In mathematics, a **principal bundle** is a mathematical object which formalizes some of the essential features of the Cartesian product *X* × *G* of a space *X* with a group *G*. In the same way as with the Cartesian product, a principal bundle *P* is equipped with

- An action of
*G*on*P*, analogous to (*x*,*g*)*h*= (*x*,*gh*) for a product space. - A projection onto
*X*. For a product space, this is just the projection onto the first factor, (*x*,*g*) →*x*.

Unlike a product space, principal bundles lack a preferred choice of identity cross-section; they have no preferred analog of (*x*,*e*). Likewise, there is not generally a projection onto *G* generalizing the projection onto the second factor, *X* × *G* → *G* which exists for the Cartesian product. They may also have a complicated topology, which prevents them from being realized as a product space even if a number of arbitrary choices are made to try to define such a structure by defining it on smaller pieces of the space.

A common example of a principal bundle is the frame bundle F*E* 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.

Principal bundles have important applications in topology and differential geometry. They have also found application in physics where they form part of the foundational framework of gauge theories. Principal bundles provide a unifying framework for the theory of fiber bundles in the sense that all fiber bundles with structure group *G* determine a unique principal *G*-bundle from which the original bundle can be reconstructed.

Read more about Principal Bundle: Formal Definition, Examples, Classification of Principal Bundles

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### Famous quotes containing the words bundle and/or principal:

““There is Lowell, who’s striving Parnassus to climb

With a whole bale of isms tied together with rhyme,

He might get on alone, spite of brambles and boulders,

But he can’t with that *bundle* he has on his shoulders,

The top of the hill he will ne’er come nigh reaching

Till he learns the distinction ‘twixt singing and preaching;”

—James Russell Lowell (1819–1891)

“So in accepting the leading of the sentiments, it is not what we believe concerning the immortality of the soul, or the like, but the universal impulse to believe, that is the material circumstance, and is the *principal* fact in this history of the globe.”

—Ralph Waldo Emerson (1803–1882)