# Principal bundle

## Contents

## Definition

### Definition with symbols

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

- A differential manifold with a differentiable map
- A Lie group action of on such that the fibres of are precisely the orbits of , 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 there exists an open neighbourhood of such that the map is the same as the projection map .

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

## Facts

### Reduction of structure group

`Further information: Reduction of structure group`

If is a subgroup of and hs a principal -bundle , we may be able to find a subbundle that is a principal -bundle. Intuitively, what we are doing is that at each , we are choosing a subset of the copy of that is the orbit of a single point under the -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 be a Lie group, a normal subgroup that is also a Lie subgroup, and the quotient group. Then any principal -bundle gives rise to a -bundle as follows: if is the -bundle, then the -bundle is simply modulo the action of . The fibre at each point is the quotient space by .

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

### Vector bundles as principal bundles

A vector bundle on a differential manifold can be viewed as a principal bundle with structure group . If is the differential manifold, the fibre at is the set of all ordered bases for .

### 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*