UDP-2,3-diacylglucosamine diphosphatase

Let $Y\to X,$ be an affine bundle modelled over a vector bundle $\bar{Y}\to X$. A connection $\mathrm{\Gamma }$ on $Y\to X$ is called the affine connection if it as a section $\mathrm{\Gamma }:Y\to J^{1}Y$ 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 $\mathrm{\Gamma }$ on $Y\to X$ is given by the tangent-valued connection form

$\mathrm{\Gamma }=d{x}^{\lambda }\otimes ({\partial }_{\lambda }+{\mathrm{\Gamma }}_{\lambda }^i{\partial }_i),{\mathrm{\Gamma }}_{\lambda }^i={\mathrm{\Gamma }}_{\lambda }{}^i{}_j(x^{\nu })y^j+{\sigma }_{\lambda }^i(x^{\nu }).$

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 $\mathrm{\Gamma }:Y\to J^{1}Y$, the corresponding linear derivative $\bar{\mathrm{\Gamma }}:\bar{Y}\to J^1\bar{Y}$ of an affine morphism $\mathrm{\Gamma }$ defines a unique linear connection on a vector bundle $\bar{Y}\to X$. With respect to linear bundle coordinates $(x^{\lambda },\stackrel{i}{\stackrel{‾}{y}})$ on $\bar{Y}$, this connection reads

$\bar{\mathrm{\Gamma }}=d{x}^{\lambda }\otimes ({\partial }_{\lambda }+{\mathrm{\Gamma }}_{\lambda }{}^i{}_j(x^{\nu })\stackrel{j}{\stackrel{‾}{y}}{\bar{\partial }}_i).$

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 $\mathrm{\Gamma }$ and an associated linear connection $\bar{\mathrm{\Gamma }}$ are connections on the same vector bundle $Y\to X$, and their difference is a basic soldering form on $\sigma ={\sigma }_{\lambda }^i(x^{\nu })d{x}^{\lambda }\otimes {\partial }_i$. 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 $\sigma ={\sigma }_{\lambda }^i(x^{\nu })d{x}^{\lambda }\otimes e_i$ where $e_i$ is a fiber basis for $Y$.

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

$T=\frac{1}{2}{T}_{\lambda \mu }^{i}d{x}^{\lambda }\wedge d{x}^{\mu }\otimes {\partial }_i,T_{\lambda \mu }^i={\partial }_{\lambda }{\sigma }_{\mu }^i-{\partial }_{\mu }{\sigma }_{\lambda }^i+{\sigma }_{\lambda }^h{\mathrm{\Gamma }}_{\mu }{}^i{}_h-{\sigma }_{\mu }^h{\mathrm{\Gamma }}_{\lambda }{}^i{}_h,$

is the torsion of $\mathrm{\Gamma }$ 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 $\theta =d{x}^{\mu }\otimes {\dot{\partial }}_{\mu }$ on $TX$ which coincides with the tautological one-form ${\theta }_X=d{x}^{\mu }\otimes {\partial }_{\mu }$ on $X$ due to the canonical vertical splitting $VTX=TX\times TX$. Given an arbitrary linear connection $\mathrm{\Gamma }$ on $TX$, the corresponding affine connection

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

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 $\mathrm{\Gamma }$, and its curvature is a sum $R+T$ of the curvature and the torsion of $\mathrm{\Gamma }$.

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.

See also

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