In studying connections on vector bundles, I am trying to find non trivial examples of connections over an elliptic curve, over $\Bbb P^1_k$ and $\Bbb A^1_k$ for a field $k$.
By a connection on a vector bundle $E$ on a scheme $X$, I mean a morphism $\nabla :E \mapsto E\otimes \Omega^1$ such that $\forall f,e \in \mathcal O_X\times E: \nabla(fe) = f \nabla(e) + df \otimes e$.
Thank you for your help, answers and references.