Let $G$ be a Lie group and $K$ a compact subgroup of $G$. Then since $K$ acts freely and properly on $G$ by translation, we can form the homogeneous manifold $K\backslash G$.
Question: is the map $K\times (K\backslash G)\rightarrow G$ defined by $(k,[g])\mapsto kg$ a diffeomorphism?
Actually I can't see why this isn't true, but I find it hard to believe because it would imply that any Lie group $G$ is always a principal $K$-bundle over $G/K$.
Edit: I meant free principal $K$-bundle.
The map you write out is not well defined (since you choose a representative of a coset and the map is not independent of this choice). What happens is that $G$ is a principal $K$-bundle over $K\backslash G$, so locally isomorphic to a product, but not globally. The latter statement basically means that locally you can choose representatives of cosets smoothly.