A note on Einstein metrics and Riemannian twistor spaces

2025/07/21 by Dameno, Davide
#Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2507.16113

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 \mathbbS⁴ and \mathbbCP². 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.

Citations

• Rigidity of Einstein manifolds with positive Yamabe invariant
• Rigidity results for Riemannian twistor spaces under vanishing curvature conditions
• On Einstein Manifolds of Positive Sectional Curvature

Related

• Rigidity results for Riemannian twistor spaces under vanishing curvature conditions
• Moving frames on the twistor space of self-dual positive Einstein 4-manifolds
• On Riemannian four-manifolds and their twistor spaces: a moving frame\n approach
• Curvature properties of the Chern connection of twistor spaces
• Four-dimensional Einstein manifolds with sectional curvature bounded from above