Every principal bundle admits a connection

Let $M$ be a smooth manifold and let $\pi:P\to M$ be a principal G-bundle with Lie structure group $G$.

Corollary (existence of principal connections)

There exists at least one principal connection on $P$.

In particular, the affine space $\mathrm{Conn}(P)$ is nonempty for every principal bundle over a smooth manifold.

Examples

1. Trivial bundle. On $P=M\times G$, the product horizontal distribution $T_xM\subset T_{(x,g)}(M\times G)$ defines a principal connection (the “zero” connection in a global gauge).
2. Hopf fibration. The principal $U(1)$-bundle $S^3\to S^2$ admits a natural connection whose horizontal distribution is the contact distribution on $S^3$; this yields the standard magnetic monopole connection on $S^2$ in local gauges.
3. Frame bundles. The frame bundle of a vector bundle (in particular, the frame bundle of $TM$) admits principal connections; choosing a Riemannian metric yields the Levi–Civita connection on the orthonormal frame bundle.