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Affine connection
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==Affine connections as Cartan connections== {{see also|Cartan connection}} Affine connections can be defined within Cartan's general framework.<ref>{{Harvnb|Cartan|1926}}.</ref> In the modern approach, this is closely related to the definition of affine connections on the frame bundle. Indeed, in one formulation, a Cartan connection is an [[absolute parallelism]] of a principal bundle satisfying suitable properties. From this point of view the {{math|'''aff'''(''n'')}}-valued one-form {{math|(''θ'', ''ω'') : T(F''M'') → '''aff'''(''n'')}} on the frame bundle (of an [[affine manifold]]) is a Cartan connection. However, Cartan's original approach was different from this in a number of ways: * the concept of frame bundles or principal bundles did not exist; * a connection was viewed in terms of parallel transport between infinitesimally nearby points;{{efn|It is difficult to make Cartan's intuition precise without invoking [[smooth infinitesimal analysis]], but one way is to regard his points being ''variable'', that is maps from some unseen parameter space into the manifold, which can then be differentiated.}} * this parallel transport was affine, rather than linear; * the objects being transported were not tangent vectors in the modern sense, but elements of an [[affine space]] with a marked point, which the Cartan connection ultimately ''identifies'' with the tangent space. ===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 <sup>j</sup>'', ''ω {{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. ===Affine space as the flat model geometry=== ====Definition of an affine space==== Informally, an '''[[affine space]]''' is a [[vector space]] without a fixed choice of [[origin (mathematics)|origin]]. It describes the geometry of [[point (mathematics)|points]] and [[Vector (geometric)|free vectors]] in space. As a consequence of the lack of origin, points in affine space cannot be added together as this requires a choice of origin with which to form the parallelogram law for vector addition. However, a vector {{mvar|v}} may be added to a point {{mvar|p}} by placing the initial point of the vector at {{mvar|p}} and then transporting {{mvar|p}} to the terminal point. The operation thus described {{math|''p'' → ''p'' + ''v''}} is the '''translation''' of {{mvar|p}} along {{mvar|v}}. In technical terms, affine {{mvar|n}}-space is a set {{math|'''A'''<sup>''n''</sup>}} equipped with a [[Group action (mathematics)|free transitive action]] of the vector group {{math|'''R'''<sup>''n''</sup>}} on it through this operation of translation of points: {{math|'''A'''<sup>''n''</sup>}} is thus a [[principal homogeneous space]] for the vector group {{math|'''R'''<sup>''n''</sup>}}. <!--More could be said here (parallel equipollence, etc.), but this discussion needs to be kept reasonably short. --> The [[general linear group]] {{math|GL(''n'')}} is the [[transformation group|group of transformations]] of {{math|'''R'''<sup>''n''</sup>}} which preserve the ''linear structure'' of {{math|'''R'''<sup>''n''</sup>}} in the sense that {{math|''T''(''av'' + ''bw'') {{=}} ''aT''(''v'') + ''bT''(''w'')}}. By analogy, the '''[[affine group]]''' {{math|Aff(''n'')}} is the group of transformations of {{math|'''A'''<sup>''n''</sup>}} preserving the ''affine structure''. Thus {{math|''φ'' ∈ Aff(''n'')}} must ''preserve translations'' in the sense that :<math>\varphi(p+v)=\varphi(p)+T(v)</math> where {{mvar|T}} is a general linear transformation. The map sending {{math|''φ'' ∈ Aff(''n'')}} to {{math|''T'' ∈ GL(''n'')}} is a [[group homomorphism]]. Its [[kernel (algebra)|kernel]] is the group of translations {{math|'''R'''<sup>''n''</sup>}}. The [[Group action (mathematics)#Orbits and stabilizers|stabilizer]] of any point {{mvar|p}} in {{mvar|A}} can thus be identified with {{math|GL(''n'')}} using this projection: this realises the affine group as a [[semidirect product]] of {{math|GL(''n'')}} and {{math|'''R'''<sup>''n''</sup>}}, and affine space as the [[homogeneous space]] {{math|Aff(''n'')/GL(''n'')}}. ====Affine frames and the flat affine connection==== An ''affine frame'' for {{mvar|A}} consists of a point {{math|''p'' ∈ ''A''}} and a basis {{math|('''e'''<sub>1</sub>,… '''e'''<sub>''n''</sub>)}} of the vector space {{math|T<sub>''p''</sub>''A'' {{=}} '''R'''<sup>''n''</sup>}}. The general linear group {{math|GL(''n'')}} acts freely on the set {{math|F''A''}} of all affine frames by fixing {{mvar|p}} and transforming the basis {{math|('''e'''<sub>1</sub>,… '''e'''<sub>''n''</sub>)}} in the usual way, and the map {{mvar|π}} sending an affine frame {{math|(''p''; '''e'''<sub>1</sub>,… '''e'''<sub>''n''</sub>)}} to {{mvar|p}} is the [[quotient map]]. Thus {{math|F''A''}} is a [[principal bundle|principal {{math|GL(''n'')}}-bundle]] over {{mvar|A}}. The action of {{math|GL(''n'')}} extends naturally to a free transitive action of the affine group {{math|Aff(''n'')}} on {{math|F''A''}}, so that {{math|F''A''}} is an {{math|Aff(''n'')}}-[[principal homogeneous space|torsor]], and the choice of a reference frame identifies {{math|F''A'' → ''A''}} with the principal bundle {{math|Aff(''n'') → Aff(''n'')/GL(''n'')}}. On {{math|F''A''}} there is a collection of {{math|''n'' + 1}} functions defined by :<math>\pi(p;\mathbf{e}_1, \dots ,\mathbf{e}_n) = p</math> (as before) and :<math>\varepsilon_i(p;\mathbf{e}_1,\dots , \mathbf{e}_n) = \mathbf{e}_i\,.</math> After choosing a basepoint for {{mvar|A}}, these are all functions with values in {{math|'''R'''<sup>''n''</sup>}}, so it is possible to take their [[exterior derivative]]s to obtain [[differential 1-form]]s with values in {{math|'''R'''<sup>''n''</sup>}}. Since the functions {{mvar|ε<sub>i</sub>}} yield a basis for {{math|'''R'''<sup>''n''</sup>}} at each point of {{math|F''A''}}, these 1-forms must be expressible as sums of the form :<math>\begin{align} \mathrm{d}\pi &= \theta^1\varepsilon_1+\cdots+\theta^n\varepsilon_n\\ \mathrm{d}\varepsilon_i &= \omega^1_i\varepsilon_1+\cdots+\omega^n_i\varepsilon_n \end{align}</math> for some collection {{math|(''θ <sup>i</sup>'', ''ω {{su|b=j|p=k}}'')<sub>1 ≤ ''i'', ''j'', ''k'' ≤ ''n''</sub>}} of real-valued one-forms on {{math|Aff(''n'')}}. This system of one-forms on the principal bundle {{math|F''A'' → ''A''}} defines the affine connection on {{mvar|A}}. Taking the exterior derivative a second time, and using the fact that {{math|d<sup>2</sup> {{=}} 0}} as well as the [[linearly independent|linear independence]] of the {{mvar|ε<sub>i</sub>}}, the following relations are obtained: :<math>\begin{align} \mathrm{d}\theta^j - \sum_i\omega^j_i\wedge\theta^i &=0\\ \mathrm{d}\omega^j_i - \sum_k \omega^j_k\wedge\omega^k_i &=0\,. \end{align}</math> These are the [[Maurer–Cartan equation]]s for the Lie group {{math|Aff(''n'')}} (identified with {{math|F''A''}} by the choice of a reference frame). Furthermore: * the [[Pfaffian system]] {{math|''θ <sup>j</sup>'' {{=}} 0}} (for all {{mvar|j}}) is [[integrability condition|integrable]], and its [[integral manifold]]s are the fibres of the principal bundle {{math|Aff(''n'') → ''A''}}. * the Pfaffian system {{math|''ω {{su|b=i|p=j}}'' {{=}} 0}} (for all {{math|''i'', ''j''}}) is also integrable, and its integral manifolds define parallel transport in {{math|F''A''}}. Thus the forms {{math|(''ω {{su|b=i|p=j}}'')}} define a flat [[connection (principal bundle)|principal connection]] on {{math|F''A'' → ''A''}}. For a strict comparison with the motivation, one should actually define parallel transport in a principal {{math|Aff(''n'')}}-bundle over {{mvar|A}}. This can be done by [[pullback bundle|pulling back]] {{math|F''A''}} by the smooth map {{math|''φ'' : '''R'''<sup>''n''</sup> × ''A'' → ''A''}} defined by translation. Then the composite {{math|''φ''′ ∗ F''A'' → F''A'' → ''A''}} is a principal {{math|Aff(''n'')}}-bundle over {{mvar|A}}, and the forms {{math|(''θ <sup>i</sup>'', ''ω {{su|b=j|p=k}}'')}} [[pullback (differential geometry)|pull back]] to give a flat principal {{math|Aff(''n'')}}-connection on this bundle. ===General affine geometries: formal definitions=== An affine space, as with essentially any smooth [[Klein geometry]], is a manifold equipped with a flat Cartan connection. More general affine manifolds or affine geometries are obtained easily by dropping the flatness condition expressed by the Maurer-Cartan equations. There are several ways to approach the definition and two will be given. Both definitions are facilitated by the realisation that 1-forms {{math|(''θ <sup>i</sup>'', ''ω {{su|b=j|p=k}}'')}} in the flat model fit together to give a 1-form with values in the Lie algebra {{math|'''aff'''(''n'')}} of the affine group {{math|Aff(''n'')}}. In these definitions, {{mvar|M}} is a smooth {{mvar|n}}-manifold and {{math|''A'' {{=}} Aff(''n'')/GL(''n'')}} is an affine space of the same dimension. ====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}}. ====Definition as a principal affine connection==== An '''affine connection''' on {{mvar|M}} is a principal {{math|Aff(''n'')}}-bundle {{mvar|Q}} over {{mvar|M}}, together with a principal {{math|GL(''n'')}}-subbundle {{mvar|P}} of {{mvar|Q}} and a principal {{math|Aff(''n'')}}-connection {{mvar|α}} (a 1-form on {{mvar|Q}} with values in {{math|'''aff'''(''n'')}}) which satisfies the following (generic) ''Cartan condition''. The {{math|'''R'''<sup>''n''</sup>}} component of pullback of {{mvar|α}} to {{mvar|P}} is a horizontal equivariant 1-form and so defines a bundle homomorphism from {{math|T''M''}} to {{math|''P'' ×<sub>GL(''n'')</sub> '''R'''<sup>''n''</sup>}}: this is required to be an isomorphism. ====Relation to the motivation==== Since {{math|Aff(''n'')}} acts on {{mvar|A}}, there is, associated to the principal bundle {{mvar|Q}}, a bundle {{math|'''''A''''' {{=}} ''Q'' ×<sub>Aff(''n'')</sub> ''A''}}, which is a fiber bundle over {{mvar|M}} whose fiber at {{mvar|x}} in {{mvar|M}} is an affine space {{math|''A''<sub>''x''</sub>}}. A [[section (fiber bundle)|section]] {{mvar|a}} of {{mvar|'''A'''}} (defining a marked point {{mvar|''a''<sub>''x''</sub>}} in {{mvar|''A''<sub>''x''</sub>}} for each {{mvar|''x'' ∈ ''M''}}) determines a principal {{math|GL(''n'')}}-subbundle {{mvar|P}} of {{mvar|Q}} (as the bundle of stabilizers of these marked points) and vice versa. The principal connection {{mvar|α}} defines an [[Ehresmann connection]] on this bundle, hence a notion of parallel transport. The Cartan condition ensures that the distinguished section {{mvar|a}} always moves under parallel transport.
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