Connection (affine bundle)
Let <math>Y\to X,</math> be an affine bundle modelled over a vector bundle <math>\overline Y\to X </math>. A connection <math>\Gamma</math> on <math>Y\to X</math> is called the affine connection if it as a section <math>\Gamma:Y\to J^1Y</math> of the jet bundle <math>J^1Y\to Y</math> of <math>Y</math> is an affine bundle morphism over <math>X</math>. In particular, this is the case of an affine connection on the tangent bundle <math>TX</math> of a smooth manifold <math>X</math>.
With respect to affine bundle coordinates <math>(x^\lambda,y^i)</math> on <math>Y</math>, an affine connection <math>\Gamma</math> on <math>Y\to X</math> is given by the tangent-valued connection form
- <math>\Gamma =dx^\lambda\otimes (\partial_\lambda + \Gamma_\lambda^i\partial_i), \qquad
\Gamma_\lambda^i=\Gamma_\lambda{}^i{}_j(x^\nu) y^j + \sigma_\lambda^i(x^\nu). </math>
An affine bundle is a fiber bundle with a general affine structure group <math> GA(m,\mathbb R) </math> of affine transformations of its typical fiber <math>V</math> of dimension <math>m</math>. Therefore, an affine connection is associated to a principal connection. It always exists.
For any affine connection <math>\Gamma:Y\to J^1Y</math>, the corresponding linear derivative <math>\overline\Gamma:\overline Y\to J^1\overline Y</math> of an affine morphism <math>\Gamma</math> defines a unique linear connection on a vector bundle <math>\overline Y\to X</math>. With respect to linear bundle coordinates <math>(x^\lambda,\overline y^i)</math> on <math>\overline Y</math>, this connection reads
- <math> \overline \Gamma=dx^\lambda\otimes(\partial_\lambda +\Gamma_\lambda{}^i{}_j(x^\nu) \overline y^j\overline\partial_i).</math>
Since every vector bundle is an affine bundle, any linear connection on a vector bundle also is an affine connection.
If <math>Y\to X</math> is a vector bundle, both an affine connection <math>\Gamma</math> and an associated linear connection <math>\overline\Gamma</math> are connections on the same vector bundle <math>Y\to X</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 <math>Y\to X</math> is a sum of a linear connection and a basic soldering form on <math>Y\to X</math>.
It should be noted that, due to the canonical vertical splitting <math>VY=Y\times Y</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 <math>e_i</math> is a fiber basis for <math>Y</math>.
Given an affine connection <math>\Gamma</math> on a vector bundle <math>Y\to X</math>, let <math>R</math> and <math>\overline R</math> be the curvatures of a connection <math>\Gamma</math> and the associated linear connection <math>\overline \Gamma</math>, respectively. It is readily observed that <math>R = \overline R + T</math>, where
- <math>T =\frac12 T_{\lambda
\mu}^i dx^\lambda\wedge dx^\mu\otimes \partial_i, \qquad 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, </math>
is the torsion of <math>\Gamma</math> with respect to the basic soldering form <math>\sigma</math>.
In particular, let us consider the tangent bundle <math>TX</math> of a manifold <math>X</math> coordinated by <math>(x^\mu,\dot x^\mu)</math>. There is the canonical soldering form <math>\theta=dx^\mu\otimes \dot\partial_\mu </math> on <math>TX</math> which coincides with the tautological one-form <math>\theta_X=dx^\mu\otimes \partial_\mu</math> on <math>X</math> due to the canonical vertical splitting <math>VTX=TX\times TX</math>. Given an arbitrary linear connection <math>\Gamma</math> on <math>TX</math>, the corresponding affine connection
- <math>A=\Gamma +\theta, \qquad
A_\lambda^\mu=\Gamma_\lambda{}^\mu{}_\nu \dot x^\nu +\delta^\mu_\lambda, </math>
on <math>TX</math> is the Cartan connection. The torsion of the Cartan connection <math>A</math> with respect to the soldering form <math>\theta</math> coincides with the torsion of a linear connection <math>\Gamma</math>, and its curvature is a sum <math>R+T</math> of the curvature and the torsion of <math>\Gamma</math>.
References
- S. Kobayashi, K. Nomizu, Foundations of Differential Geometry, Vols. 1 & 2, Wiley-Interscience, 1996, ISBN 0-471-15733-3.
- Sardanashvily, G., Advanced Differential Geometry for Theoreticians. Fiber bundles, jet manifolds and Lagrangian theor, Lambert Academic Publishing, 2013, ISBN 978-3-659-37815-7; arXiv: 0908.1886.