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 subgroup of finite index

The reduction of structure group to a subgroup of finite index is of particular interest because for a subgroup of finite index, knowing the value of $P'_{m}$ at one point $m$ suffices to determine the value of $P'$. In other words, there are as many possible reductions as the index (number of cosets) of the subgroup. In the particular case of a normal subgroup of finite index, the reductions correspond to elements of the quotient group.

Smallest structure group is holonomy group

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