A circle bundle is a fiber bundle whose fibers are circles. It may also have the structure of a principal bundle if there is an action of that preserves the fibers, and is locally trivial. That is, if every point has a...
A bundle map is a map between bundles along with a compatible map between the base manifolds. Suppose and are two bundles, then is a bundle map if there is a map such that for all . In particular, the fiber bundle of...
The term "bundle" is an abbreviated form of the full term fiber bundle. Depending on context, it may mean one of the special cases of fiber bundles, such as a vector bundle or a principal bundle. Bundles are so named because they...
Given a principal bundle , with fiber a Lie group and base manifold , and a group representation of , say , then the associated vector bundle is (1) In particular, it is the quotient space where . This construction...
Given a group action and a principal bundle , the associated fiber bundle on is (1) In particular, it is the quotient space where . For example, the torus has a action given by (2) and the frame bundle...
An anchor is the bundle map from a vector bundle to the tangent bundle TB satisfying 1. and 2. where and are smooth sections of , is a smooth function of , and the bracket is the "Jacobi-Lie bracket"...
