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 Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In mathematics, the Cartesian product is a direct product of sets. ... Look up group in Wiktionary, the free dictionary. ...

1. An action of G on P, analogous to (x,g)h = (x, gh) for the product space.
2. A projection onto X, which is just the projection onto the first factor for a product space: (x,g) → x.

Unlike a product space, however, 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 (x,g) → g onto the second factor. 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; they are fibre bundles. In mathematics, a symmetry group describes all symmetries of objects. ... A Möbius strip, an object with only one surface and one edge; such shapes are an object of study in topology. ... In mathematics, in particular in topology, a fiber bundle is a space which locally looks like a product of two spaces but may possess a different global structure. ...

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. In mathematics, the idea of a frame in the theory of smooth manifolds is understood in terms meaning it can vary from point to point. ... In mathematics, a vector bundle is a geometrical construct where to every point of a topological space (or manifold, or algebraic variety) we attach a vector space in a compatible way, so that all those vector spaces, glued together, form another topological space (or manifold or variety). ... In mathematics, a subset B of a vector space V is said to be a basis of V if it satisfies one of the four equivalent conditions: B is both a set of linearly independent vectors and a generating set of V. B is a minimal generating set of V... In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. ...

In formal terms, a principal G-bundle is a fiber bundle P on a topological space X equipped with a free and transitive action of a topological group G on the fibers of P. The fibers are then principal homogeneous spaces for the right action of G on itself. Principal G-bundles are fiber bundles with structure group G as well, in the sense that they admit a local trivialization in which the transition maps are given by transformations in G. In mathematics, in particular in topology, a fiber bundle (or fibre bundle) is a space which locally looks like a product of two spaces but may possess a different global structure. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics, a symmetry group describes all symmetries of objects. ... In mathematics, a topological group G is a group that is also a topological space such that the group multiplication G × G → G and the inverse operation G → G are continuous maps. ... In mathematics, a principal homogeneous space, or G-torsor, for a group G is a set X on which G acts freely and transitively. ... In mathematics, in particular in topology, a fiber bundle is a space which locally looks like a product of two spaces but may possess a different global structure. ... In algebraic topology, a fibration is a continuous mapping Y → X satisfying the homotopy lifting property. ... In topology, an atlas describes how a complicated space is glued together from simpler pieces. ...

Principal bundles have important applications in topology and differential geometry. They have also found application in the 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. A Möbius strip, an object with only one surface and one edge; such shapes are an object of study in topology. ... In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... The first few hydrogen atom electron orbitals shown as cross-sections with color-coded probability density Physics (Greek: (phúsis), nature and (phusiké), knowledge of nature) is the branch of science concerned with the discovery and characterization of universal laws which govern matter, energy, space, and time. ... In physics, gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ...

Formal definition

A principal G-bundle is a fiber bundle π : P → X together with a continuous right action P × G → P by a topological group G such that G preserves the fibers of P and acts freely and transitively on them. The abstract fiber of the bundle is taken to be G itself. (One often requires the base space X to be a Hausdorff space and possibly paracompact). In mathematics, in particular in topology, a fiber bundle (or fibre bundle) is a space which locally looks like a product of two spaces but may possess a different global structure. ... In topology, a continuous function is generally defined as one for which preimages of open sets are open. ... In mathematics, a symmetry group describes all symmetries of objects. ... In mathematics, a topological group G is a group that is also a topological space such that the group multiplication G × G → G and the inverse operation G → G are continuous maps. ... In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ... In mathematics, a paracompact space is a topological space in which every open cover admits an open locally finite refinement. ...

It follows that the orbits of the G-action are precisely the fibers of π : P → X and the orbit space P/G is homeomorphic to the base space X. To say that G acts freely and transitively on the fibers means that the fibers take on the structure of G-torsors (i.e., they are spaces with a transitive free group action, hence we are given a family of principal homogeneous spaces over the base space). A G-torsor is a space which is homeomorphic to G but lacks a group structure since there is no preferred choice of an identity element. In mathematics, groups are often used to describe symmetries of objects. ... This word should not be confused with homomorphism. ... In mathematics, a principal homogeneous space, or G-torsor, for a group G is a set X on which G acts freely and transitively. ... In mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a binary operation on that set. ...

A principal G-bundle can also be characterized as a G-bundle π : P → X with fiber G where the structure group acts on the fiber by left multiplication. Since right multiplication by G on the fiber commutes with the action of the structure group, there exists an invariant notion of right multiplication by G on P. The fibers of π then become right G-torsors for this action.

One can also define principal G-bundles in the category of smooth manifolds. Here π : P → X is required to be a smooth map between smooth manifolds, G is required to be a Lie group, and the corresponding action on P should be smooth. In mathematics, categories allow one to formalize notions involving abstract structure and processes that preserve structure. ... In mathematics, a manifold M is a type of space, characterized in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ... In mathematics, a smooth function is one that is infinitely differentiable, i. ... In mathematics, a Lie group, named after Norwegian mathematician Sophus Lie (IPA pronunciation: , sounds like Lee), is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. ...

Examples

The prototypical example of a smooth principal bundle is the frame bundle of a smooth manifold M, often denoted FM or GL(M). Here the fiber over a point x in M is the set of all frames (i.e. ordered bases) for the tangent space T_xM. The general linear group GL(n,R) acts simply-transitively on these frames. These fibers can be glued together in a natural way so as to obtain a principal GL(n,R)-bundle over M. In mathematics, the idea of a frame in the theory of smooth manifolds is understood in terms meaning it can vary from point to point. ... The tangent space of a manifold is a concept which facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other. ... In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. ...

Variations on the above example include the orthonormal frame bundle of a Riemannian manifold. Here the frames are required to be orthonormal with respect to the metric. The structure group is the orthogonal group O(n). In mathematics, a frame bundle is a principal fiber bundle F(E) associated to any vector bundle E. The fiber of F(E) over a point x is the set of all ordered bases, or frames, for Ex. ... In Riemannian geometry, a Riemannian manifold is a real differentiable manifold in which each tangent space is equipped with an inner product in a manner which varies smoothly from point to point. ... In linear algebra, two vectors v and w are said to be orthonormal if they are both orthogonal (according to a given inner product) and normalized. ... In mathematics, the metric tensor is a symmetric tensor field of rank 2 that is used to measure distance in a space. ... In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication. ...

More generally, if E is any vector bundle of rank k over M, then the bundle of frames of E is a principal GL(k,R)-bundle, sometimes denoted F(E).

A normal (regular) covering space p : C → X is a principal bundle where the structure group π₁(X) / p _* π₁(C) acts on C via the monodromy action. In particular, the universal cover of X is a principal bundle over X with structure group π₁(X). In mathematics, specifically topology, a covering map is a continuous surjective map p : C → X, with C and X being topological spaces, which has the following property: to every x in X there exists an open neighborhood U such that p -1(U) is a union of mutually disjoint... In mathematics, specifically topology, a covering map is a continuous surjective map p : C → X, with C and X being topological spaces, which has the following property: to every x in X there exists an open neighborhood U such that p -1(U) is a union of mutually disjoint... In mathematics, specifically topology, a covering map is a continuous surjective map p : C → X, with C and X being topological spaces, which has the following property: to every x in X there exists an open neighborhood U such that p -1(U) is a union of mutually disjoint...

Let G be any Lie group and let H be a closed subgroup (not necessarily normal). Then G is a principal H-bundle over the (left) coset space G/H. Here the action of H on G is just right multiplication. The fibers are the left cosets of H (in this case there is a distinguished fiber, the one containing the identity, which is naturally isomorphic to H). In mathematics, a normal subgroup N of a group G is a subgroup invariant under conjugation; that is, for each element n in N and each g in G, the element g−1ng is still in N. The statement N is a normal subgroup of G is written: . There are... In mathematics, in particular in the theory of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a manifold or topological space X on which G acts by symmetry in a transitive way; it is not assumed that the action of G is faithful. ...

Consider the projection π: S¹ → S¹ given by z ↦ z². This principal Z₂-bundle is the associated bundle of the Möbius strip. Besides the trivial bundle, this is the only principle Z₂-bundle over S^-1. In mathematics, the theory of fiber bundles with a structure group (a topological group) allows an operation of creating an associated bundle, in which the typical fiber of a bundle changes from to , which are both topological spaces with a group action of . ... A Möbius strip made with a piece of paper and tape. ...

Projective spaces provide more interesting examples of principal bundles. Recall that the n-sphere Sⁿ is a two-fold covering space of real projective space RPⁿ. The natural action of O(1) on Sⁿ gives it the structure of a principal O(1)-bundle over RPⁿ. Likewise, S²ⁿ⁺¹ is a principal U(1)-bundle over complex projective space CPⁿ and S⁴ⁿ⁺³ is a principal Sp(1)-bundle over quaternionic projective space HPⁿ. We then have a series of principal bundles for each positive n: This article does not cite its references or sources. ... A sphere is a perfectly symmetrical geometrical object. ... In mathematics, real projective space, or RPn is the projective space of lines in Rn+1. ... In mathematics, the unitary group of degree n, denoted U(n), is the group of n×n unitary matrices, with the group operation that of matrix multiplication. ... In mathematics, complex projective space, or CPn, is the projective space of (complex) lines in Cn+1. ... In mathematics, the name symplectic group can refer to two different, but closely related, types of mathematical groups. ... In mathematics, quaternionic projective space is an extension of the ideas of real projective space and complex projective space, to the case where coordinates lie in the ring of quaternions H. Quaternionic projective space of dimension n is usually denoted by HPn and is a closed manifold of (real) dimension...

Here S(V) denotes the unit sphere in V (equipped with the Euclidean metric). For all of these examples the n = 1 cases give the so-called Hopf bundles. In mathematics, the Hopf bundle (or Hopf fibration), named after Heinz Hopf, is an important example of a fiber bundle. ...

Trivializations and cross sections

One of the most important questions regarding fiber bundles is whether or not they are trivial (i.e. isomorphic to a product bundle). For principal bundles there is a convenient characterization of triviality: In mathematics, in particular in topology, a fiber bundle (or fibre bundle) is a space which locally looks like a product of two spaces but may possess a different global structure. ...

Theorem. A principal bundle is trivial if and only if it admits a global cross section.

The same is not true for other fiber bundles. Vector bundles, for instance, always have a zero section whether they are trivial or not. In mathematics, a vector bundle is a geometrical construct where to every point of a topological space (or manifold, or algebraic variety) we attach a vector space in a compatible way, so that all those vector spaces, glued together, form another topological space (or manifold or variety). ...

The same theorem applies to local trivializations of a principal bundles. Let π : P → X be a principal G-bundle. An open set U in X admits a local trivialization if and only if there exists a local section on U. Given a local trivialization one can define an associated local section by In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U...

where e is the identity in G. Conversely, given a section s one defines a trivialization Φ by In mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a binary operation on that set. ...

The fact that G acts simply transitively on the fibers of P guarantees that this map is a bijection. One can check that it is also a homeomorphism. The local trivializations defined by a local section are G-equivariant in the following sense. If we write in the form Φ(p) = (π(p),φ(p)) then the map satisfies A bijective function. ... In the mathematical field of topology a homeomorphism or topological isomorphism (from the Greek words homeos = identical and morphe = shape) is a special isomorphism between topological spaces which respects topological properties. ... In mathematics, an equivariant map is a function between two sets that commutes with the action of a group. ...

Equivariant trivializations therefore preserve the G-torsor structure of the fibers. In terms of the associated local section s the map φ is given by

The local version of the cross section theorem then states that the equivariant local trivializations of a principal bundle are in one-to-one correspondence with local sections.

Given an equivariant local trivialization ({U_i}, {Φ_i}) of P, we have local sections s_i on each U_i. On overlaps these must be related by the action of the structure group G. In fact, the relationship is provided by the transition functions In mathematics, a transition function has several different meanings: In topology, a transition function is a homeomorphism from one coordinate chart to another. ...

For any x in U_i ∩ U_j we have

Characterization of smooth principal bundles

If π : P → X is a smooth principal G-bundle then G acts freely and properly on P so that the orbit space P/G is diffeomorphic to the base space X. It turns out that these properties completely characterize smooth principal bundles. That is, if P is a smooth manifold, G a Lie group and μ : P × G → P a smooth, free, and proper right action then In mathematics, a continuous function between topological spaces is called proper if inverse images of compact subsets are compact. ... In mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. ...

• P/G is a smooth manifold,
• the natural projection π : P → P/G is a smooth submersion, and
• P is a smooth principal G-bundle over P/G.

In mathematics, a submersion is a differentiable map between differentiable manifolds whose derivative is everywhere surjective. ...

Reduction of the structure group

Given a subgroup , one may consider the bundle P / H whose fibers are homeomorphic to the coset space G / H. If the new bundle admits a global section, then one says that the section is a reduction of the structure group from G to H . The reason for this name is that the (fiberwise) inverse image of the values of this section form a subbundle of P which is a principal H-bundle. If H is the identity, then a section of P itself is a reduction of the structure group to the identity. Reductions of the structure group do not in general exist. In mathematics, in particular in the theory of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a manifold or topological space X on which G acts by symmetry in a transitive way; it is not assumed that the action of G is faithful. ...

Many topological questions about the structure of a bundle may be rephrased as to questions about the admissibility of the reduction of the structure group. For example:

• A 2n-dimensional real manifold admits an almost-complex structure if the frame bundle on the manifold, whose fibers are , can be reduced to the group .

• An n-dimensional manifold admits n vector fields that are linearly independent at each point if its frame bundle is parallelizable, that is, if the frame bundle admits a global section.

• An n-dimensional real manifold admits a k-plane field if the frame bundle can be reduced to the structure group .

See also the articles reduction of the structure group and G-structure for a related discussion. In mathematics, an almost complex manifold is a smooth manifold equipped with smooth linear complex structure on each tangent space. ... In mathematics, the idea of a frame in the theory of smooth manifolds is understood in terms meaning it can vary from point to point. ... In mathematics, the idea of a frame in the theory of smooth manifolds is understood in terms meaning it can vary from point to point. ... This is a glossary of terms specific to differential geometry and differential topology. ... In mathematics, the theory of fiber bundles with a structure group (a topological group) allows an operation of creating an associated bundle, in which the typical fiber of a bundle changes from to , which are both topological spaces with a group action of . ... In differential geometry, a G-structure on a n-manifold M, for a given structure group G (which is a Lie subgroup of the general linear group GL(n)) is a G-subbundle of the frame bundle on M. The notion of G-structures includes many other structures on manifolds...

Associated vector bundles and frames

If P is a principal G-bundle and V is a linear representation of G, then one can construct a vector bundle with fibre V, as the quotient of the product P×V by the diagonal action of G. This is a special case of the associated bundle construction, and E is called an associated vector bundle to P. If the representation of G on V is faithful, so that G is a subgroup of the general linear group GL(V), then E is a G-bundle and P provides a reduction of structure group of the frame bundle of E from GL(V) to G. This is the sense in which principal bundles provide an abstract formulation of the theory of frame bundles. Representation theory is the branch of mathematics that studies properties of abstract groups via their representations as linear transformations of vector spaces. ... In mathematics, the theory of fiber bundles with a structure group (a topological group) allows an operation of creating an associated bundle, in which the typical fiber of a bundle changes from to , which are both topological spaces with a group action of . ... To meet Wikipedias quality standards, this article or section may require cleanup. ... In mathematics, a faithful representation ρ of a group G on a vector space V is a linear representation in which different elements g of G are represented by distinct linear mappings ρ(g). ...

See also

• associated bundle
• vector bundle
• G-structure
• gauge theory

In mathematics, the theory of fiber bundles with a structure group (a topological group) allows an operation of creating an associated bundle, in which the typical fiber of a bundle changes from to , which are both topological spaces with a group action of . ... In mathematics, a vector bundle is a geometrical construct where to every point of a topological space (or manifold, or algebraic variety) we attach a vector space in a compatible way, so that all those vector spaces, glued together, form another topological space (or manifold or variety). ... In differential geometry, a G-structure on a n-manifold M, for a given structure group G (which is a Lie subgroup of the general linear group GL(n)) is a G-subbundle of the frame bundle on M. The notion of G-structures includes many other structures on manifolds... In physics, gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ...

References

• Bleecker, David (1981). Gauge Theory and Variational Principles. Addison-Wesley Publishing. ISBN 0-486-44546-1 (Dover edition). 
• Jost, Jürgen (2005). Riemannian Geometry and Geometric Analysis, (4th ed.), New York: Springer. ISBN 3-540-25907-4. 
• Sharpe, R. W. (1997). Differential Geometry: Cartan's Generalization of Klein's Erlangen Program. New York: Springer. ISBN 0-387-94732-9. 
• Steenrod, Norman (1951). The Topology of Fibre Bundles. Princeton: Princeton University Press. ISBN 0-691-00548-6.