It proves that such jets form a fibre bundle, analogous to the tangent bundle, which is an associated bundle of a jet group.
12.
In a paper, written together with Ehresmann, he introduced the notion of an associated bundle and proved results known today as the exact homotopy sequence of a fibration.
13.
If, in addition, a right action is given on the fibre of the principal bundle, we describe how to construct any associated bundle by means of a fibre product construction.
14.
For this reason, in the category of fibre bundles with a structure group " G ", the principal connection contains all relevant information for " G "-connections on the associated bundles.
15.
The above notion can be extended to the associated bundle P \ times _ G V where " V " is a vector space transforming covariantly under some representation of " G ".
16.
Hence, unless there is an overriding reason to consider connections on associated bundles ( as there is, for instance, in the case of Cartan connections ) one usually works directly with the principal connection.
17.
The reason for the name is that a solder form solders ( or attaches ) the abstract principal bundle to the manifold " M " by identifying an associated bundle with the tangent bundle.
18.
One important application to note is " change of fiber " : if the " f " " ij " are all you need to make a bundle, then there are many ways to make an associated bundle.
19.
By applying the preceding ideas to the local 1-parameter group of diffeomorphisms defined by a vector field on " M ", and differentiating with respect to the parameter, a notion of Lie derivative on any associated bundle is obtained.
20.
In this way, a principal " G "-bundle equipped with a right action is often thought of as part of the data specifying a fibre bundle with structure group " G ", since to a fibre bundle one may construct the principal bundle via the associated bundle construction.
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