A note on Einstein metrics and Riemannian twistor spaces
Abstract
Abstract Inspired by the problem of classifying four-dimensional Einstein manifolds with positive scalar curvature, we prove that an Einstein four-manifold whose associated twistor space has fibrewise constant scalar curvature is half conformally flat: in particular, the only compact Einstein four-manifolds with positive scalar curvature satisfying this twistorial condition are $${{\mathbb {S}}}^4$$ S 4 and $$\mathbb {C}\mathbb {P}^2$$ C P 2 , both endowed with their standard metrics. 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.
Fields of Study & Subjects
Geometric Analysis and Curvature Flows
Advanced Differential Geometry Research
Geometry and complex manifolds
Publication details
- Journal / source
- Mathematische Zeitschrift
- Publisher
- Springer Science and Business Media LLC
- Volume
- 313
- Issue
- 4
- Publication date
- 30 July 2026
- Language
- EN