Theorem: Existence of principal connections on smooth manifolds

Every principal bundle over a smooth manifold admits a principal connection, using partitions of unity.

Theorem: Existence of principal connections on smooth manifolds

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

Theorem

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

Equivalently: every principal bundle over a smooth manifold admits a $G$ -equivariant horizontal distribution $H\subset TP$ complementary to the vertical subbundle.

Examples

Trivial bundle. On $P=M\times G$ , the product distribution $T M\oplus 0$ defines a connection; in connection-form language, this is the “flat” connection.
Frame bundle connections. Applying this theorem to $P=\mathrm{Fr}(E)$ gives existence of connections on any vector bundle $E\to M$ ; compare connections on Fr(E) induced by covariant derivatives .
Hopf fibration. The principal $U(1)$ -bundle $S^3\to S^2$ admits the standard connection whose horizontal spaces are orthogonal complements of the fiber circles (with respect to the round metric on $S^3$ ).