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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:

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.