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 $\Gamma :Y\to J^{1}Y$ of the jet bundle $J^{1}Y\to Y$ 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 $\Gamma :Y\to J^{1}Y$, 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

$T={\frac 12}T_{{\lambda \mu }}^{i}dx^{\lambda }\wedge dx^{\mu }\otimes \partial _{i},\qquad T_{{\lambda \mu }}^{i}=\partial _{\lambda }\sigma _{\mu }^{i}-\partial _{\mu }\sigma _{\lambda }^{i}+\sigma _{\lambda }^{h}\Gamma _{\mu }{}^{i}{}_{h}-\sigma _{\mu }^{h}\Gamma _{\lambda }{}^{i}{}_{h},$

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

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

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.