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| In [[geometry]], the notion of a '''connection''' makes precise the idea of transporting data along a curve or family of curves in a ''parallel'' and consistent manner. There are a variety of kinds of connections in modern geometry, depending on what sort of data one wants to transport. For instance, an [[affine connection]], the most elementary type of connection, gives a means for transporting [[tangent space|tangent vectors]] to a [[manifold]] from one point to another along a curve. An affine connection is typically given in the form of a [[covariant derivative]], which gives a means for taking [[directional derivative]]s of vector fields: the infinitesimal transport of a [[vector field]] in a given direction.
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| Connections are of central importance in modern geometry in large part because they allow a comparison between the local geometry at one point and the local geometry at another point. [[Differential geometry]] embraces several variations on the connection theme, which fall into two major groups: the infinitesimal and the local theory. The local theory concerns itself primarily with notions of [[parallel transport]] and [[holonomy]]. The infinitesimal theory concerns itself with the differentiation of geometric data. Thus a covariant derivative is a way of specifying a [[derivative]] of a vector field along another vector field on a manifold. A [[Cartan connection]] is a way of formulating some aspects of connection theory using [[differential forms]] and [[Lie group]]s. An [[Ehresmann connection]] is a connection in a [[fibre bundle]] or a [[principal bundle]] by specifying the allowed directions of motion of the field. A [[Koszul connection]] is a connection generalizing the derivative in a [[vector bundle]].
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| Connections also lead to convenient formulations of ''geometric invariants'', such as the [[curvature]] (see also [[Riemann curvature tensor|curvature tensor]] and [[curvature form]]), and [[torsion tensor]].
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| ==Motivation: the unsuitability of coordinates==
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| [[File:Connection-on-sphere.png|frame|Parallel transport (of black arrow) on a sphere. Blue, respectively red arrows represent parallel transports in different directions but ending at the same lower right point. The fact that they end up not pointing in the same direction is a function of the curvature of the sphere.]]
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| Consider the following problem. Suppose that a tangent vector to the sphere ''S'' is given at the north pole, and we are to define a manner of consistently moving this vector to other points of the sphere: a means for ''parallel transport''. Naïvely, this could be done using a particular [[coordinate system]]. However, unless proper care is applied, the parallel transport defined in one system of coordinates will not agree with that of another coordinate system. A more appropriate parallel transportation system exploits the symmetry of the sphere under rotation. Given a vector at the north pole, one can transport this vector along a curve by rotating the sphere in such a way that the north pole moves along the curve without axial rolling. This latter means of parallel transport is the [[Levi-Civita connection]] on the sphere. If two different curves are given with the same initial and terminal point, and a vector ''v'' is rigidly moved along the first curve by a rotation, the resulting vector at the terminal point will be ''different from'' the vector resulting from rigidly moving ''v'' along the second curve. This phenomenon reflects the [[curvature]] of the sphere. A simple mechanical device that can be used to visualize parallel transport is the [[South Pointing Chariot]].
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| For instance, suppose that ''S'' is given coordinates by the [[stereographic projection]]. Regard ''S'' as consisting of unit vectors in '''R'''<sup>3</sup>. Then ''S'' carries a pair of coordinate patches: one covering a neighborhood of the north pole, and the other of the south pole. The mappings
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| :<math>
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| \begin{align}
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| \varphi_0(x,y) & =\left(\frac{2x}{1+x^2+y^2}, \frac{2y}{1+x^2+y^2}, \frac{1-x^2-y^2}{1+x^2+y^2}\right)\\[8pt]
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| \varphi_1(x,y) & =\left(\frac{2x}{1+x^2+y^2}, \frac{2y}{1+x^2+y^2}, \frac{x^2+y^2-1}{1+x^2+y^2}\right)
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| \end{align}
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| </math>
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| cover a neighborhood ''U''<sub>0</sub> of the north pole and ''U''<sub>1</sub> of the south pole, respectively. Let ''X'', ''Y'', ''Z'' be the ambient coordinates in '''R'''<sup>3</sup>. Then φ<sub>0</sub> and φ<sub>1</sub> have inverses
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| :<math>
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| \begin{align}
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| \varphi_0^{-1}(X,Y,Z)&=\left(\frac{X}{Z+1}, \frac{Y}{Z+1}\right), \\[8pt]
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| \varphi_1^{-1}(X,Y,Z)&=\left(\frac{-X}{Z-1}, \frac{-Y}{Z-1}\right),
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| \end{align}
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| </math>
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| so that the coordinate transition function is [[inversion in a circle|inversion in the circle]]:
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| :<math>\varphi_{01}(x,y) = \varphi_0^{-1}\circ\varphi_1(x,y)=\left(\frac{x}{x^2+y^2},\frac{y}{x^2+y^2}\right)</math>
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| Let us now represent a vector field in terms of its components relative to the coordinate derivatives. If ''P'' is a point of ''U''<sub>0</sub> ⊂ ''S'', then a vector field may be represented by
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| :<math>v(P) = J_{\varphi_0}(\varphi_0^{-1}(P))\cdot {\bold v}_0(\varphi_0^{-1}(P))\qquad(1) </math>
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| where <math>J_{\varphi_0}</math> denotes the [[Jacobian matrix]] of φ<sub>0</sub>, and '''v'''<sub>0</sub> = '''v'''<sub>0</sub>(''x'', ''y'') is a vector field on '''R'''<sup>2</sup> uniquely determined by ''v''. Furthermore, on the overlap between the coordinate charts ''U''<sub>0</sub> ∩ ''U''<sub>1</sub>, it is possible to represent the same vector field with respect to the φ<sub>1</sub> coordinates:
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| :<math>v(P) = J_{\varphi_1}(\varphi_1^{-1}(P))\cdot {\bold v}_1(\varphi_1^{-1}(P)). \qquad (2) </math>
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| To relate the components '''v'''<sub>0</sub> and '''v'''<sub>1</sub>, apply the [[chain rule]] to the identity φ<sub>1</sub> = φ<sub>0</sub> o φ<sub>01</sub>:
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| :<math>J_{\varphi_1}(\varphi_1^{-1}(P)) = J_{\varphi_0}(\varphi_0^{-1}(P))\cdot J_{\varphi_{01}}(\varphi_1^{-1}(P)). \, </math>
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| Applying both sides of this matrix equation to the component vector '''v'''<sub>1</sub>(φ<sub>1</sub><sup>−1</sup>(''P'')) and invoking (1) and (2) yields
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| :<math>{\bold v}_0(\varphi_0^{-1}(P)) = J_{\varphi_{01}}(\varphi_1^{-1}(P))\cdot {\bold v}_1(\varphi_1^{-1}(P)). \qquad (3)</math> | |
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| We come now to the main question of defining how to transport a vector field parallelly along a curve. Suppose that ''P''(''t'') is a curve in ''S''. Naïvely, one may consider a vector field parallel if the coordinate components of the vector field are constant along the curve. However, an immediate ambiguity arises: in ''which'' coordinate system should these components be constant?
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| For instance, suppose that ''v''(''P''(''t'')) has constant components in the ''U''<sub>1</sub> coordinate system. That is, the functions '''v'''<sub>1</sub>(''φ''<sub>1</sub><sup>−1</sup>(''P''(''t''))) are constant. However, applying the [[product rule]] to (3) and using the fact that ''d'''''v'''<sub>1</sub>/''dt'' = 0 gives
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| :<math>\frac{d}{dt}{\bold v}_0(\varphi_0^{-1}(P(t)))=\left(\frac{d}{dt}J_{\varphi_{01}}(\varphi_1^{-1}(P(t)))\right)\cdot {\bold v}_1(\varphi_1^{-1}(P(t))).</math>
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| But <math>\left(\frac{d}{dt}J_{\varphi_{01}}(\varphi_1^{-1}(P(t)))\right)</math> is always a non-singular matrix (provided that the curve ''P''(''t'') is not stationary), so '''v'''<sub>1</sub> and '''v'''<sub>0</sub> ''cannot ever be'' simultaneously constant along the curve.
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| ===Resolution===
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| The problem observed above is that the usual [[directional derivative]] of [[vector calculus]] does not behave well under changes in the coordinate system when applied to the components of vector fields. This makes it quite difficult to describe how to parallelly translate vector fields, if indeed such a notion makes any sense at all. There are two fundamentally different ways of resolving this problem.
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| The first approach is to examine what is required for a generalization of the directional derivative to "behave well" under coordinate transitions. This is the tactic taken by the [[covariant derivative]] approach to connections: good behavior is equated with [[covariance and contravariance of vectors|covariance]]. Here one considers a modification of the directional derivative by a certain [[linear operator]], whose components are called the [[Christoffel symbols]], which involves no derivatives on the vector field itself. The directional derivative ''D''<sub>'''u'''</sub>'''v''' of the components of a vector '''v''' in a coordinate system φ in the direction '''u''' are replaced by a ''covariant derivative'':
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| :<math>\nabla_{\bold u} {\bold v} = D_{\bold u} {\bold v} + \Gamma(\varphi)\{{\bold u},{\bold v}\}</math>
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| where Γ depends on the coordinate system φ and is [[Bilinear form|bilinear]] in '''u''' and '''v'''. In particular, Γ does not involve any derivatives on '''u''' or '''v'''. In this approach, Γ must transform in a prescribed manner when the coordinate system φ is changed to a different coordinate system. This transformation is not [[tensor]]ial, since it involves not only the ''first derivative'' of the coordinate transition, but also its ''second derivative''. Specifying the transformation law of Γ is not sufficient to determine Γ uniquely. Some other normalization conditions must be imposed, usually depending on the type of geometry under consideration. In [[Riemannian geometry]], the [[Levi-Civita connection]] requires compatibility of the [[Christoffel symbols]] with the [[Riemannian metric|metric]] (as well as a certain symmetry condition). With these normalizations, the connection is uniquely defined.
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| The second approach is to use [[Lie group]]s to attempt to capture some vestige of symmetry on the space. This is the approach of [[Cartan connection]]s. The example above using rotations to specify the parallel transport of vectors on the sphere is very much in this vein.
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| ==Historical survey of connections==
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| Historically, connections were studied from an [[infinitesimal]] perspective in [[Riemannian geometry]]. The infinitesimal study of connections began to some extent with [[Christoffel]]. This was later taken up more thoroughly by [[Gregorio Ricci-Curbastro]] and [[Tullio Levi-Civita]] {{harv|Levi-Civita|Ricci|1900}} who observed in part that a connection in the infinitesimal sense of Christoffel also allowed for a notion of [[parallel transport]].
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| The work of Levi-Civita focused exclusively on regarding connections as a kind of [[differential operator]] whose parallel displacements were then the solutions of [[differential equation]]s. As the twentieth century progressed, [[Élie Cartan]] developed a new notion of connection. He sought to apply the techniques of [[Pfaffian system]]s to the geometries of [[Felix Klein]]'s [[Erlangen program]]. In these investigations, he found that a certain infinitesimal notion of connection (a [[Cartan connection]]) could be applied to these geometries and more: his connection concept allowed for the presence of [[curvature]] which would otherwise be absent in a classical Klein geometry. (See, for example, {{harv|Cartan|1926}} and {{harv|Cartan|1983}}.) Furthermore, using the dynamics of [[Gaston Darboux]], Cartan was able to generalize the notion of parallel transport for his class of infinitesimal connections. This established another major thread in the theory of connections: that a connection is a certain kind of [[differential form]].
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| The two threads in connection theory have persisted through the present day: a connection as a differential operator, and a connection as a differential form. In 1950, [[Jean-Louis Koszul]] {{harv|Koszul|1950}} gave an algebraic framework for regarding a connection as a differential operator by means of the [[Koszul connection]]. The Koszul connection was both more general than that of Levi-Civita, and was easier to work with because it finally was able to eliminate (or at least to hide) the awkward [[Christoffel symbols]] from the connection formalism. The attendant parallel displacement operations also had natural algebraic interpretations in terms of the connection. Koszul's definition was subsequently adopted by most of the differential geometry community, since it effectively converted the ''analytic'' correspondence between covariant differentiation and parallel translation to an ''algebraic'' one.
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| In that same year, [[Charles Ehresmann]] {{harv|Ehresmann|1950}}, a student of Cartan's, presented a variation on the connection as a differential form view in the context of [[principal bundle]]s and, more generally, [[fibre bundle]]s. [[Ehresmann connection]]s were, strictly speaking, not a generalization of Cartan connections. Cartan connections were quite rigidly tied to the underlying [[differential topology]] of the manifold because of their relationship with [[Cartan's equivalence method]]. Ehresmann connections were rather a solid framework for viewing the foundational work of other geometers of the time, such as [[Shiing-Shen Chern]], who had already begun moving away from Cartan connections to study what might be called [[gauge connection]]s. In Ehresmann's point of view, a connection in a principal bundle consists of a specification of ''horizontal'' and ''vertical'' vector fields on the total space of the bundle. A parallel translation is then a lifting of a curve from the base to a curve in the principal bundle which is horizontal. This viewpoint has proven especially valuable in the study of [[holonomy]].
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| ==Possible approaches==
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| *A rather direct approach is to specify how a [[covariant derivative]] acts on elements of the [[module (mathematics)|module]] of [[vector field]]s as a [[differential operator]]. More generally, a similar approach applies for [[connection (vector bundle)|connections]] in any [[vector bundle]].
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| *Traditional index notation specifies the connection by components; see [[Christoffel symbols]]. (''Note'': this has three indices, but is '''''not''''' a [[tensor]]).
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| *In [[pseudo-Riemannian]] and [[Riemannian geometry]] the [[Levi-Civita connection]] is a special connection associated to the [[metric tensor]].
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| *These are examples of [[affine connection]]s. There is also a concept of [[projective connection]], of which the [[Schwarzian derivative]] in [[complex analysis]] is an instance. More generally, both affine and projective connections are types of [[Cartan connection]]s.
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| *Using [[principal bundle]]s, a connection can be realized as a [[Lie algebra]]-valued [[differential form]]. See [[connection (principal bundle)]].
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| *An approach to connections which makes direct use of the notion of transport of "data" (whatever that may be) is the [[Ehresmann connection]].
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| *The most abstract approach may be that suggested by [[Alexander Grothendieck]], where a [[Grothendieck connection]] is seen as [[descent (category theory)|descent]] data from infinitesimal neighbourhoods of the [[diagonal]]; see {{harv|Osserman|2004}}.
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| ==References==
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| {{Reflist}}<!--added under references heading by script-assisted edit-->
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| *{{citation|last1=Levi-Civita|first1=T.|last2=Ricci|first2=G.|title=Méthodes de calcul différential absolu et leurs applications|journal=Math. Ann. B|volume=54|year=1900|pages=125–201|doi=10.1007/BF01454201}}
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| *{{citation|last=Cartan|first=Élie|authorlink=Élie Cartan|title=Sur les varietes a connexion projective|journal=Bulletin de la Société Mathématique|volume=52|year=1924|pages=205–241}}
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| *{{citation| first = Élie | last = Cartan | title = Espaces à connexion affine, projective et conforme | journal = Acta Math. | volume = 48 | year = 1926 | pages = 1–42 | doi=10.1007/BF02629755}}
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| *{{citation|last=Cartan|first=Élie|authorlink=Élie Cartan|title=Geometry of Riemannian spaces|publisher=Math Sci Press|year=1983|url=http://books.google.com/?id=-YvvVfQ7xz4C&printsec=frontcover|isbn=978-0-915692-34-7}}
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| *{{citation|last=Ehresmann|first=C.|title=Les connexions infinitésimales dans un espace fibré différentiable|series=Colloque de Toplogie, Bruxelles|year=1950|pages=29–55}}
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| *{{citation|last=Koszul|first=J. L.|title=Homologie et cohomologie des algebres de Lie|journal=Bulletin de la Société Mathématique|volume=78|year=1950|pages=65–127}}
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| *{{springer|id=C/c025140|author=Lumiste, Ü.|title=Connection}}
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| *{{citation|last=Osserman|first=B.|title=Connections, curvature, and p-curvature|url=http://math.berkeley.edu/~osserman/math/connections.pdf|format=PDF|year=2004}}
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| *{{Citation | last = Mangiarotti | first = L.
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| | last2 = [[Gennadi Sardanashvily|Sardanashvily]]
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| | first2 = G. | year = 2000
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| | title = Connections in Classical and Quantum Field Theory
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| | publisher = World Scientific | isbn = 981-02-2013-8
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| }}.
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| * {{citation | first=Shigeyuki|last= Morita | title=Geometry of Differential Forms | publisher=AMS | year=2001 | isbn = 0-8218-1045-6}}
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| == External links ==
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| *[http://www.map.mpim-bonn.mpg.de/Connections Connections] at the Manifold Atlas
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| ==See also==
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| *[[Affine connection]]
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| *[[Cartan connection]]
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| *[[Ehresmann connection]]
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| *[[Connection (fibred manifold)]]
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| *[[Connection (principal bundle)]]
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| *[[Connection (vector bundle)]]
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| *[[Connection (affine bundle)]]
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| *[[Connection (composite bundle)]]
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| *[[Connection (algebraic framework)]]
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| [[Category:Differential geometry]]
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| [[Category:Connection (mathematics)| ]]
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| [[Category:Differential topology]]
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