In differential geometry, a connection on an affine bundle is a specialisation to affine bundles of the more general notion of a connection on a principal bundle. Let be an affine bundle modelled over a vector bundle . A connection on is called an affine connection if, as a section of the jet bundle of , it is an affine bundle morphism over .
The term "affine connection" as used in this article should not be confused with its more common usage, namely a connection on the tangent bundle of a smooth manifold , though as discussed below, the latter can be considered as a special example of the former.
With respect to affine bundle coordinates on , an affine connection on is given by the tangent-valued connection form
An affine bundle is a fiber bundle with a general affine structure group of affine transformations of its typical fiber of dimension . Therefore, an affine connection is associated to a principal connection. It always exists.
For any affine connection , the corresponding linear derivative of an affine morphism defines a unique linear connection on a vector bundle . With respect to linear bundle coordinates on , this connection reads
Since every vector bundle is an affine bundle, any linear connection on a vector bundle also is an affine connection.
If is a vector bundle, both an affine connection and an associated linear connection are connections on the same vector bundle , and their difference is a basic soldering form on
Thus, every affine connection on a vector bundle is a sum of a linear connection and a basic soldering form on .
Due to the canonical vertical splitting , this soldering form is brought into a vector-valued form
where is a fiber basis for .
Given an affine connection on a vector bundle , let and be the curvatures of a connection and the associated linear connection , respectively. It is readily observed that , where
is the torsion of with respect to the basic soldering form .
In particular, consider the tangent bundle of a manifold coordinated by . There is the canonical soldering form
on which coincides with the tautological one-form
on due to the canonical vertical splitting . Given an arbitrary linear connection on , the corresponding affine connection
on is the Cartan connection. The torsion of the Cartan connection with respect to the soldering form coincides with the torsion of a linear connection , and its curvature is a sum of the curvature and the torsion of .
See also Connection (fibred manifold) Affine connection Connection (vector bundle) Connection (mathematics) Affine gauge theory
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