Suppose I define a principal $G$-bundle as a map $\pi: P \to M$ with a smooth right action of $G$ on $P$ that acts freely and transitively on the fibers of $\pi$. Does it follow that $P$ is locally isomorphic to $M \times G$ with the obvious right action of $G$ on $M \times G$? Let's suppose $M$ is a manifold.
I know that fiber bundles over a contractible set are trivial and a manifold is locally contractible, but I believe this statements refers to locally trivial fiber bundles and so will not apply to this case.
A related question is: if we have a fibration such that the base space is contractible and all fibers are homeomorphic, does it follow that the fibration is just the product of the base with the fiber?