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<p>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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\({{\mathbb {S}}}^4\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">S</mi> </mrow> <mn>4</mn> </msup> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathbb {C}\mathbb {P}^2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">C</mi> <msup> <mrow> <mi mathvariant="double-struck">P</mi> </mrow> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation>, 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.</p>

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A note on Einstein metrics and Riemannian twistor spaces

  • 期刊论文

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  • 发表日期：

    2026年7月30日

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  • Davide Dameno

摘要

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.