Connection (mathematics)

In differential geometry, connection (spelt as connexion by the British) is a way of specifying covariant differentiation on a manifold. The theory of connections leads to invariants of curvature, 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 (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 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.