Principal bundle

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Definition

Definition with symbols

Let M be a differential manifold and G a Lie group. A princial bundle on M with structure group G is defined as the following data:

  • A differential manifold P with a differentiable map \pi:P \to M
  • A Lie group action of G on M such that the fibres of \pi are precisely the orbits of G, and also the group action is free (that is no two group elements have the same effect on any point)

satisfying the following condition called local triviality:

Given any point m \in M there exists an open neighbourhood U of M such that the map \pi:\pi^{-1}(U) \to U is the same as the projection map U \times G \to U.

Intuitively a principal G-bundle means a copy of G at each point, varying smoothly with the point. However, the copy of G at each point, does not have any natural "origin" or identity element.

Facts

Reduction of structure group

Further information: Reduction of structure group

If H is a subgroup of G and M hs a principal G-bundle P, we may be able to find a subbundle P' that is a principal H-bundle. Intuitively, what we are doing is that at each m \in M, we are choosing a subset of the copy of G that is the orbit of a single point under the H-action. The tricky part is to make a choice that varies smoothly with the point.

The process of finding a subbundle that is principal for a subgroup is termed reduction of the structure group.

Lifting of structure group

Further information: Lifting of structure group

Let G be a Lie group, N a normal subgroup that is also a Lie subgroup, and G/N the quotient group. Then any principal G-bundle gives rise to a G/N-bundle as follows: if \pi:P \to M is the G-bundle, then the G/N-bundle is simply P modulo the action of N. The fibre at each point is the quotient space by N.

The question of lifting the structure group is the other way around: given a principal bundle over H = G/N, find a principal G-bundle that would give rise to this on quotienting by N.

Vector bundles as principal bundles

A vector bundle on a differential manifold can be viewed as a principal bundle with structure group GL(n,\R). If M is the differential manifold, the fibre at m \in M is the set of all ordered bases for T_pM.

Connections on principal bundles

Further information: Connection on principal bundle

Typically the term connection is used for a vector bundle on a differential manifold, and it is intended to be a rule that allows us to differentiate sections of the vector bundle alogn vector fields (viz, sections of the tangent bundle).

We can generalize this to define a notion of connection on a principal bundle. Fill this in later