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 S U ( ∞ ) 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 gives a 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 .