Reduction of structure group

Definition

Let $M$ be a differential manifold and $G$ a Lie group. Let $P$ be a principal bundle over $M$ with structure group $G$ . Suppose $H$ is a Lie subgroup of $G$ .

A reduction of structure group to $H$ is defined as the following data: A subbundle $P^{'} \subseteq P$ , viz at each point $m \in M$ , a subset $P'_{m}$ of $P_{m}$ , such that if we restrict the action of $G$ on $P_{m}$ to the subgroup $H$ , then $H$ acts freely and transitively on $P'_{m}$ .

In other words, if we view the principal $G$ -bundle as a copy of $G$ at each point, then the reduction of structure group involves choosing smoothly, for each fibre, a subset which serves as a copy of $H$ .

Facts

Reduction to a normal subgroup

The reduction of structure group to a normal subgroup is of particular interest because for a normal subgroup, there is essentially only one kind of reduction. In other words, any two reductions to a normal subgroup Fill this in later

Smallest structure group is holonomy group

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