Connections and covariant derivatives

Question

Let $A$ be a connection on a principal $G$-bundle $P$, let $\chi :G\rightarrow GL(V)$ be a representation of $G$, and let $E:=P\times _\chi V$ be the associated gauge bundle. Then, there is a covariant derivative on $E$ corresponding to the connection $A$ on $P$. What property uniquely characterizes this covariant derivative? In what way does it 'mesh' with the structure of the connection A on P? Sure, you can define this covariant derivative, but covariant derivatives exist anyways. What's so special about this one? (I'm looking for something along the lines of the properties that characterize the Levi-Civita covariant derivative: given a metric $g_{ab}$ on a manifold, there is a unique torsion-free covariant derivative $\nabla _a$ on the tangent bundle such that $\nabla _ag_{bc}=0$.)

Answer 1

Let $d_A$ denote the covariant derivative on $P$ induced by $A$. Then $d_A$ is a map $$d_A: \Omega^0_G(P; V) \longrightarrow \Omega^1_G(P; V),$$ where $\Omega^k_G(P; V)$ denotes the space of basic $V$-valued $k$-forms on $P$. Now for each $k$ there is an isomorphism $$\psi_k: \Omega^k_G(P; V) \xrightarrow{~\cong~} \Omega^k(M; P \times_\chi V),$$ where $M$ is the base space of $P$. Then the induced covariant derivative $\nabla^A$ on $P \times_\chi V$ induced by $A$ and $\chi$ is the map making the following diagram commute: $$\require{AMScd} \begin{CD} \Omega^0_G(P; V) @>{d_A}>> \Omega^1_G(P; V) \\ @V{\psi_0}VV @VV{\psi_1}V \\ \Omega^0(M; P \times_\chi V) @>>{\nabla^A}> \Omega^1(M; P \times_\chi V). \end{CD}$$