Let $\lambda = (P,\pi,M;G)$ be a smooth principal $G$-bundle (projection $\pi : P \to M)$, $V$ a finite dimensional vector space, and $\rho : G \to GL(V)$ a smooth representation of $G$ in $V$.


We can associate to $\lambda$ a vector bundle on $M$ with total space $P \times\_{\rho} V$, wich I note $\lambda \times\_{\rho} V$. If $\lambda$ is equipped with a principal connection $H$ whose 1-form is $\omega$, there is a unique connection $\nabla^H$ on $\lambda \times\_{\rho} V$. 


If $(U\_i,G,\Phi\_i)$ is a trivialisation of $\lambda$, it induces a trivialisation $(U\_i,V,\Psi\_i)$ on $\lambda \times\_{\rho} V$. Let $s\_i$ be the local section of $\lambda$ defined by $s\_i(x) = \Phi\_i^{-1}(x,e)$, then the local 1-form of $H$ on $M$ is $\omega\_i = s\_i^\* \omega$, and there is a simple relation between the local connection form $\Gamma\_i$ of $\nabla^H$ and local 1-form $\omega\_i$ :
$$ \Gamma\_i = \rho\_\* \omega\_i$$


On the other way, if we have a vector bundle $\xi = (E,\pi,M)$ with fiber-type $V$, we associate to $\xi$ its frame bundle, i.e. a $GL(V)$-principal bundle which I note $\lambda\_F(\xi)$, whose total space $P$ is the set of couples $(x,F\_x)$ where the isomorphism $F\_x : V \to E\_x$ is a frame of $E\_x$. 


If there is a connection $\nabla$ on $\xi$, whose horizontal distribution is $H(\nabla)$ we can associate to it a unique connection $H\_{\nabla}$ on $\lambda\_F(\xi)$ : this horizontal distribution can be defined at a point $p \in P$ as
$${H\_{\nabla}}\_{p} = \{Y \in T\_{p}P : \forall v \in V,\, T\_{p}Q\_{v}(Y) \in H(\nabla)\_{[Q\_{v}(p)]}\}.$$
where $Q\_v : P \to E, \ (x,F\_x) \mapsto F\_x(v)$.


**My question** : is there a simple expression relating the given local $\Gamma\_i$ of $\nabla$ and the local 1-form of the induced connection $H\_{\nabla}$ on $\lambda\_F(\xi)$ ?



Connection 1-form of the frame bundle associated to a vector bundle with a connection
I think I have the answer, and if I am not wrong it's so simple that I even regret to have asked this question :
$$ \omega\_i = \Gamma\_i$$
This comes form the fact that the space of linear connection on $\xi$ and the space of principal connection on $\lambda\_F(\xi)$ are affine isomorphic, and the correspondances
$$H \mapsto \nabla^H \quad \text{and} \quad \nabla\mapsto H(\nabla)$$
realizes this isomorphism, once we make the canonical identification
$$\xi \leadsto \lambda\_F(\xi) \leadsto \lambda\_F(\xi) \times\_{\rho} V \simeq \xi$$
Here the structural group of $\lambda\_F(\xi)$ is $GL(V)$ and the representation $\rho$ is the identity. So the induces connection on $\lambda\_F(\xi)$ induces a connection on $\lambda\_F(\xi) \times\_{\rho} V \simeq \xi$ such that in the correspondind trivializations
$$\Gamma\_i = \rho\_\* \omega\_i = \omega\_i$$,
and this $\Gamma\_i$ should be the same than the connection form of the initial linear connection on $\xi$.


