Connection (affine bundle)

Let $Y → X$ be an affine bundle modelled over a vector bundle $\overline{Y} → X$. A connection $Γ$ on $Y → X$ is called the affine connection if it as a section $Γ : Y → J^{1}Y$ of the jet bundle $J^{1}Y → Y$ of $Y$ is an affine bundle morphism over $X$. In particular, this is an affine connection on the tangent bundle $TX$ of a smooth manifold $X$. (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 $(x^{λ}, y^{i})$ on $Y$, an affine connection $Γ$ on $Y → X$ is given by the tangent-valued connection form


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

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


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

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

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

Due to the canonical vertical splitting $VY = Y × Y$, 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 $Γ$ on a vector bundle $Y → X$, let $R$ and $\overline{R}$ be the curvatures of a connection $Γ$ and the associated linear connection $\overline{Γ}$, respectively. It is readily observed that $R = \overline{R} + T$, where


 * $$\begin{align}

T &=\tfrac12 T_{\lambda\mu}^i dx^\lambda\wedge dx^\mu\otimes \partial_i\,, \\ 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\,, \end{align}$$

is the torsion of $Γ$ with respect to the basic soldering form $σ$.

In particular, consider the tangent bundle $TX$ of a manifold $X$ coordinated by $(x^{μ}, ẋ^{μ})$. 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 $VTX = TX × TX$. Given an arbitrary linear connection $Γ$ on $TX$, the corresponding affine connection


 * $$\begin{align}

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

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