Connection on a principal bundle
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Let $G$ be a Lie group with Lie algebra $\frak{g}$ and | Let $G$ be a Lie group with Lie algebra $\frak{g}$ and | ||
− | $F \to M$ a [[Principal bundle|principal bundle]] for $G$ over a | + | $F \to M$ a [[Principal bundle of smooth manifolds|principal bundle]] for $G$ over a |
smooth manifold $M$. A ''connection'' on $F$ is a ''distribution'' | smooth manifold $M$. A ''connection'' on $F$ is a ''distribution'' | ||
(a subbundle of the tangent bundle) $\mathcal{H} \subset TF$ | (a subbundle of the tangent bundle) $\mathcal{H} \subset TF$ | ||
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The decomposition $TF = \mathcal{V} \oplus \mathcal{H}$ can be given by the projection | 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 | $\pi_\mathcal{V} : TF \to \mathcal{V}$ onto the vertical distribution. Since each vertical | ||
− | space $\mathcal{V}_f$ can be identified with $\frak{g}$ (see [[Principal bundle]]), | + | space $\mathcal{V}_f$ can be identified with $\frak{g}$ (see [[Principal bundle of smooth manifolds|Principal bundle]]), this map $\pi_V$ can be viewed as a $\frak{g}$-valued 1-form on $F$, |
− | 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''. | a linear map $\omega : TF \to \frak{g}$; this is called the ''connection form''. | ||
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A connection $\mathcal{H}$ on a $G$-principal bundles $F$ induces a distribution on any associated | 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 | + | bundle $E = (F \times E_o)/G$ (see [[Principal bundle of smooth manifolds|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 [[Connections]]). |
− | 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 [[Connections]]). | + | |
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== References == | == References == | ||
{{#RefList:}} | {{#RefList:}} | ||
[[Category:Definitions]] | [[Category:Definitions]] | ||
+ | [[Category:Connections and curvature]] |
Revision as of 10:21, 15 May 2013
The user responsible for this page is Jost Eschenburg. No other user may edit this page at present. |
This page has not been refereed. The information given here might be incomplete or provisional. |
1 Definition
Let be a Lie group with Lie algebra and a principal bundle for over a smooth manifold . A connection on is a distribution (a subbundle of the tangent bundle) on , called the "horizontal distribution", which is -invariant and complementary to the vertical distribution on .
The decomposition can be given by the projection onto the vertical distribution. Since each vertical space can be identified with (see Principal bundle), this map can be viewed as a -valued 1-form on , a linear map ; this is called the connection form.
The -valued 2-form is called curvature form and measures the non-integrability of the distribution , see the theory page Connections for details.
A connection on a -principal bundles induces a distribution on any associated bundle (see Principal bundle) since passes trivially to and by -invariance to . The induced distribution is called a connection on . If is a vector bundle (the action of on is linear), the connection on is closely related to a covariant derivative (see Connections).