Connection on associated bundle

Question

Let $M$ be a compact Riemannian manifold without boundary, and consider $E$ be a vector bundle over M with metric structure on the fibers $F$. Now consider two connections $\nabla$ and $\nabla'$ on $E$ compatible with metric structure $\langle-,-\rangle$. Denote by $T$ the difference $\nabla - \nabla'$, which is an element in $\Omega^1(M, \mathrm{End}(E))$. Now, consider the $G$-bundle principal $P$ associated to $E$. Similarly, if $\omega$ and $\omega'$ are two connection 1-forms on $P$, the difference between these connections is a 1-form $\theta \in \Omega^1(P, \mathfrak{g})$ that satisfies the equivariance condition: $$ R_g^* \theta = \mathrm{Ad}(g^{-1})\theta, \quad \forall g \in G. $$ The $\mathfrak{g}$-valued $1$-form $\theta$ is relates to $T$ in the associated vector bundle $E$, as $\theta$ encodes the change in the horizontal distribution of the principal bundle, which propagates to the associated bundle.

I would like to find an equation that relates the two forms, as in the case of connections. More explicitly, given a connection $\nabla$ on $E$ acting on the sections of $E$, we can associate this connection with horizontal vector fields acting on $G$-equivariant functions on the associated $G$-principal bundle $P$. That is,

$$ \nabla_{X}s = X^{h}f, $$

where $f$ is a $G$-equivariant function and $X^{h}$ is horizontal lift of vector field $X$.

I would like to achieve something similar for 1-forms with values in $End(E)$, but I would prefer not to resort to local coordinates. I found some expressions in Kobayashi's book, of the form: $$ T(X)(s) = \theta(X^h) \cdot s, $$ where $X^h$ is the horizontal lift of $X$ under the connection $\omega$, and $s$ is a section of $E$.

Answer 1

Note that $\theta$ being the difference of two connection one-forms, it is also horizontal: for every $\xi\in\mathfrak{g}$, $\theta(X_\xi)=0$, where $X_\xi$ is the fundamental vector field of the $G$-action of $P$ (i.e. $\theta$ vanishes on any section of the vertical bundle of $P$).

Therefore $\theta$ is basic, and thus determines a $1$-form $\bar\theta$ on $M$ with values in the adjoint bundle $ad(P)$ (that is the vector bundle associated with the adjoint representation).

Now I guess that your structure group $G$ comes with a morphism $G\to GL(r)$, where $r$ is the rank of $E$. This induces a morphism of representations $\mathfrak{g}\to End(\mathbb{R}^r)$. Passing to associated vector bundles, you get a bundle map $ad(P)\to End(E)$ (recall that $E$ is the vector bundle associated with the representation $\mathbb{R}^r$), and thus a linear map $\mu:\Omega^1(M,ad(P))\to \Omega^1(M,End(E))$.

The result you're looking after is $\mu(\bar\theta)=T$.

Remark: note that $\bar\theta(X)=\theta(X^h)$, and $\mu(\alpha)(X)(s)=\alpha(X)\cdot s$. Hence $\mu(\bar\theta)=T$ precisely becomes $\theta(X^h)\cdot s=T(X)(s)$.