Question regarding reduction of structure group

Question

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 :

1. 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.

2. 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.

Answer 1

$P_H$ is not a principal $H$-bundle, since as you correctly say, it has fiber isomorphic to $G/H$. In fact, one can show that $P_H$ can be identified with $P/H$: this is proved for example in "Foundations of Differential Geometry - Volume I" by Kobayashi and Nomizu, Chapter I, Proposition 5.5.

Instead, associated with a $G$-equivariant map $\phi:P\to G/H$, the corresponding $H$-subbundle is $\phi^{-1}(eH)$. Using $G$-equivariance it's straightforward to show this is actually invariant under the right $H$-action, and that it defines a $H$-bundle over $M$. The "reduction" of the bundle is simply the inclusion map $i:\phi^{-1}(eH)\to P$.

To answer your second question, for an arbitrary trivialisation of $P$, there's no guarantee the transition functions will take values in $H$. However if you define a local trivialisation of $\phi^{-1}(eH)$, in the form of local sections $s_U:U\subset M\to \phi^{-1}(eH)\subset P$, and use these to also define a local trivialisation of $P$, then the transition functions will be $H$-valued.