Connection (affine bundle)

Let $Y\to X,$ be an affine bundle modelled over a vector bundle $\overline Y\to X$. A connection $\Gamma$ on $Y\to X$ is called the affine connection if it as a section $\Gamma:Y\to J^1Y$ of the jet bundle $J^1Y\to Y$ of $Y$ is an affine bundle morphism over $X$. In particular, this is the case of an affine connection on the tangent bundle $TX$ of a smooth manifold $X$.

With respect to affine bundle coordinates $(x^\lambda,y^i)$ on $Y$, an affine connection $\Gamma$ on $Y\to X$ is given by the tangent-valued connection form

$\Gamma =dx^\lambda\otimes (\partial_\lambda + \Gamma_\lambda^i\partial_i), \qquad \Gamma_\lambda^i=\Gamma_\lambda{}^i{}_j(x^\nu) y^j + \sigma_\lambda^i(x^\nu).$

An affine bundle is a fiber bundle with a general affine structure group $GA(m,\mathbb R)$ 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 $\Gamma:Y\to J^1Y$, the corresponding linear derivative $\overline\Gamma:\overline Y\to J^1\overline Y$ of an affine morphism $\Gamma$ defines a unique linear connection on a vector bundle $\overline Y\to X$. With respect to linear bundle coordinates $(x^\lambda,\overline y^i)$ on $\overline Y$, this connection reads

$\overline \Gamma=dx^\lambda\otimes(\partial_\lambda +\Gamma_\lambda{}^i{}_j(x^\nu) \overline y^j\overline\partial_i).$

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

If $Y\to X$ is a vector bundle, both an affine connection $\Gamma$ and an associated linear connection $\overline\Gamma$ are connections on the same vector bundle $Y\to X$, 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 $Y\to X$ is a sum of a linear connection and a basic soldering form on $Y\to X$.

It should be noted that, due to the canonical vertical splitting $VY=Y\times Y$, this soldering form is brought into a vector-valued form $\sigma= \sigma_\lambda^i(x^\nu) dx^\lambda\otimes e_i$ where $e_i$ is a fiber basis for $Y$.

Given an affine connection $\Gamma$ on a vector bundle $Y\to X$, let $R$ and $\overline R$ be the curvatures of a connection $\Gamma$ and the associated linear connection $\overline \Gamma$, respectively. It is readily observed that $R = \overline R + T$, where

$T =\frac12 T_{\lambda \mu}^i dx^\lambda\wedge dx^\mu\otimes \partial_i, \qquad T_{\lambda \mu}^i = \partial_\lambda\sigma_\mu^i - \partial_\mu\sigma_\lambda^i + \sigma_\lambda^h \Gamma_\mu{}^i{}_h - \sigma_\mu^h \Gamma_\lambda{}^i{}_h,$

is the torsion of $\Gamma$ with respect to the basic soldering form $\sigma$.

In particular, let us consider the tangent bundle $TX$ of a manifold $X$ coordinated by $(x^\mu,\dot x^\mu)$. There is the canonical soldering form $\theta=dx^\mu\otimes \dot\partial_\mu$ on $TX$ which coincides with the tautological one-form $\theta_X=dx^\mu\otimes \partial_\mu$ on $X$ due to the canonical vertical splitting $VTX=TX\times TX$. Given an arbitrary linear connection $\Gamma$ on $TX$, the corresponding affine connection

$A=\Gamma +\theta, \qquad A_\lambda^\mu=\Gamma_\lambda{}^\mu{}_\nu \dot x^\nu +\delta^\mu_\lambda,$

on $TX$ is the Cartan connection. The torsion of the Cartan connection $A$ with respect to the soldering form $\theta$ coincides with the torsion of a linear connection $\Gamma$, and its curvature is a sum $R+T$ of the curvature and the torsion of $\Gamma$.

See also

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

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.