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 $π : P \to M$
A Lie group action of $G$ on $M$ such that the fibres of $π$ 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 $π : π^{- 1} (U) \to U$ 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 $m \in M < m a t h >, w e a r e c h o o s i n g a s u b s e t o f t h e c o p y o f < m a t h > G$ 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.

Vector bundles as principal bundles

A vector bundle on a differential manifold can be viewed as a principal bundle with structure group $G L (n, R)$ . 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