Almost Einstein and Poincaré-Einstein manifolds in Riemannian signature
type: Item
instance of: scholarly article
title: Almost Einstein and Poincaré-Einstein manifolds in Riemannian signature
zbMATH Open document ID: 1194.53038
author: Rod Gover
published in: Journal of Geometry and Physics
publication date: 2010-02-26T00:00:00Z
full work available at URL: https://arxiv.org/abs/0803.3510
review text: A real function \(s\) on a Riemannian manifold \((M,g)\) is called an almost Einstein structure if the trace-free part of \[ \text{Hess}^g(s)\;+\;sP^g \] vanishes identically, where \(P^g\) denotes the Schouten tensor. This PDE is conformally covariant and for any solution \(s\) the metric \(s^{-2}g\) in the conformal class \([g]\) on \(M\) is Einstein. However, \(s\) might have zeros. Thus, in general, the underlying conformal manifold \((M,[g])\) is called almost Einstein (=AE). This extends the notion of Einstein spaces to conformal geometry. The idea for this comes from tractor calculus. In his work the author describes the singular set \(\Sigma\) for an AE structure \(s\). The notion of almost scalar constant spaces is introduced as well. In the negative scalar curvature case any singular set \(\Sigma\neq\emptyset\) is a smooth hypersurface and the model of AE manifolds is closely linked to the Poincaré-Einstein model. The author compares the conformal geometry of \(\Sigma\) and \(M\). Moreover, he derives curvature conditions for the existence of AE structures. This relates to Dirichlet type problems of finding AE structures on an ambient space for a given scale singularity \(\Sigma\). Also, the relation of the Fefferman-Graham ambient spaces for \(\Sigma\) and \(M\) is discussed.
reviewed by: Felipe Leitner
Mathematics Subject Classification ID: 53C25
Mathematics Subject Classification ID: 53A30
Mathematics Subject Classification ID: 53C20
zbMATH DE Number: 5675292
zbMATH Keywords: Poincaré-Einstein manifolds
zbMATH Keywords: Einstein manifolds
zbMATH Keywords: conformal differential geometry
MaRDI profile type: Publication
Wikidata QID: Q115353345
OpenAlex ID: W2962921685
arXiv ID: 0803.3510
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DOI: 10.1016/J.GEOMPHYS.2009.09.016
description: scientific article