Principal bundle of smooth manifolds
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1 Definition
An -principal bundle for a Lie group is a smooth bundle where acts on from the right, the action is free (that is if for some then ), and the -orbits are precisely the fibres, for every with . The group is called the structure group of the principal bundle .
The mapping , is a diffeomorphism which is equivariant where acts on itself by right translations. The tangent space of the fibre, is called vertical space; by means of the differential it can be identified with the Lie algebra . The vertical spaces together form an integrable distribution on , called 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 .
For further information, see [Kobayashi&Nomizu1963].
2 Examples
A main example is the frame bundle of a manifold whose fibre over is the set of all bases of the tangent space . The group acts on as follows: is sending a basis onto the basis . Moreover, if is equipped with a Riemannian metric, there is the orthogonal frame bundle where consists of the set of orthonormal bases on ; this is acted on by the orthogonal group in a similar way. If is a Kähler manifold (a Riemannian manifold with a parallel and orthogonal almost complex structure on its tangent bundle), we have the principal bundle of unitary frames (orthonormal frames of type ) with structure group .
A different type of examples comes from homogeneous spaces. If is a manifold and a Lie group acting transitively on by diffeomorphisms, is a principal bundle over in various ways: Fixing we have the bundle , . Its fibre over is the isotropy group , while the fibre over is . Thus is an -principal bundle where acts on by right multiplication. When we identify with the coset space using the -equivariant map , the principal bundle is just the canonical projection .
We may embed the principal bundle into the frame bundle as follows. Fixing any basis of for some , we map (viewed as a diffeomorphism on ) onto the basis of . Thus the structure group of becomes a subgroup of , the structure group of .
3 References
- [Kobayashi&Nomizu1963] S. Kobayashi and K. Nomizu, Foundations of differential geometry. Vol I, Interscience Publishers, a division of John Wiley & Sons, New York-London, 1963. MR1393940 (97c:53001a) Zbl 0508.53002