Principal bundle

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In mathematics, principal bundles are an abstract version of the notion of a frame bundle, which is a fiber bundle whose fiber over a point consists of a collection of ordered bases for a vector space (the fiber of a vector bundle over that point). Principal bundles are characterized by admitting an action of a group G which acts freely and transitively on the fibers, generalizing the action of the general linear group on the set of all bases for a vector space.

More precisely, principal G-bundle is a fiber bundle for which the fibers are all G-torsors (also known as principal homogeneous spaces) for the group 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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[edit] 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).

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. 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.

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.

[edit] 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 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).

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).

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).

Consider the projection π: S1 → S1 given by z ↦ z2. This principal Z2-bundle is the associated bundle of the Möbius strip.

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:

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

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.

[edit] 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:

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.

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 \Phi : \pi^{-1}(U) \to U \times G one can define an associated local section s : U \to \pi^{-1}(U) by

s(x) = \Phi^{-1}(x,e)\,

where e is the identity in G. Conversely, given a section s one defines a trivialization Φ by

\Phi^{-1}(x,g) = s(x)\cdot g

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 \Phi : \pi^{-1}(U) \to U \times G in the form Φ(p) = (π(p),φ(p)) then the map \varphi : P \to G satisfies

\varphi(p\cdot g) = \varphi(p)g.

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

\varphi(s(x)\cdot g) = g.

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 ({Ui}, {Φi}) of P, we have local sections si on each Ui. On overlaps these must be related by the action of the structure group G. In fact, the relationship is provided by the transition functions

t_{ij} = U_i \cap U_j \to G\,

For any x in Ui ∩ Uj we have

s_j(x) = s_i(x)\cdot t_{ij}(x).

[edit] 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

  • 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.

[edit] Reduction of the structure group

Given a subgroup H \subset G, 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.

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:

  • 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 GL(k,\mathbb{R}) \subset GL(n,\mathbb{R}).

See also the articles reduction of the structure group and G-structure for a related discussion.

[edit] 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 E=P\times_G 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 is 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.

[edit] See also

[edit] 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. 
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