Principal bundle of smooth manifolds
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1 Definition
A -principal bundle for a Lie group is a smooth bundle where acts on from the right, the action is free (that is [ for some ]), and the -orbits are the fibres, for every . Then the mapping , is a diffeomorphism which is equivariant with respect to the action of on itself by right translations. The tangent space of the fibre, is called the vertical space; by means of the differential it can be identified with the Lie algebra . The vertical spaces together form an integrable distribution on , the vertical distribution.
If is any -space, i.e. a manifold on which acts from the left, we have a free -action on given by . Then the orbit space is a bundle over with fibres diffeomorphic to ; it is called an associated bundle to the principal bundle . In particular, if is a vector space and the -action on is linear (a representation of ), then is a vector bundle over , associated to the principal bundle .