# When is a quotient of a principal bundle is a principal bundle?

Suppose $$\pi:P\rightarrow M$$ is a principal $$G$$ bundle.

Let $$H$$ be a Lie group acting freely and properly on $$P$$ and on $$M$$ so that $$P/H$$ and $$M/H$$ are manifolds. Further assume this action is such that it defines a map $$P/H\rightarrow M/H$$.

Is it always true that the induced map $$P/H\rightarrow M/H$$ is also a principal $$G$$ bundle?

I think it is true, may be with some extra conditions.

Can some one please say what extra conditions (if at all) I need to confirm $$P/H\rightarrow M/H$$ is a principal $$G$$ bundle?

In order for $$P/H\to M/H$$ to be a principal $$G$$-bundle, we need to have a well-defined, free $$G$$-action on $$P/H$$. This is the case if and only if $$H$$ acts on $$P$$ by morphisms of $$G$$-spaces, that is, the $$H$$ action commutes with the $$G$$ action.
• Ok. I am assuming $G$ action commutes with $H$ action... Then, it is true right? With no extra conditions it is true.. – user537667 Feb 26 at 11:31