# Principal *S*^{1}-bundles

Chapter

- 368 Downloads

Let

*P*and*M*be*C*^{∞}manifolds,*π*:*P*→*M*a*C*^{∞}map of*P*onto*M*and*G*a Lie group acting on*P*to the right. Then (*P, G, M*) is called a*principal G-bundle*if- 1.
*G*acts freely on*P*, - 2.
π(

*p*_{1}) = π(*p*_{2}) if and only if there exists*g*∈*G*such that*p*_{1}*g*=*p*_{2}, - 3.
*P*is locally trivial over*M*, i.e., for every*m*∈*M*there exists a neighborhood*U*of*m*and a map*F*_{ u }: π_{-1}(*U*) →*G*such that*F*_{ u }(*pg*) = (*F*_{ u }(*p*))*g*and such that the map Ψ : π_{-1}(*U*) →*U*×*G*taking*p*to (π(*p*),*F*_{ u }(*p*)) is a diffeomorphism.

## Keywords

Short Exact Sequence Principal Bundle Connection Form Horizontal Lift Bundle Isomorphism
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Preview

Unable to display preview. Download preview PDF.

## Copyright information

© Springer Science+Business Media New York 2002