Metrics compatible with a symmetric connection in dimension three
type: Item
instance of: scholarly article
title: Metrics compatible with a symmetric connection in dimension three
zbMATH Open document ID: 0861.53013
DOI: 10.1016/0393-0440(95)00014-3
published in: Journal of Geometry and Physics
publication date: 1997-05-12T00:00:00Z
review text: The paper is dealing with the problem of characterizing symmetric connections \(\nabla\) that are Levi-Civita connections of a (pseudo-Riemannian) metric \(g\), mainly in the case of 3-dimensional manifolds. After reexamining the classical prescriptive solution of the problem, the main result provides necessary and sufficient conditions for the existence of a metric \(g\) compatible with a given \(\nabla\). They are expressed in terms of the curvature tensor \(R\), the Ricci tensor \(K\), commutators of \(R\) and \(\nabla R\), commutators of \(R\) and \(\nabla^2 R\), the endomorphism \[ \nabla_V \nabla_W R(X,Y)-\nabla_{\nabla_WV} R(X,Y), \] where \(X\), \(Y\), \(V\), \(W\) are arbitrary vector fields of the manifold under consideration. In particular, the second derivatives of \(R\) characterize the exceptional metrics. An explicit metric satisfying these conditions is also given. It should be noted that the problem has two main cases according to the rank of \(K\) (1 or 2).
reviewed by: E. Vassiliou
Mathematics Subject Classification ID: 53B05
Mathematics Subject Classification ID: 53B20
Mathematics Subject Classification ID: 53B30
zbMATH DE Number: 894314
zbMATH Keywords: Levi-Civita connections
author: Gerard Thompson
MaRDI profile type: Publication
full work available at URL: 0393-0440(95)00014-3
OpenAlex ID: W2010014568
cites work: Q4720334
cites work: Conditions for the local existence of metric in a generic affine manifold
cites work: Sufficient conditions for a curvature tensor to be Riemannian and for determining its metric
cites work: Conditions for a symmetric connection to be a metric connection
cites work: Connections and symmetries in spacetime
cites work: Invariant characterizations of two-dimensional affine and metric spaces
cites work: Non-uniqueness of the metric in Lorentzian manifolds
cites work: Conditions on a connection to be a metric connection
cites work: Normal form of a metric admitting a parallel field of planes
cites work: CANONICAL FORM FOR A RIEMANNIAN SPACE WITH A PARALLEL FIELD OF NULL PLANES
description: scientific article; zbMATH DE number 894314