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Let $H\subset G$ be a subgroup and $\pi:P\to B$ be a principal $H$-bundle.

$G$ has a left $H$ action and one can define a principal $G$-bundle $\pi':P\times_H G\to B$ where $P\times_H G$ is quotiening out the diagonal $H$-action of $P\times G$

This latter bundle $\pi'$ is called the extension of the structure group from $H$ to $G$ of the bundle $\pi$, I guess.

There also exists another term reduction of the structure group. Is this just the ''dual''? i.e. is $\pi$ the reduction of the structure group from $G$ to $H$ of the bundle $\pi'$?

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You can define the diagonal H -action of P×G? – Alcides de carvalho jr Sep 21 '15 at 22:56
up vote 1 down vote accepted

Yes, see .

For example, when $G=GL(k)$ (so you have a vector bundle), the possibility of reducing the structure group to $GL^+(k)$ means that the bundle is orientable. Reducing the structure group to $O(k)$ means introducing a metric.

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