Connection on a principal bundle
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The $\frak{g}$-valued 2-form $\Omega := d\omega + [\omega,\omega]$ is called ''curvature 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 theory page [[ | + | and measures the non-integrability of the distribution $\mathcal{H}$, see the theory page [[Connections]] for details. |
− | for details. | + | |
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]]) 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 | 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 [[ | + | on $E$ is closely related to a [[covariant derivative]] (see [[Connections]]). |
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Revision as of 13:07, 15 March 2013
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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).