Anti-self-dual Hermitian metrics and Painlevé III
type: Item
instance of: scholarly article
title: Anti-self-dual Hermitian metrics and Painlevé III
zbMATH Open document ID: 1076.53059
author: Shoji Okumura
published in: Osaka Journal of Mathematics
publication date: 2005-02-18T00:00:00Z
review text: Cohomogeneity-one metrics with an \(\text{SU}(2)\) isometry group transitive on codimension one surfaces with self-dual Weyl tensor have been studied by many authors. It was shown that the self-dual Einstein equations for such metrics reduce to a particular Painlevé VI equation. Painlevé III equations are degenerated from Painlevé VI. In this paper, the author studies the \(\text{SU}(2)\)-invariant anti-self-dual metrics which are specified by the solutions of Painlevé III. Diagonal and non-diagonal metrics are studied. The author shows that the metric is specified by a solution of Painlevé III if and only if there exists an \(\text{SU}(2)\)-invariant Hermitian structure.
reviewed by: Andrew Bucki
Mathematics Subject Classification ID: 53C25
Mathematics Subject Classification ID: 32L25
Mathematics Subject Classification ID: 34C30
zbMATH DE Number: 2134867
zbMATH Keywords: anti-self-dual metric
zbMATH Keywords: Painlevé equation
zbMATH Keywords: Hermitian metric
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2134867