Connection (affine bundle)

Let be an affine bundle modelled over a vector bundle . A connection on is called the affine connection if it as a section of the jet bundle of is an affine bundle morphism over . In particular, this is an affine connection on the tangent bundle of a smooth manifold . (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 on , an affine connection on is given by the tangent-valued connection form

: $\begin\Gamma &=dx^\lambda\otimes \left(\partial_\lambda + \Gamma_\lambda^i\partial_i\right)\,, \\ \Gamma_\lambda^i&=_j\left(x^\nu\right) y^j + \sigma_\lambda^i\left(x^\nu\right)\,. \end$

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

: $\overline \Gamma=dx^\lambda\otimes\left(\partial_\lambda +_j\left(x^\nu\right) \overline y^j\overline\partial_i\right)\,.$

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
: $\sigma= \sigma_\lambda^i(x^\nu) dx^\lambda\otimes\partial_i \,.$
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
: $\sigma= \sigma_\lambda^i(x^\nu) dx^\lambda\otimes e_i$
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

: $\begin T &=\tfrac12 T_^i dx^\lambda\wedge dx^\mu\otimes \partial_i\,, \\ T_^i &= \partial_\lambda\sigma_\mu^i - \partial_\mu\sigma_\lambda^i + \sigma_\lambda^h _h - \sigma_\mu^h _h\,, \end$

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
:$\theta=dx^\mu\otimes \dot\partial_\mu$
on which coincides with the tautological one-form
:$\theta_X=dx^\mu\otimes \partial_\mu$
on due to the canonical vertical splitting . Given an arbitrary linear connection on , the corresponding affine connection

: $\begin A&=\Gamma +\theta\,, \\ A_\lambda^\mu&=_\nu \dot x^\nu +\delta^\mu_\lambda\,, \end$

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

References

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Differential geometry
Connection (mathematics)

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