Abstract

In this seminar, I review the various curvature conditions that one might wish to impose on a Bianchi-type IX metric, and the direct route to the self-dual Einstein metrics obtained from solutions of the Painlevé VI equation.

In four-dimensions there are several ‘nice’ conditions which one may impose on a Riemannian metric. For example, one may require it to be Kähler, or Einstein, or to have anti-self-dual (ASD) Weyl tensor. These possibilities are set out in figure 5.1, which I have used before (Tod 1995, 1997). The overlapping regions in the figure also correspond to interesting conditions: a Kähler metric with zero-scalar curvature (‘scalar-flat’ in the terminology of (LeBrun 1991)) has ASD Weyl tensor; ASD Einstein is known as quaternionic Kähler; ASD Ricci-flat is hyper-Kähler. Elsewhere in the top circle of the diagram are hyper-complex metrics, which have three integrable complex structures with the algebra of the (unit, complex) quaternions but are not Kähler.

Conditions on the curvature are most readily imposed using Cartan calculus, so suppose that (e0, e1, e2, e3) is a normalised basis of 1-forms in some Riemannian 4-manifold.