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

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 $SU(\infty )$-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.

arXiv comment9 pages

Cited by in corpus (12)

• Twistor Approach to String Compactifications: a Review
• Self-dual Einstein Spaces, Heavenly Metrics and Twistors
• Geometry of Killing spinors in neutral signature
• Killing-Yano tensor and supersymmetry of the self-dual Plebanski-Demianski solution
• Killing Symmetries in $\mathcal{H}$-Spaces with $\Lambda$
• Heisenberg symmetry and hypermultiplet manifolds
• Isometries, gaugings and N=2 supergravity decoupling
• Euclidean supersymmetric solutions with the self-dual Weyl tensor
• Membrane Instantons from Toda Field Theory
• One-sided type-D Ricci-flat metrics
• Homothetic Killing Vectors in Expanding HH-Spaces with Lambda
• Einstein Metrics, Projective Structures and the $SU(\infty )$ Toda Equation