Einstein Metrics, Projective Structures and the $SU(\infty )$ 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 $SU(\infty )$ 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 $T^{\ast }N$ of any projective structure $(N,[\nabla ])$. We show that there is a canonical Einstein of metric on an ${\mathbb{R}}^{\ast }$--bundle over $T^{\ast }N$, with a connection whose curvature is the pull--back of the natural symplectic structure from $T^{\ast }N$.

Dedicated to Joseph Krasil'shchik on the occasion of his 70th birthday. Final version, to appear in JGP