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

### Other articles related to "bundle, principal bundle, principal, bundles":

... is an equivariant lift of the oriented orthonormal frame

**bundle**FSO(M) → M with respect to the double covering ρ Spin(n) → SO(n) ... In other words, a pair (P,FP) is a spin structure on the

**principal bundle**π FSO(M) → M when a) πP P → M is a

**principal**Spin(n)-

**bundle**over M, b) FP P → FSO(M) is an equivariant ... The

**principal bundle**πP P → M is also called the

**bundle**of spin frames over M ...

... Let be a fibre

**bundle**with fibre ... between this quotient space and the fibre

**bundle**... If the structure group of the

**bundle**is known to reduce, you could replace with the reduced structure group ...

... If a vector

**bundle**E is equipped with a Riemannian

**bundle**metric then each fiber Ex is not only a vector space but an inner product space ... The orthonormal frame

**bundle**of E, denoted FO(E), is the set of all orthonormal frames at each point x in the base space X ... a method entirely analogous to that of the ordinary frame

**bundle**...

... Let M be a paracompact topological manifold and E an oriented vector

**bundle**on M of dimension n equipped with a fibre metric ... A spinor

**bundle**of E is a prescription for consistently associating a spin representation to every point of M ... topological obstructions to being able to do it, and consequently, a given

**bundle**E may not admit any spinor

**bundle**...

**Principal Bundle**s

... The classifying space has the property that any G

**principal bundle**over a paracompact manifold B is isomorphic to a pullback of the

**principal bundle**... is true, as the set of isomorphism classes of

**principal**G

**bundles**over the base B identifies with the set of homotopy classes of maps B → BG ...

### 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)

“The *principal* office of history I take to be this: to prevent virtuous actions from being forgotten, and that evil words and deeds should fear an infamous reputation with posterity.”

—Tacitus (c. 55–c. 120)