Connection (affine bundle)

In differential geometry, the classical notion of a connection (mathematics), most commonly encountered in the context of vector bundles, can be generalised to affine bundles. Let Template:Math be an affine bundle modelled over a vector bundle Template:Math. A connection Template:Math on Template:Math is called an affine connection if, as a section Template:Math of the jet bundle Template:Math of Template:Math, it is an affine bundle morphism over Template:Math.

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 Template:Math of a smooth manifold Template:Math, though as discussed below, the latter can be considered as a special example of the former.

With respect to affine bundle coordinates Template:Math on Template:Math, an affine connection Template:Math on Template:Math is given by the tangent-valued connection form

<math>\begin{align}\Gamma &=dx^\lambda\otimes \left(\partial_\lambda + \Gamma_\lambda^i\partial_i\right)\,, \\ \Gamma_\lambda^i&={{\Gamma_\lambda}^i}_j\left(x^\nu\right) y^j + \sigma_\lambda^i\left(x^\nu\right)\,. \end{align}</math>

An affine bundle is a fiber bundle with a general affine structure group Template:Math of affine transformations of its typical fiber Template:Math of dimension Template:Math. Therefore, an affine connection is associated to a principal connection. It always exists.

For any affine connection Template:Math, the corresponding linear derivative Template:Math of an affine morphism Template:Math defines a unique linear connection on a vector bundle Template:Math. With respect to linear bundle coordinates Template:Math on Template:Math, this connection reads

<math> \overline \Gamma=dx^\lambda\otimes\left(\partial_\lambda +{{\Gamma_\lambda}^i}_j\left(x^\nu\right) \overline y^j\overline\partial_i\right)\,.</math>

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

If Template:Math is a vector bundle, both an affine connection Template:Math and an associated linear connection Template:Math are connections on the same vector bundle Template:Math, and their difference is a basic soldering form on

<math>\sigma= \sigma_\lambda^i(x^\nu) dx^\lambda\otimes\partial_i \,.</math>

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

Due to the canonical vertical splitting Template:Math, this soldering form is brought into a vector-valued form

<math>\sigma= \sigma_\lambda^i(x^\nu) dx^\lambda\otimes e_i </math>

where Template:Math is a fiber basis for Template:Math.

Given an affine connection Template:Math on a vector bundle Template:Math, let Template:Math and Template:Math be the curvatures of a connection Template:Math and the associated linear connection Template:Math, respectively. It is readily observed that Template:Math, where

<math>\begin{align}

T &=\tfrac12 T_{\lambda\mu}^i dx^\lambda\wedge dx^\mu\otimes \partial_i\,, \\ T_{\lambda \mu}^i &= \partial_\lambda\sigma_\mu^i - \partial_\mu\sigma_\lambda^i + \sigma_\lambda^h {{\Gamma_\mu}^i}_h - \sigma_\mu^h {{\Gamma_\lambda}^i}_h\,, \end{align}</math>

is the torsion of Template:Math with respect to the basic soldering form Template:Math.

In particular, consider the tangent bundle Template:Math of a manifold Template:Math coordinated by Template:Math. There is the canonical soldering form

<math>\theta=dx^\mu\otimes \dot\partial_\mu </math>

on Template:Math which coincides with the tautological one-form

<math>\theta_X=dx^\mu\otimes \partial_\mu</math>

on Template:Math due to the canonical vertical splitting Template:Math. Given an arbitrary linear connection Template:Math on Template:Math, the corresponding affine connection

<math>\begin{align}

A&=\Gamma +\theta\,, \\ A_\lambda^\mu&={{\Gamma_\lambda}^\mu}_\nu \dot x^\nu +\delta^\mu_\lambda\,, \end{align}</math>

on Template:Math is the Cartan connection. The torsion of the Cartan connection Template:Math with respect to the soldering form Template:Math coincides with the torsion of a linear connection Template:Math, and its curvature is a sum Template:Math of the curvature and the torsion of Template:Math.

See also

• Connection (fibred manifold)
• Affine connection
• Connection (vector bundle)
• Connection (mathematics)
• Affine gauge theory

References

• Sardanashvily, G. (2013). Advanced Differential Geometry for Theoreticians. Fiber bundles, jet manifolds and Lagrangian theory. Lambert Academic Publishing. ISBN 978-3-659-37815-7. Bibcode: 2009arXiv0908.1886S.