Circle bundle

In mathematics, a circle bundle is a fiber bundle where the fiber is the circle ${\displaystyle \scriptstyle \mathbf {S} ^{1}}$.

Oriented circle bundles are also known as principal U(1)-bundles. In physics, circle bundles are the natural geometric setting for electromagnetism. A circle bundle is a special case of a sphere bundle.

Circle bundles over surfaces are an important example of 3-manifolds. A more general class of 3-manifolds is Seifert fiber spaces, which may be viewed as a kind of "singular" circle bundle, or as a circle bundle over a two-dimensional orbifold.

The Maxwell equations correspond to an electromagnetic field represented by a 2-form F, with ${\displaystyle \pi ^{\!*}F}$ being cohomologous to zero. In particular, there always exists a 1-form A such that

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