Principal bundle

Remarks

The product $M \times G$ with $(x, h) \cdot g = (x, hg)$ is a principal $G$-bundle, the trivial one. The map $\R \to S^1$, $t \mapsto e^{2\pi i t}$, with $\Z$ acting by $t \cdot n = t + n$, is a principal $\Z$-bundle. The Hopf map $S^3 \to S^2$ is a principal $U(1)$-bundle.

Non-example: the circle group acts on $S^2$ by rotations about the vertical axis. This action preserves the fibers of the identity map $S^2 \to S^2$, but these fibers are points, not copies of the circle. More generally, an action with a fixed point cannot be part of a principal bundle structure, because clause 2 forces $G$ to act freely.

Clause 2 implies that $G$ acts freely and transitively on each fiber. Conversely, for a free and proper smooth action of a Lie group $G$ on a manifold $P$, the quotient map $P \to P/G$ is a principal $G$-bundle. The same definition with topological groups and homeomorphisms gives topological principal bundles.