A note on Einstein metrics and Riemannian twistor spaces

TL;DR

This paper investigates Einstein four-manifolds with specific twistor space properties, showing they are either the 4-sphere or complex projective plane, and classifies those with Ricci parallel twistor spaces.

Contribution

It proves that certain Einstein four-manifolds with constant scalar curvature on fibers are half conformally flat and classifies complete four-manifolds with Ricci parallel twistor spaces.

Findings

Only $ ext{S}^4$ and $ ext{CP}^2$ satisfy the twistorial condition among compact Einstein four-manifolds.

Classified complete four-manifolds with Ricci parallel twistor spaces.

Established a link between twistor space properties and conformal flatness.

Abstract

Inspired by the problem of classifying Einstein manifolds with positive scalar curvature, we prove that an Einstein four-manifold whose associated twistor space has scalar curvature constant on the fibers of the twistor bundle is half conformally flat: in particular, the only compact Einstein four-manifolds with positive scalar curvature satisfying this twistorial condition are $\mathbb{S}^4$ and $\mathbb{CP}^2$. We also generalize a well-known result due to Friedrich and Grunewald, providing a classification of complete four-manifolds whose twistor space is Ricci parallel.