Gravity and duality between coordinates and matter fields

We use the duality between the local Cartezian coordinates and the solution s of the Klein-Gordon equation to parametrize locally the spacetime in term s of wave functions and prepotentials. The components of metric, metric con nection, curvature as well as the Einstein equation are given in this param etrization. We also discuss the local duality between coordinates and quant um fields and the metric in this later reparametrization. (C) 2000 Elsevier Science B.V. All rights reserved.