I will describe a link between Riemannian geometry in 4 dimensions and a certain class of 6-dimensional symplectic manifolds. The bridge is given by “definite connections”. These are SO(3)-connections over 4-manifolds whose curvature is non-zero on every 2-plane. They are a special case of Weinstein’s “fat connections”. I will explain how a definite connection over M^4 can be seen as a “potential” for both a symplectic structure on a 2-sphere bundle over M (Weinstein’s original construction) and a Riemannian metric on M itself. This leads to a new variational description of Einstein metrics. It also opens the way to using symplectic techniques to prove things in Riemannian geometry and vice versa. I will describe some results which have been proved this way, as well as some hopes and dreams for the future. Various parts of this are joint with Dima Panov and Kirill Krasnov.