A note on Einstein metrics and Riemannian twistor spaces

Key Points

An einstein four-manifold with a scalar curvature constant on the fibers is half conformally flat.
The compact einstein four-manifolds with positive scalar curvature that fit this condition are specifically $ ext{S}^4$ and $ ext{CP}^2$.
A generalization of Friedrich and Grunewald's result offers a classification of complete four-manifolds with ricci parallel twistor spaces.
The findings imply constraints on the structure of einstein manifolds within the context of scalar curvature.

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 S⁴ and CP². 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.

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Cite This Study

Davide Dameno (Mon,) studied this question.

synapsesocial.com/papers/68d473b531b076d99fa6c602 https://doi.org/https://doi.org/10.48550/arxiv.2507.16113

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