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.
Relationship to electrodynamics
Given a circle bundle P over M and its projection
one has the homomorphism
- The Hopf fibration is an example of a non-trivial circle bundle.
- The unit normal bundle of a surface is another example of a circle bundle.
- The unit normal bundle of a non-orientable surface is a circle bundle that is not a principal bundle. Orientable surfaces have principal unit tangent bundles.
- Another method for constructing circle bundles is using a complex line bundle and taking the associated sphere (circle in this case) bundle. Since this bundle has an orientation induced from we have that it is a principal -bundle. Moreover, the characteristic classes from Chern-Weil theory of the -bundle agree with the characteristic classes of .
- For example consider the analytification a complex plane curve
Since and the characteristic classes pull back non-trivially, we have that the line bundle associated to the sheaf has chern class .
The isomorphism classes of circle bundles over a manifold M are in one-to-one correspondence with the homotopy classes of maps . There is an extension of groups, , where . Circle bundles classified by maps into are known as principal -bundles, and are classified by an element of the second integral cohomology group of M, since . This isomorphism is realized by the Euler class. A circle bundle is a principal bundle if and only if the associated map is null-homotopic, which is true if and only if the bundle is fibrewise orientable.
- Weisstein, Eric W. "Circle Bundle". MathWorld.
- Chern, Shiing-shen (1977), "Circle bundles", Lecture Notes in Mathematics, 597/1977, Springer Berlin/Heidelberg, pp. 114–131, ISBN 978-3-540-08345-0.
- https://mathoverflow.net/questions/144092/is-every-orientable-circle-bundle-principal. Missing or empty