Let $\pi : E \to \mathbb{R}^3$ define a vector bundle with a connection form $\nabla$ on $\mathbb{R}^3$.
The text that i'm am reading then goes on to say that $\nabla$ is $\mathbb{R}$-invariant in say the $x_1$ direction, that is, invariant under the action of the additive group $\mathbb{R}$ of translations in the $x_1$ direction. I'm unsure exactly what this means.
Writing the connection form in component form as $\nabla =A_1 dx_1 +A_2 dx_2 + A_3 dx_3$ does this mean that each $A_i$ is $x_1$-invariant? And if so, how would you write this mathematically? Like is it equivalent to saying $\nabla_{\frac{\partial}{\partial x_1}} A_i = 0$?