Connection on a principal bundle

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1 Introduction

Let $G$${{Stub}} == Introduction == <wikitex include>; Let $G$ be a Lie group with Lie algebra $\frak{g}$ and $F \to M$ a [[Principal bundle|principal bundle]] for $G$ over a smooth manifold $M$. A ''connection'' on $F$ is a ''distribution'' (a subbundle of the tangent bundle) $\mathcal{H} \subset TF$ on $F$, called the "horizontal distribution", which is $G$-invariant and complementary to the vertical distribution $\mathcal{V}$ on $F$. The decomposition $TF = \mathcal{V} \oplus \mathcal{H}$ can be given by the projection $\pi_\mathcal{V} : TF \to \mathcal{V}$ onto the vertical distribution. Since each vertical space $\mathcal{V}_f$ can be identified with $\frak{g}$ (see [[Principal bundle]]), this map $\pi_V$ can be viewed as a $\frak{g}$-valued 1-form on $F$, a linear map $\omega : TF \to \frak{g}$; this is called the ''connection form''. The $\frak{g}$-valued 2-form $\Omega := d\omega + [\omega,\omega]$ is called ''curvature form'' and measures the non-integrability of the distribution $\mathcal{H}$, see the page [[Connection]] for details. A connection $\mathcal{H}$ on a $G$-principal bundles $F$ induces a distribution on any associated bundle $E = (F \times E_o)/G$ (see [[Principal bundle]]) since $\mathcal{H}$ passes trivially to $F \times E_o$ and by $G$-invariance to $E$. The induced distribution is called a ''connection on $E$''. If $E_o$ is a vector bundle (the action of $G$ on $E_o$ is linear), the connection on $E$ is closely related to a [[covariant derivative]] (see [[Connection]]). </wikitex> == References == {{#RefList:}} [[Category:Definitions]]G$ be a Lie group with Lie algebra $\frak{g}$$\frak{g}$ and $F \to M$$F \to M$ a principal bundle for $G$$G$ over a smooth manifold $M$$M$. A connection on $F$$F$ is a distribution (a subbundle of the tangent bundle) $\mathcal{H} \subset TF$$\mathcal{H} \subset TF$ on $F$$F$, called the "horizontal distribution", which is $G$$G$-invariant and complementary to the vertical distribution $\mathcal{V}$$\mathcal{V}$ on $F$$F$.

The decomposition $TF = \mathcal{V} \oplus \mathcal{H}$$TF = \mathcal{V} \oplus \mathcal{H}$ can be given by the projection $\pi_\mathcal{V} : TF \to \mathcal{V}$$\pi_\mathcal{V} : TF \to \mathcal{V}$ onto the vertical distribution. Since each vertical space $\mathcal{V}_f$$\mathcal{V}_f$ can be identified with $\frak{g}$$\frak{g}$ (see Principal bundle), this map $\pi_V$$\pi_V$ can be viewed as a $\frak{g}$$\frak{g}$-valued 1-form on $F$$F$, a linear map $\omega : TF \to \frak{g}$$\omega : TF \to \frak{g}$; this is called the connection form.

The $\frak{g}$$\frak{g}$-valued 2-form $\Omega := d\omega + [\omega,\omega]$$\Omega := d\omega + [\omega,\omega]$ is called curvature form and measures the non-integrability of the distribution $\mathcal{H}$$\mathcal{H}$, see the page Connection for details.

A connection $\mathcal{H}$$\mathcal{H}$ on a $G$$G$-principal bundles $F$$F$ induces a distribution on any associated bundle $E = (F \times E_o)/G$$E = (F \times E_o)/G$ (see Principal bundle) since $\mathcal{H}$$\mathcal{H}$ passes trivially to $F \times E_o$$F \times E_o$ and by $G$$G$-invariance to $E$$E$. The induced distribution is called a connection on $E$$E$. If $E_o$$E_o$ is a vector bundle (the action of $G$$G$ on $E_o$$E_o$ is linear), the connection on $E$$E$ is closely related to a covariant derivative (see Connection).