Existence of a $G$-invariant metric on a principal bundle

Question

Given a smooth principal $G$-bundle $\pi: P \rightarrow M$, I want to show the existence of connections by showing that $P$ admits $G$-invariant metrics. I was thinking in a kind of averaging procedure like in Maschke's theorem's proof. Do I need $P$ to be compact for this? Thanks

Answer 1

Hint: You just need to suppose that $G$ is endowed with a finite Haar $\mu$measure which is the case if $G$ is compact, for $x\in P,u,v\in T_xP$, take any differentiable metric $\langle,\rangle$ on $P$ and set $\langle u,v\rangle_G=\int_G\langle Dg_x(u),Dg_x(v)\rangle d\mu$