# Connection 1-form of the frame bundle associated to a vector bundle with a connection

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 $\lambda_F(\xi)$ ?

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 $\xi$.