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. Connections are of central importance in modern geometry in large part because they allow a comparison between the local geometry at one point to 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 groups. An Ehresmann connection is a connection in a fibre bundle or a principal bundle using osculating spaces of the derivative of a field. A Koszul connection is a connection generalizing the derivative in a vector bundle.

Connections also lead to convenient formulations of geometric invariants, such as the curvature (see also curvature tensor and curvature form), and torsion tensor.

General concept: the unsuitability of coordinates

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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.

For instance, suppose that S is given coordinates by the stereographic projection. Regard S as consisting of unit vectors in R³. 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

$$

\begin{matrix} \phi_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+x^2}\right)\\ \phi_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+x^2}\right) \end{matrix} cover a neighborhood U₀ of the north pole and U₁ of the south pole, respectively. Let X, Y, Z be the ambient coordinates in R³. Then φ₀ and φ₁ have inverses

$$

\begin{matrix} \phi_0^{-1}(X,Y,Z)&=\left(\frac{X}{Z+1}, \frac{Y}{Z+1}\right)\\ &\\ \phi_1^{-1}(X,Y,Z)&=\left(\frac{-X}{Z-1}, \frac{-Y}{Z-1}\right) \end{matrix} , so that the coordinate transition function is inversion in the circle:

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

Let us now represent a vector field in terms of its components relative to the coordinate derivatives. If P is a point of U₀ ⊂ S, then a vector field may be represented by

$v(P)\; =\; D\backslash phi\_0(\backslash phi\_0^\{-1\}(P))\backslash cdot\; \{\backslash bold\; v\}\_0(\backslash phi\_0^\{-1\}(P))\backslash ,\backslash ,\backslash ,$ (1)

where Dφ₀ denotes the Jacobian matrix of φ₀, and v₀=v₀(x,y) is a vector field on R² uniquely determined by v. Furthermore, on the overlap between the coordinate charts U₀ ∩ U₁, it is possible to represent the same vector field with respect to the φ₁ coordinates:

$v(P)\; =\; D\backslash phi\_1(\backslash phi\_1^\{-1\}(P))\backslash cdot\; \{\backslash bold\; v\}\_1(\backslash phi\_1^\{-1\}(P))\backslash ,\backslash ,\backslash ,$ (2).

To relate the components v₀ and v₁, apply the chain rule to the identity φ₁ = φ₀ o φ₀₁:

$D\backslash phi\_1(\backslash phi\_1^\{-1\}(P))\; =\; D\backslash phi\_0(\backslash phi\_0^\{-1\}(P))\backslash cdot\; D\backslash phi\_\{01\}(\backslash phi\_1^\{-1\}(P))$.

Applying both sides of this matrix equation to the component vector v₁(φ₁^-1(P)) and invoking (1) and (2) yields

$\{\backslash bold\; v\}\_0(\backslash phi\_0^\{-1\}(P))\; =\; D\backslash phi\_\{01\}(\backslash phi\_1^\{-1\}(P))\backslash cdot\; \{\backslash bold\; v\}\_1(\backslash phi\_1^\{-1\}(P))$. (3)

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?

For instance, suppose that v(P(t)) has constant components in the U₁ coordinate system. That is, the functions v₁(φ₁^-1(P(t))) are constant. However, applying the product rule to (3) and using the fact that dv₁/dt = 0 gives

$\backslash frac\{d\}\{dt\}\{\backslash bold\; v\}\_0(\backslash phi\_0^\{-1\}(P(t)))=\backslash left(\backslash frac\{d\}\{dt\}D\backslash phi\_\{01\}(\backslash phi\_1^\{-1\}(P(t)))\backslash right)\backslash cdot\; \{\backslash bold\; v\}\_1(\backslash phi\_1^\{-1\}(P(t)))$.

But $\backslash left(\backslash frac\{d\}\{dt\}D\backslash phi\_\{01\}(\backslash phi\_1^\{-1\}(P(t)))\backslash right)$ is always a non-singular matrix (provided that the curve P(t) is not stationary), so v₁ and v₀ cannot ever be simultaneously constant along the curve.

Resolution

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.

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. 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_uv of the components of a vector v in a coordinate system φ in the direction u are replaced by a covariant derivative:

$\backslash nabla\_\{\backslash bold\; u\}\; \{\backslash bold\; v\}\; =\; D\_\{\backslash bold\; u\}\; \{\backslash bold\; v\}\; +\; \backslash Gamma(\backslash phi)\backslash \{\{\backslash bold\; u\},\{\backslash bold\; v\}\backslash \}$

where Γ depends on the coordinate system φ and is 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 tensorial, 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 metric (as well as a certain symmetry condition). With these normalizations, the connection is uniquely defined.

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

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-CivitaLevi-Civita, T. and Ricci, G. "Méthodes de calcul différential absolu et leurs applications", Math. Ann. B, 54 (1900) 125-201. who observed in part that a connection in the infinitesimal sense of Christoffel also allowed for a notion of parallel transport.

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 equations. As the twentieth century progressed, Elie Cartan developed a new notion of connection. He sought to apply the techniques of Pfaffian systems 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, Cartan, E. "Sur les varietes a connexion projective", Bulletin de la Société Mathématique, 52 (1924) 205-241., Cartan, E., Geometry of Riemannian spaces, Math Sci Press, 1983.) Furthermore, using the dynamics of Darboux, Cartan was able to generalize 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.

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 gave an algebraic framework for regarding a connection as a differential operator by means of the Koszul connectionKoszul, J. L. "Homologie et cohomologie des algebres de Lie", Bulletin de la Société Mathématique 78 (1950) 65-127.. 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.

In that same year, Charles Ehresmann, a student of Cartan's, presented a variation on the connection as a differential form view in the context of principal bundles and, more generally, fibre bundlesEhresmann, C. "Les connexions infinitésimales dans un espace fibré différentiable", Colloque de Toplogie, Bruxelles (1950) 29-55.. Ehresmann connections 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 connections. 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.

Possible approaches

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• A rather direct module-style approach to covariant differentiation, stating the conditions allowing vector fields to act as differential operators on vector bundle sections. See covariant derivative, and Koszul connection for the most general approach along these lines.
• Traditional index notation specifies the connection by components; see Christoffel symbols. (Note: this has three indices, but is not a tensor).
• In pseudo-Riemannian and Riemannian geometry there is a way of deriving a connection from the metric tensor. See Levi-Civita connection.
• The connections referred to above are linear or affine connections. There is also a concept of projective connection, of which the Schwarzian derivative in complex analysis is an instance.
• Using principal bundles, a connection can be realized as a Lie algebra-valued differential forms. See connection form and Cartan connection.
• The most abstract approach may be that suggested by Alexander Grothendieck, where a connection is seen as descent data from infinitesimal neighbourhoods of the diagonal.Osserman, B., "Connections, curvature, and p-curvature", preprint.

References

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Differential geometry | Connection (mathematics)

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