# Einstein Metrics, Projective Structures and the $SU(\infty)$ Toda Equation

## TL;DR

This paper explores the relationship between two-dimensional projective structures with vector fields and solutions to the $SU(
fty)$ Toda equation, providing explicit examples, twistor space constructions, and a link to Einstein metrics.

## Contribution

It establishes a new explicit correspondence between projective structures and Toda solutions, introduces novel solutions, and connects these to Einstein metrics on cotangent bundles.

## Key findings

- Constructed new explicit solutions to the $SU(
abla)$ Toda equation.
- Developed mini-twistor spaces for these solutions.
- Demonstrated a projective-to-Einstein metric correspondence.

## 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 $T^*N$ of any projective structure $(N, [\nabla])$. 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 $T^*N$.