Principal Bundle

Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, a principal bundle is a mathematical object which formalizes some of the essential features of a Cartesian product X × G of a space X with a group G. Analogous to the Cartesian product, a principal bundle P is equipped with a projection onto X, which is just the projection onto the first factor for a product space. A common example of a principal bundle is the frame bundle FE of a vector bundle E, which consists of all ordered bases of the vector space attached to each point. The group G in this case is the general linear group, which acts in the usual way on ordered bases. Since there is no preferred way to choose an ordered basis of a vector space, a frame bundle lacks a canonical choice of identity cross-section.