Connection (mathematics)

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In differential geometry, a connection (also connexion) or covariant derivative is a way of specifying a derivative of a vector field along another vector field on a manifold. That is an application to tangent bundles; there are more general connections, used in differential geometry and other fields of mathematics to formulate intrinsic differential equations. Connection may refer to a connection on any vector bundle, or also a connection on a principal bundle.

Connections give rise to parallel transport along a curve on a manifold. A connection also leads to invariants of curvature (see also curvature tensor and curvature form), and the so-called torsion.

General concept

The general concept can be summarized as follows: given a fiber bundle

with E the total space and B the base space, the tangent space at any point of E has a canonical "vertical" subspace ( see vertical space), the subspace tangent to the fiber. The connection fixes a choice of "horizontal" subspace (see horizontal space) at each point of E so that the tangent space of E is a direct sum of vertical and horizontal subspaces. Usually more requirements are imposed on the choice of "horizontal" subspaces, but they depend on the type of the bundle.

Given a the induced bundle has an induced connection. If B' = I is a segment then the connection on B gives a trivialization on the induced bundle over I, i.e. a choice of smooth one-parameter family of isomorphisms between the fibers over I. This family is called parallel displacement along the curve and it gives an equivalent description of connection (which in case of Levi-Civita connection on a Riemannian manifold is called parallel transport).

There are many ways to describe a connection; in one particular approach, a connection can be locally described as a matrix of 1-forms on the base space which is the multiplant of the difference between the covariant derivative and the ordinary partial derivative in a coordinate chart. That is, partial derivatives are not an intrinsic notion on a manifold: a connection 'fixes up' the concept and permits discussion in geometric terms.

Possible approaches

There are quite a number of possible approaches to the connection concept. They include the following:

• A rather direct module-style approach to covariant differentiation, stating the conditions allowing vector fields to act as differential operators on vector bundle sections.
• Traditional index notation specifies the connection by components; see Covariant derivative (three indices, but this is not a tensor).
• In Riemannian geometry there is a way of deriving a connection from the metric tensor (Levi-Civita connection).
• Using principal bundles and 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.

The connections referred to above are linear or affine connections. There is also a concept of projective connection; the most commonly-met form of this is the Schwarzian derivative in complex analysis.

See also: Gauss-Manin connection