Smooth principal $G$-bundle

Let $G$ be a finite-dimensional real Lie group, $M$ a smooth manifold, and $P$ a smooth manifold. A smooth right principal $G$-bundle is a right principal $G$-bundle $\pi :P\to M$ in the topological sense of Principal g bundle and associated fiber bundle, with the following additional smooth structure: $\pi$ and the right $G$-action are smooth, and the principal-bundle charts can be chosen as equivariant diffeomorphisms ${\pi }^{-1}(U)\cong U\times G$. Thus each chart respects the projection to $U$ and the right action on the $G$ factor. The smoothness convention is that of Smooth maps between manifolds with boundary; boundaryless manifolds are the default in this item.

The topological principal-bundle clause retains its free, transitive right action on every fibre and its local triviality. Smoothness adds compatible differentiable structure; it does not select a connection or assert that a global trivialization exists. The empty-base case is allowed and then $P$ is empty by the cited topological definition.