Principal bundle trivialization by a global section ($B\times G⟶P,(x,g)\mapsto s(x)g$)

A principal bundle is trivial exactly when it has a global section of a fiber bundle. Given such a section, the displayed map is a smooth equivariant isomorphism: the free transitive action on each fiber gives the unique inverse group coordinate, and local trivializations make it smooth. Conversely, a product trivialization has the section $x\mapsto (x,e)$.