Induced connection on an associated bundle via horizontals

Let $\pi:P\to M$ be a principal G-bundle with principal connection $\omega$, and let $E=P\times_G F$ be the associated bundle for a left $G$-space $F$. Write $q:P\times F\to E$ for the quotient map and denote by $\pi_E:E\to M$ the bundle projection.

Construction (horizontal distribution on E). For $[p,u]\in E$, define the horizontal subspace

where $H_p=\ker(\omega_p)\subset T_pP$ is the horizontal subspace of $P$ and $0\in T_uF$ is the zero vector. This is well-defined (independent of the representative $(p,u)$) because $H$ is $G$-equivariant and the quotient identifies $(pg,g^{-1}u)$ with $(p,u)$.

Then $TE$ splits as

giving an Ehresmann connection on $E$.

In the vector bundle case (fiber a vector space), this induced connection coincides with the usual notion of connection on a vector bundle .

Examples

1. For $E=P\times_G G$ with left multiplication, the induced connection recovers the original principal connection under the identification $E\cong P$.
2. For $E=\mathrm{Ad}(P)$, the construction produces a natural connection on the adjoint bundle used in gauge theory to differentiate gauge parameters.
3. If $P$ is trivial over $U$ and $A$ is the local gauge potential, the induced horizontals on $U\times F$ are determined by $A$ acting through the infinitesimal action of $\mathfrak g$ on $F$.