A Note on Riemannian Anti-self-dual Einstein metrics with Symmetry

2006/09/11 by Paul Tod, Tod, Paul · 1 citation
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #FOS: Physical sciences #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #High Energy Physics - Theory (hep-th)

paper · pdf · doi:10.48550/arxiv.hep-th/0609071

openalex publication_date 2006/09/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We give a complete proof of the result stated in an earlier article, that the general Einstein metric with a symmetry, an anti-self-dual Weyl tensor and nonzero scalar curvature is determined by a solution of the SU(∞)-Toda field equation. We consider the two canonical forms found for solutions to the same problem by Przanowski (J.Math.Phys. 32(1991) 1004-1010) and show that his Class A will reduce to the Toda equation with respect to a second complex structure, different from that in which the metric is first given.

Cited by

• Poincaré-Einstein 4-manifolds with conformally Kähler geometry

Related

• Einstein Metrics, Projective Structures and the SU(∞) Toda Equation
• The General Self-dual solution of the Einstein Equations
• One-sided type-D Ricci-flat metrics
• One-sided type-D metrics with aligned Einstein-Maxwell
• Anti-self-dual gravity from asymmetric heavenly equation standpoint