Let $G$ be a Lie group and $M$ be a smooth manifold.
I have to show that a $G$-principal bundle admits a reduction to some subgroup $H$ of $G$ if and only if there exists a Čech cocycle $\{(U_{\alpha})_{\alpha \in I}, \psi_{\alpha\beta} : U_{\alpha} \cap U_{\beta} \rightarrow G\}$ defining it so that $\psi_{\alpha\beta}$ takes values in $H$.
This exercise originates from the book "Lectures on Kähler Geometry" by Andrei Moroianu.
I know the following definitions :
In the lecture we have defined a $G$-principal bundle over $M$ as a smooth manifold $P$ endowed with a smooth submersion $\pi : P \rightarrow M$ and a group action of $G$ on $P$ to the right that restricts to a free transitive action on a fiber $\pi^{-1}(x)$. Let $H$ be a subgroup of $G$ and let $f:H \rightarrow G$ be a Lie group morphism. A reduction of $P$ to $H$ is an $H$-principal bundle $Q$ together with a bundle morphism $Q \rightarrow P$ whose associated Lie group morphism is $f$.
Unfortunately, it is not clear to me how to prove the above claim. Does somebody have an idea ? Can someone give me any hints ?
Thanks for your help.
Consider $Q$ defined as follows: It is the quotient of $\cup_\alpha(U_\alpha\times H)$ by the relation $(x,y)\simeq (x,\psi_{\alpha\beta}(x)y), x\in U_\alpha\cap U_\beta$. The projections $p_\alpha:U_\alpha\times H\rightarrow U_\alpha$ can be glued to define a projection $p_Q:Q\rightarrow M$ and the canonical embedding: $U_\alpha\times H\rightarrow U_\alpha\times G$ induces a morphism $Q\rightarrow P$.