Editing Affine connection (section)

===Explanations and historical intuition===

The points just raised are easiest to explain in reverse, starting from the motivation provided by surface theory. In this situation, although the planes being rolled over the surface are tangent planes in a naive sense, the notion of a [[tangent space]] is really an [[differential (infinitesimal)|infinitesimal]] notion,{{efn|Classically, the [[tangent space]] was viewed as an infinitesimal approximation, while in modern differential geometry, tangent spaces are often defined in terms of differential objects such as derivations.<ref>{{Harvnb|Kobayashi|Nomizu|1996|loc=Volume 1, sections 1.1–1.2}}</ref>}} whereas the planes, as [[affine subspace]]s of {{math|'''R'''<sup>3</sup>}}, are [[Infinity|infinite]] in extent. However these affine planes all have a marked point, the point of contact with the surface, and they are tangent to the surface at this point. The confusion therefore arises because an affine space with a marked point can be identified with its tangent space at that point. However, the parallel transport defined by rolling does not fix this origin: it is [[affine transformation|affine]] rather than linear; the linear parallel transport can be recovered by applying a translation.

Abstracting this idea, an affine manifold should therefore be an {{mvar|n}}-manifold {{mvar|M}} with an affine space {{math|''A''<sub>''x''</sub>}}, of dimension {{mvar|n}}, ''attached'' to each {{math|''x'' ∈ ''M''}} at a marked point {{math|''a''<sub>''x''</sub> ∈ ''A''<sub>''x''</sub>}}, together with a method for transporting elements of these affine spaces along any curve {{mvar|C}} in {{mvar|M}}. This method is required to satisfy several properties:
# for any two points {{math|''x'', ''y''}} on {{mvar|C}}, parallel transport is an [[affine transformation]] from {{math|''A''<sub>''x''</sub>}} to {{math|''A''<sub>''y''</sub>}};
# parallel transport is defined infinitesimally in the sense that it is differentiable at any point on {{mvar|C}} and depends only on the tangent vector to {{mvar|C}} at that point;
# the derivative of the parallel transport at {{mvar|x}} determines a [[linear isomorphism]] from {{math|T<sub>''x''</sub>''M''}} to {{math|T<sub>''a<sub>x</sub>''</sub>''A''<sub>''x''</sub>}}.

These last two points are quite hard to make precise,<ref>For details, see {{Harvtxt|Lumiste|2001b|ignore-err=yes}}. The following intuitive treatment is that of {{Harvtxt|Cartan|1923}} and {{Harvtxt|Cartan|1926}}.</ref> so affine connections are more often defined infinitesimally. To motivate this, it suffices to consider how affine [[frames of reference]] transform infinitesimally with respect to parallel transport. (This is the origin of Cartan's [[method of moving frames]].) An affine frame at a point consists of a list {{math|(''p'', '''e'''<sub>1</sub>,… '''e'''<sub>''n''</sub>)}}, where {{math|''p'' ∈ ''A''<sub>''x''</sub>}}{{efn|This can be viewed as a choice of origin: actually it suffices to consider only the case {{math|''p'' {{=}} ''a''<sub>''x''</sub>}}; Cartan implicitly identifies this with {{mvar|x}} in {{mvar|M}}.}} and the {{math|'''e'''<sub>''i''</sub>}} form a basis of {{math|T<sub>''p''</sub>(''A''<sub>''x''</sub>)}}.  The affine connection is then given symbolically by a first order [[differential system]]

:<math>(*) \begin{cases}
\mathrm{d}{p} &= \theta^1\mathbf{e}_1 + \cdots + \theta^n\mathbf{e}_n \\
\mathrm{d}\mathbf{e}_i &= \omega^1_i\mathbf{e}_1 + \cdots + \omega^n_i\mathbf{e}_n
\end{cases} \quad i=1,2,\ldots,n</math>

defined by a collection of [[differential forms|one-forms]] {{math|(''θ&thinsp;<sup>j</sup>'', ''ω&thinsp;{{su|b=i|p=j}}'')}}. Geometrically, an affine frame undergoes a displacement travelling along a curve {{mvar|γ}} from {{math|''γ''(''t'')}} to {{math|''γ''(''t'' + ''δt'')}} given (approximately, or infinitesimally) by

:<math>\begin{align}
p(\gamma(t+\delta t)) - p(\gamma(t)) &= \left(\theta^1\left(\gamma'(t)\right)\mathbf{e}_1 + \cdots + \theta^n\left(\gamma'(t)\right)\mathbf{e}_n\right)\mathrm \delta t \\
\mathbf{e}_i(\gamma(t+\delta t)) - \mathbf{e}_i(\gamma(t)) &= \left(\omega^1_i\left(\gamma'(t)\right)\mathbf{e}_1 + \cdots + \omega^n_i\left(\gamma'(t)\right) \mathbf{e}_n\right)\delta t\,.
\end{align}</math>

Furthermore, the affine spaces {{math|''A''<sub>''x''</sub>}} are required to be tangent to {{mvar|M}} in the informal sense that the displacement of {{math|''a''<sub>''x''</sub>}} along {{mvar|γ}} can be identified (approximately or infinitesimally) with the tangent vector {{math|''γ''′(''t'')}} to {{mvar|γ}} at {{math|''x'' {{=}} ''γ''(''t'')}} (which is the infinitesimal displacement of {{mvar|x}}). Since
:<math>a_x (\gamma(t + \delta t)) - a_x (\gamma(t)) = \theta\left(\gamma'(t)\right) \delta t \,,</math>
where {{mvar|θ}} is defined by {{math|''θ''(''X'') {{=}} ''θ''<sup>1</sup>(''X'')'''e'''<sub>1</sub> + … +  ''θ''<sup>''n''</sup>(''X'')'''e'''<sub>''n''</sub>}}, this identification is given by {{mvar|θ}}, so the requirement is that {{mvar|θ}} should be a linear isomorphism at each point.

The tangential affine space {{math|''A''<sub>''x''</sub>}} is thus identified intuitively with an ''infinitesimal affine neighborhood'' of {{mvar|x}}.

The modern point of view makes all this intuition more precise using principal bundles (the essential idea is to replace a frame or a ''variable'' frame by the space of all frames and functions on this space). It also draws on the inspiration of [[Felix Klein]]'s [[Erlangen programme]],<ref>Cf. R. Hermann (1983), Appendix 1–3 to {{Harvtxt|Cartan|1951}}, and also {{Harvtxt|Sharpe|1997}}.</ref> in which a ''geometry'' is defined to be a [[homogeneous space]]. Affine space is a geometry in this sense, and is equipped with a ''flat'' Cartan connection. Thus a general affine manifold is viewed as ''curved'' deformation of the flat model geometry of affine space.