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Ingeometry, the notion of aconnection makes precise the idea of transporting local geometric objects, such astangent vectors ortensors in thetangent space, along a curve or family of curves in aparallel and consistent manner. There are various kinds of connections in modern geometry, depending on what sort of data one wants to transport. For instance, anaffine connection, the most elementary type of connection, gives a means for parallel transport oftangent vectors on amanifold from one point to another along a curve. An affine connection is typically given in the form of acovariant derivative, which gives a means for takingdirectional derivatives of vector fields, measuring the deviation of avector field from being parallel in a given direction.

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 ofparallel transport andholonomy. The infinitesimal theory concerns itself with the differentiation of geometric data. Thus a covariant derivative is a way of specifying aderivative of a vector field along another vector field on a manifold. ACartan connection is a way of formulating some aspects of connection theory usingdifferential forms andLie groups. AnEhresmann connection is a connection in afibre bundle or aprincipal bundle by specifying the allowed directions of motion of the field. AKoszul connection is a connection which defines directional derivative for sections of avector bundle more general than the tangent bundle.

Connections also lead to convenient formulations ofgeometric invariants, such as thecurvature (see alsocurvature tensor andcurvature form), andtorsion tensor.

Consider the following problem. Suppose that a tangent vector to the sphereS 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 forparallel transport. Naively, this could be done using a particularcoordinate 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 theLevi-Civita connection on the sphere. If two different curves are given with the same initial and terminal point, and a vectorv is rigidly moved along the first curve by a rotation, the resulting vector at the terminal point will bedifferent from the vector resulting from rigidly movingv along the second curve. This phenomenon reflects thecurvature of the sphere. A simple mechanical device that can be used to visualize parallel transport is thesouth-pointing chariot.

For instance, suppose thatS is a sphere given coordinates by thestereographic projection. RegardS as consisting of unit vectors inR³. ThenS carries a pair ofcoordinate patches corresponding to the projections from north pole and south pole. The mappings

cover a neighborhoodU₀ of the north pole andU₁ of the south pole, respectively. LetX,Y,Z be the ambient coordinates inR³. Then φ₀ and φ₁ have inverses

so that the coordinate transition function isinversion in the circle:

${\displaystyle {\phi }_{01}(x,y)={\phi }_0^{-1}\circ {\phi }_1(x,y)=\left(\frac{x}{x^2+y^2},\frac{y}{x^2+y^2}\right)}$

Let us now represent avector field${\displaystyle v}$ on S (an assignment of a tangent vector to each point in S) in local coordinates. IfP is a point ofU₀ ⊂S, then a vector field may be represented by thepushforward of a vector fieldv₀ onR² by${\displaystyle {\phi }_0}$:

${\displaystyle v(P)=J_{{\phi }_0}\left({\phi }_0^{-1}(P)\right)\cdot {\mathbf{v}}_0\left({\phi }_0^{-1}(P)\right)}$

1

where${\displaystyle J_{{\phi }_0}}$ denotes theJacobian matrix of φ₀ (${\displaystyle d{{\phi }_0}_x(\mathbf{u})=J_{{\phi }_0}(x)\cdot \mathbf{u}}$), andv₀ = v₀(x, y) is a vector field onR² uniquely determined byv (since the pushforward of alocal diffeomorphism at any point is invertible). Furthermore, on the overlap between the coordinate chartsU₀ ∩U₁, it is possible to represent the same vector field with respect to the φ₁ coordinates:

${\displaystyle v(P)=J_{{\phi }_1}\left({\phi }_1^{-1}(P)\right)\cdot {\mathbf{v}}_1\left({\phi }_1^{-1}(P)\right).}$

2

To relate the componentsv₀ andv₁, apply thechain rule to the identity φ₁ = φ₀ o φ₀₁:

${\displaystyle J_{{\phi }_1}\left({\phi }_1^{-1}(P)\right)=J_{{\phi }_0}\left({\phi }_0^{-1}(P)\right)\cdot J_{{\phi }_{01}}\left({\phi }_1^{-1}(P)\right).}$

Applying both sides of this matrix equation to the component vectorv₁(φ₁⁻¹(P)) and invoking (1) and (2) yields

${\displaystyle {\mathbf{v}}_0\left({\phi }_0^{-1}(P)\right)=J_{{\phi }_{01}}\left({\phi }_1^{-1}(P)\right)\cdot {\mathbf{v}}_1\left({\phi }_1^{-1}(P)\right).}$

3

We come now to the main question of defining how to transport a vector field parallelly along a curve. Suppose thatP(t) is a curve inS. 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: inwhich coordinate system should these components be constant?

For instance, suppose thatv(P(t)) has constant components in theU₁ coordinate system. That is, the functionsv₁(φ₁⁻¹(P(t))) are constant. However, applying theproduct rule to (3) and using the fact thatdv₁/dt = 0 gives

${\displaystyle \frac{d}{dt}{\mathbf{v}}_0\left({\phi }_0^{-1}(P(t))\right)=\left(\frac{d}{dt}{J}_{{\phi }_{01}}\left({\phi }_1^{-1}(P(t))\right)\right)\cdot {\mathbf{v}}_1\left({\phi }_1^{-1}\left(P(t)\right)\right).}$

But${\displaystyle \left(\frac{d}{dt}{J}_{{\phi }_{01}}\left({\phi }_1^{-1}(P(t))\right)\right)}$ is always a non-singular matrix (provided that the curveP(t) is not stationary), sov₁ andv₀cannot ever be simultaneously constant along the curve.

The problem observed above is that the usualdirectional derivative ofvector 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 translate vector fields in a parallel manner, if indeed such a notion makes any sense at all. There are two fundamentally different ways of resolving this problem.

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 thecovariant derivative approach to connections: good behavior is equated withcovariance. Here one considers a modification of the directional derivative by a certainlinear operator, whose components are called theChristoffel symbols, which involves no derivatives on the vector field itself. The directional derivativeD_uv of the components of a vectorv in a coordinate systemφ in the directionu are replaced by acovariant derivative:

${\displaystyle {\mathrm{\nabla }}_{\mathbf{u}}\mathbf{v}=D_{\mathbf{u}}\mathbf{v}+\mathrm{\Gamma }(\phi )\{\mathbf{u},\mathbf{v}\}}$

where Γ depends on the coordinate systemφ and isbilinear inu andv. In particular, Γ does not involve any derivatives onu orv. In this approach, Γ must transform in a prescribed manner when the coordinate systemφ is changed to a different coordinate system. This transformation is nottensorial, since it involves not only thefirst derivative of the coordinate transition, but also itssecond 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. InRiemannian geometry, theLevi-Civita connection requires compatibility of theChristoffel symbols with themetric (as well as a certain symmetry condition). With these normalizations, the connection is uniquely defined.

The second approach is to useLie groups to attempt to capture some vestige of symmetry on the space. This is the approach ofCartan connections. The example above using rotations to specify the parallel transport of vectors on the sphere is very much in this vein.

Historically, connections were studied from aninfinitesimal perspective inRiemannian geometry. The infinitesimal study of connections began to some extent withElwin Christoffel. This was later taken up more thoroughly byGregorio Ricci-Curbastro andTullio Levi-Civita (Levi-Civita & Ricci 1900) who observed in part that a connection in the infinitesimal sense of Christoffel also allowed for a notion ofparallel transport.

The work of Levi-Civita focused exclusively on regarding connections as a kind ofdifferential operator whose parallel displacements were then the solutions ofdifferential equations. As the twentieth century progressed,Élie Cartan developed a new notion of connection. He sought to apply the techniques ofPfaffian systems to the geometries ofFelix Klein'sErlangen program. In these investigations, he found that a certain infinitesimal notion of connection (aCartan connection) could be applied to these geometries and more: his connection concept allowed for the presence ofcurvature which would otherwise be absent in a classical Klein geometry. (See, for example, (Cartan 1926) and (Cartan 1983).) Furthermore, using the dynamics ofGaston 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 ofdifferential form.

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 (Koszul 1950) gave an algebraic framework for regarding a connection as a differential operator by means of theKoszul 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 awkwardChristoffel 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 theanalytic correspondence between covariant differentiation and parallel translation to analgebraic one.

In that same year,Charles Ehresmann (Ehresmann 1950), a student of Cartan's, presented a variation on the connection as a differential form view in the context ofprincipal bundles and, more generally,fibre bundles.Ehresmann connections were, strictly speaking, not a generalization of Cartan connections. Cartan connections were quite rigidly tied to the underlyingdifferential topology of the manifold because of their relationship withCartan's equivalence method. Ehresmann connections were rather a solid framework for viewing the foundational work of other geometers of the time, such asShiing-Shen Chern, who had already begun moving away from Cartan connections to study what might be calledgauge connections. In Ehresmann's point of view, a connection in a principal bundle consists of a specification ofhorizontal andvertical 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 ofholonomy.

• A rather direct approach is to specify how acovariant derivative acts on elements of themodule ofvector fields as adifferential operator. More generally, a similar approach applies forconnections in anyvector bundle.
• Traditional index notation specifies the connection by components; seeChristoffel symbols. (Note: this has three indices, but isnot atensor).
• Inpseudo-Riemannian andRiemannian geometry theLevi-Civita connection is a special connection associated to themetric tensor.
• These are examples ofaffine connections. There is also a concept ofprojective connection, of which theSchwarzian derivative incomplex analysis is an instance. More generally, both affine and projective connections are types ofCartan connections.
• Usingprincipal bundles, a connection can be realized as aLie algebra-valueddifferential form. Seeconnection (principal bundle).
• An approach to connections which makes direct use of the notion of transport of "data" (whatever that may be) is theEhresmann connection.
• The most abstract approach may be that suggested byAlexander Grothendieck, where aGrothendieck connection is seen asdescent data from infinitesimal neighbourhoods of thediagonal; see (Osserman 2004).

• Affine connection
• Cartan connection
• Ehresmann connection
• Grothendieck connection
• Levi-Civita connection
• Connection form
• Connection (fibred manifold)
• Connection (principal bundle)
• Connection (vector bundle)
• Connection (affine bundle)
• Connection (composite bundle)
• Connection (algebraic framework)
• Gauge theory (mathematics)
• Connes connection

• Levi-Civita, T.; Ricci, G. (1900),"Méthodes de calcul différentiel absolu et leurs applications",Mathematische Annalen,54 (1–2):125–201,doi:10.1007/BF01454201,S2CID 120009332
• Cartan, Élie (1924), "Sur les variétés à connexion projective",Bulletin de la Société Mathématique de France,52:205–241,doi:10.24033/bsmf.1053
• Cartan, Élie (1926), "Les groupes d'holonomie des espaces généralisés",Acta Mathematica,48 (1–2):1–42,doi:10.1007/BF02629755
• Cartan, Élie (1983),Geometry of Riemannian spaces, Math Sci Press,ISBN 978-0-915692-34-7
• Ehresmann, C. (1950),Les connexions infinitésimales dans un espace fibré différentiable, Colloque de Toplogie, Bruxelles, pp. 29–55
• Koszul, J. L. (1950), "Homologie et cohomologie des algèbres de Lie",Bulletin de la Société Mathématique de France,78:65–127,doi:10.24033/bsmf.1410
• Lumiste, Ü. (2001) [1994],"Connection",Encyclopedia of Mathematics,EMS Press
• Osserman, B. (2004),Connections, curvature, and p-curvature(PDF), archived fromthe original(PDF) on 2006-12-21, retrieved2007-02-04
• Mangiarotti, L.;Sardanashvily, G. (2000),Connections in Classical and Quantum Field Theory, World Scientific,ISBN 981-02-2013-8.
• Morita, Shigeyuki (2001),Geometry of Differential Forms, AMS,ISBN 0-8218-1045-6

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