The Indefinite Self-Dual Metrics and Painleve Equations

TL;DR

This paper classifies certain SU(2)-invariant anti-self-dual metrics with indefinite signature, linking them to solutions of Painleve equations and elucidating their geometric significance.

Contribution

It provides a classification of these metrics based on Painleve equations and explains their geometric interpretation, advancing understanding of indefinite self-dual geometries.

Findings

Metrics are characterized by solutions to Painleve VI, V, III, or II.

Each Painleve type corresponds to a specific geometric structure.

The geometric meaning of each Painleve solution type is clarified.

Abstract

We classify the SU(2)-invariant anti-self-dual metrics with a signature (+,+,-,-). The metrics are specified by a solution of Painleve VI, V, III or II. Moreover we show the geometric meaning of the metrics specified by each type of Painlev\'e functions.