Connection (mathematics)

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In differential geometry, connection (spelt also as connexion) is a way of specifying covariant differentiation on a manifold. The theory of connections leads to invariants of curvature (see also curvature tensor), and the so-called torsion. That is an application to tangent bundles; there are more general connections, in differential geometry. A connection may refer to a connection on any vector bundle or a connection on a principal bundle.

In one particular approach, a connection is a Lie algebra valued 1-form which is the multiplant of the difference between the covariant derivative and the ordinary partial derivative. That is, partial derivatives are not an intrinsic notion on a manifold: a connection 'fixes up' the concept and permits discussion in geometric terms. Connections give rise to parallel transport.

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 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 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.