Connection (mathematics)

In geometry, the notion of a connection makes precise the idea of transporting local geometric objects, such as tangent vectors or tensors in the tangent space, along a curve or family of curves in a parallel and consistent manner.

There are various kinds of connections in modern geometry, depending on what sort of data one wants to transport.

An affine connection is typically given in the form of a covariant derivative, which gives a means for taking directional derivatives of vector fields, measuring the deviation of a vector field from being parallel in a given direction.

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.

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.

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.

A simple mechanical device that can be used to visualize parallel transport is the south-pointing chariot.

), and v0 = v0(x, y) is a vector field on R2 uniquely determined by v (since the pushforward of a local diffeomorphism at any point is invertible).

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: To relate the components v0 and v1, apply the chain rule to the identity φ1 = φ0 o φ01: Applying both sides of this matrix equation to the component vector v1(φ1−1(P)) and invoking (1) and (2) yields We come now to the main question of defining how to transport a vector field parallelly along a curve.

The first approach is to examine what is required for a generalization of the directional derivative to "behave well" under coordinate transitions.

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).

The second approach is to use Lie groups to attempt to capture some vestige of symmetry on the space.

Historically, connections were studied from an infinitesimal perspective in Riemannian geometry.

The infinitesimal study of connections began to some extent with Elwin Christoffel.

As the twentieth century progressed, Élie 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.

Furthermore, using the dynamics of Gaston Darboux, Cartan was able to generalize the notion of parallel transport for his class of infinitesimal connections.

The attendant parallel displacement operations also had natural algebraic interpretations in terms of the connection.

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 of principal bundles and, more generally, fibre bundles.