A circle bundle is a fiber bundle in which the fiber is isomorphic to \(S^1\) as a topological group. Consider circle bundles over a circle, which are of the form \begin{align*} S^1 \to E \xrightarrow{\pi} S^1 \end{align*}

Definition: A principal \(G{\hbox{-}}\) bundle is a fiber bundle \(F \to E \to B\) for which \(G\) acts freely and transitively on each fiber \(F_b:= \pi^{-1}(b]]\).