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Connection (affine bundle)

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Let YX be an affine bundle modelled over a vector bundle YX. A connection Γがんま on YX is called the affine connection if it as a section Γがんま : Y → J1Y of the jet bundle J1YY of Y is an affine bundle morphism over X. In particular, this is an affine connection on the tangent bundle TX of a smooth manifold X. (That is, the connection on an affine bundle is an example of an affine connection; it is not, however, a general definition of an affine connection. These are related but distinct concepts both unfortunately making use of the adjective "affine".)

With respect to affine bundle coordinates (xλらむだ, yi) on Y, an affine connection Γがんま on YX is given by the tangent-valued connection form

An affine bundle is a fiber bundle with a general affine structure group GA(m, ℝ) of affine transformations of its typical fiber V of dimension m. Therefore, an affine connection is associated to a principal connection. It always exists.

For any affine connection Γがんま : Y → J1Y, the corresponding linear derivative Γがんま : Y → J1Y of an affine morphism Γがんま defines a unique linear connection on a vector bundle YX. With respect to linear bundle coordinates (xλらむだ, yi) on Y, this connection reads

Since every vector bundle is an affine bundle, any linear connection on a vector bundle also is an affine connection.

If YX is a vector bundle, both an affine connection Γがんま and an associated linear connection Γがんま are connections on the same vector bundle YX, and their difference is a basic soldering form on

Thus, every affine connection on a vector bundle YX is a sum of a linear connection and a basic soldering form on YX.

Due to the canonical vertical splitting VY = Y × Y, this soldering form is brought into a vector-valued form

where ei is a fiber basis for Y.

Given an affine connection Γがんま on a vector bundle YX, let R and R be the curvatures of a connection Γがんま and the associated linear connection Γがんま, respectively. It is readily observed that R = R + T, where

is the torsion of Γがんま with respect to the basic soldering form σしぐま.

In particular, consider the tangent bundle TX of a manifold X coordinated by (xμみゅー, μみゅー). There is the canonical soldering form

on TX which coincides with the tautological one-form

on X due to the canonical vertical splitting VTX = TX × TX. Given an arbitrary linear connection Γがんま on TX, the corresponding affine connection

on TX is the Cartan connection. The torsion of the Cartan connection A with respect to the soldering form θしーた coincides with the torsion of a linear connection Γがんま, and its curvature is a sum R + T of the curvature and the torsion of Γがんま.

See also

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References

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  • Kobayashi, S.; Nomizu, K. (1996). Foundations of Differential Geometry. Vol. 1–2. Wiley-Interscience. ISBN 0-471-15733-3.
  • Sardanashvily, G. (2013). Advanced Differential Geometry for Theoreticians. Fiber bundles, jet manifolds and Lagrangian theory. Lambert Academic Publishing. arXiv:0908.1886. Bibcode:2009arXiv0908.1886S. ISBN 978-3-659-37815-7.