On metrics satisfying equation \(R_{ij}-\frac{1}{2}Kg_{ij}=T_{ij}\) for constant tensors \(T\)
type: Item
instance of: scholarly article
title: On metrics satisfying equation \(R_{ij}-\frac{1}{2}Kg_{ij}=T_{ij}\) for constant tensors \(T\)
zbMATH Open document ID: 1006.53504
DOI: 10.1016/S0393-0440(01)00043-2
author: Romildo Pina
author: Keti Tenenblat
published in: Journal of Geometry and Physics
publication date: 2002-06-23T00:00:00Z
review text: The interesting problem is to find necessary and sufficient conditions on a symmetric tensor \(T_{ij}\) on a compact manifold so that one can find a metric \(g_{ij}\) satisfying \(R_{ij}- \frac 12 Kg_{ij}= T_{ij}\), where \(R_{ij}\) is the Ricci tensor and \(K\) the scalar curvature of \(g_{ij}\). The authors give a necessary and sufficient condition for a constant symmetric tensor \(T_{ij}\) on \(\mathbb{R}^n\), \(n\geq 3\) for which there exist metrics \(\overline g\), conformal to a pseudo-Euclidean metric \(g\), such that \(\overline{R_{ij}} -\frac 12 \overline K\overline{g_{ij}} =T_{ij}\), where \(\overline{R_{ij}}\) and \(\overline{K}\) are the Ricci tensor and the scalar curvature of \(\overline g\). All solutions \(\overline g\) are given explicitly and it is shown that there are no complete metrics \(\overline g\) conformal and nonhomothetic to \(g\).
Mathematics Subject Classification ID: 53C21
Mathematics Subject Classification ID: 53C80
Mathematics Subject Classification ID: 53C50
zbMATH DE Number: 1758075
zbMATH Keywords: Ricci tensor
zbMATH Keywords: compact manifold
zbMATH Keywords: scalar curvature
MaRDI profile type: Publication
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Wikidata QID: Q126351790
full work available at URL: s0393-0440(01)00043-2
OpenAlex ID: W2041473294
description: scientific article; zbMATH DE number 1758075