Consider a connection $d$ on a principle bundle $E\to M$. It gives gives you a curvature by the usual $$ d^2s=Fs \quad \forall s\in \Gamma(E) $$
I was wondering if the connection is completely determined by $F$.
For an abelian group it looks like it's true, but in general it seems a difficult problem.
This is actually for physics so you could use YM's equation $d\star F=0$ but I don't believe it is needed.
pushing a little further, I wanted also to know if this works for GR, is the connection completely constrained by the Riemman Tensor?