Let $\pi:P \to M$ be a principal $G$-bundle. Given a subgroup $H$ of $G$ one can consider the fibre bundle $P_H :=P\times_{G}G/H$ whose fibres are the coset space $G/H$.
We define a reduction of structure group to be a section of $P_H \to M$ which is also the same as a $G$-equivariant map from $P \to G/H$.
Here are my questions :
- I would expect $P_H$ to be a principal $H$-bundle from the name. But the fibres are $G/H$. Am I understanding wrong?
Another interpretation of $G$-bundles are by specifying transition functions from $U\cap V \to G$ satisfying cocycle conditions.
- I would expect reduction to $H$ should somehow mean the transition functions land in $H$ instead of the full $G$. How can I formulate this?
All spaces above are smooth manifolds and groups are Lie groups.