ω}} takes values in {{math|'''gl'''(''n'')}}. Conditions 1 and 2 are equivalent to {{mvar|ω}} being a principal {{math|GL(''n'')}}-connection and {{mvar|θ}} being a horizontal equivariant 1-form, which induces a [[bundle homomorphism]] from {{math|T''M''}} to the [[associated bundle]] {{math|''P'' ×<sub>GL(''n'')</sub> '''R'''<sup>''n''</sup>}}. Condition 3 is equivalent to the fact that this bundle homomorphism is an isomorphism. (However, this decomposition is a consequence of the rather special structure of the affine group.) Since {{mvar|P}} is the [[frame bundle]] of {{math|''P'' ×<sub>GL(''n'')</sub> '''R'''<sup>''n''</sup>}}, it follows that {{mvar|θ}} provides a bundle isomorphism between {{mvar|P}} and the frame bundle {{math|F''M''}} of {{mvar|M}}; this recovers the definition of an affine connection as a principal {{math|GL(''n'')}}-connection on {{math|F''M''}}.

The 1-forms arising in the flat model are just the components of {{mvar|θ}} and {{mvar|ω}}.