Principal bundle

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 $GL(n,\mathbb {R} )$. If $M$ is the differential manifold, the fibre at $m\in M$ is the set of all ordered bases for $T_{p}M$.

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