Principal bundle

In mathematics, a principal G-bundle is a special kind of fiber bundle for which the fibers are all G-torsors (also known as principal homogeneous spaces) for the action of a topological group G. Principal G-bundles are G-bundles in the sense that the group G also serves as the structure group of the bundle.

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.

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Formal definition

A principal G-bundle is a fiber bundle π : PX together with a continuous right action P × GP by a topological group G such that G preserves the fibers of P and acts freely and transitively on them. (One often requires the base space X to be Hausdorff and possibly paracompact). The abstract fiber of the bundle is taken to be G itself.

It follows that the orbits of the G-action are precisely the fibers of π : PX and the orbit space P/G is homeomorphic 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. 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.

The local trivializations of a principal G-bundle are required to be G-equivariant maps so as to preserve the G-torsor structure of the fibers. Specifically, this means that if

[itex]\phi : \pi^{-1}(U) \to U \times G\,[itex]

is a local trivialization of the form [itex]\phi(p) = (\pi(p),\psi(p))[itex] then

[itex]\phi(p\cdot g) = (\pi(p),\psi(p)g).[itex]

One can also define principal G-bundles in the category of smooth manifolds. Here π : PX 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.

Examples

The most common example of a smooth principal bundle is the frame bundle of a smooth manifold M. Here the fiber over a point x in M is the set of all frames (i.e. ordered bases) for the tangent space TxM. 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.

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).

A normal (regular) covering space p : CX is a principal bundle where the structure group [itex]\pi_1(X)/p_{*}\pi_1(C)[itex] acts on C via the monodromy action. In particular, the universal cover of X is a principal bundle over X with structure group [itex]\pi_1(X)[itex].

Let G be any Lie group and let H be a closed subgroup. Then G is a principal H-bundle over G/H (the left coset space of H). Here the action of H on G is just right multiplication.

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 an 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:

[itex]\mbox{O}(1) \to S(\mathbb{R}^{n+1}) \to \mathbb{RP}^n[itex]
[itex]\mbox{U}(1) \to S(\mathbb{C}^{n+1}) \to \mathbb{CP}^n[itex]
[itex]\mbox{Sp}(1) \to S(\mathbb{H}^{n+1}) \to \mathbb{HP}^n[itex]

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.

Characterization of principal bundles

If π : PX 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 and Lie group and μ : P × GP a smooth, free, and proper right action then

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

References

• Jurgen Jost, Riemannian Geometry and Geometric Analysis, (2002) Springer-Verlag, Berlin ISBN 3-540-4267-2 See section 1.7.
• David Bleecker, Gauge Theory and Variational Principles, (1981), Addison-Wesley Publishing, ISBN 0-201-10096-7 See Chapter 1.

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