= Principal bundle trivialization by a global section
{title2=$B\times G\longrightarrow P,\quad(x,g)\mapsto s(x)g$}
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 $x\mapsto(x,e)$.
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