Editing Affine connection (section)

==Formal definition as a differential operator==

{{see also|Covariant derivative|Connection (vector bundle)}}

Let {{mvar|M}} be a smooth [[manifold]] and let {{math|Γ(T''M'')}} be the space of [[vector fields]] on {{mvar|M}}, that is, the space of [[section (fiber bundle)|smooth sections]] of the [[tangent bundle]] {{math|T''M''}}. Then an '''affine connection''' on {{mvar|M}} is a [[bilinear map]]

: <math>\begin{align}
\Gamma(\mathrm{T}M)\times \Gamma(\mathrm{T}M) & \rightarrow \Gamma(\mathrm{T}M)\\
(X,Y) & \mapsto \nabla_X Y\,,\end{align}</math>
such that for all {{mvar|f}} in the set of smooth functions on {{math|''M''}}, written {{math|''C''<sup>∞</sup>(''M'', '''R''')}}, and all vector fields {{math|''X'', ''Y''}} on {{mvar|M}}:
# {{math|∇''<sub>fX</sub>Y'' {{=}} ''f'' ∇''<sub>X</sub>Y''}}, that is, {{math|∇}} is {{math|''C''<sup>∞</sup>(''M'', '''R''')}}-''linear'' in the first variable;
# {{math|∇<sub>''X''</sub>(''fY'') {{=}} (∂<sub>''X''</sub>&thinsp;''f'') ''Y'' + ''f'' ∇''<sub>X</sub>Y''}}, where {{math|∂<sub>''X''</sub>}} denotes the [[directional derivative]]; that is, {{math|∇}} satisfies ''Leibniz rule'' in the second variable.

===Elementary properties===

* It follows from property 1 above that the value of {{math|∇<sub>''X''</sub>''Y''}} at a point {{math|''x'' ∈ ''M''}} depends only on the value of {{mvar|X}} at {{mvar|x}} and not on the value of {{mvar|X}} on {{math|1=''M'' − {''x''}<nowiki/>}}. It also follows from property 2 above that the value of {{math|∇<sub>''X''</sub>''Y''}} at a point {{math|''x'' ∈ ''M''}} depends only on the value of {{mvar|Y}} on a neighbourhood of {{mvar|x}}.
* If {{math|∇<sup>1</sup>, ∇<sup>2</sup>}} are affine connections then the value at {{mvar|x}} of {{math|∇{{su|p=1|b=''X''}}''Y'' − ∇{{su|p=2|b=''X''}}''Y''}} may be written {{math|Γ<sub>''x''</sub>(''X''<sub>''x''</sub>, ''Y''<sub>''x''</sub>)}} where <math display="block">\Gamma_x : \mathrm{T}_xM \times \mathrm{T}_xM \to \mathrm{T}_xM</math> is bilinear and depends smoothly on {{mvar|x}} (i.e., it defines a smooth [[bundle homomorphism]]). Conversely if {{math|∇}} is an affine connection and {{math|Γ}} is such a smooth bilinear bundle homomorphism (called a [[connection form]] on {{mvar|M}}) then {{math|∇ + Γ}} is an affine connection.
* If {{mvar|M}} is an open subset of {{math|'''R'''<sup>''n''</sup>}}, then the tangent bundle of {{mvar|M}} is the [[trivial bundle]] {{math|''M'' × '''R'''<sup>''n''</sup>}}. In this situation there is a canonical affine connection {{math|d}} on {{mvar|M}}: any vector field {{mvar|Y}} is given by a smooth function {{mvar|V}} from {{mvar|M}} to {{math|'''R'''<sup>''n''</sup>}}; then {{math|d<sub>''X''</sub>''Y''}} is the vector field corresponding to the smooth function {{math|d''V''(''X'') {{=}} ∂<sub>''X''</sub>''Y''}} from {{mvar|M}} to {{math|'''R'''<sup>''n''</sup>}}. Any other affine connection {{math|∇}} on {{mvar|M}} may therefore be written {{math|∇ {{=}} d + Γ}}, where {{math|Γ}} is a connection form on {{mvar|M}}.
* More generally, a [[local trivialization]] of the tangent bundle is a [[bundle map|bundle isomorphism]] between the restriction of {{math|T''M''}} to an open subset {{mvar|U}} of {{mvar|M}}, and {{math|''U'' × '''R'''<sup>''n''</sup>}}. The restriction of an affine connection {{math|∇}} to {{mvar|U}} may then be written in the form {{math|d + Γ}} where {{math|Γ}} is a connection form on {{mvar|U}}.