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Associated Fiber Bundle


Given a group action G×F->F and a principal bundle pi:A->M, the associated fiber bundle on M is

 pi^~:A×F/G->M.
(1)

In particular, it is the quotient space A×F/G where (a,x)∼(ga,g^(-1)x).

For example, the torus T={(e^(is),e^(it))} has a S^1 action given by

 phi(e^(itheta))(e^(is),e^(it))=(e^(i(s+theta)),e^(i(t+theta)))
(2)

and the frame bundle on the sphere,

 pi:SO(3)->S^2,
(3)

is a principal S^1 bundle. The associated fiber bundle is a fiber bundle on the sphere, with fiber the torus. It is an example of a four-dimensional manifold.


See also

Bundle, Fiber Bundle, Group Action, Principal Bundle, Quotient Space

This entry contributed by Todd Rowland

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Cite this as:

Rowland, Todd. "Associated Fiber Bundle." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/AssociatedFiberBundle.html

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