Einstein metrics, projective structures and the SU (∞) Toda equation

Abstract

We establish an explicit correspondence between two–dimensional projective structures admitting a projective vector field, and a class of solutions to the SU (∞) Toda equation.We give several examples of new, explicit solutions of the Toda equation, and construct their mini–twistor spaces. Finally we discuss the projective-to-Einstein correspondence, which givesa neutral signature Einstein metric on a cotangent bundle T∗ N of any projective structure (N,[∇]). We show that there is a canonical Einstein of metric on an R∗–bundle over T∗ N, with a connection whose curvature is the pull–back of the natural symplectic structure from T∗ N