One component of the curvature tensor of a Lorentzian manifold
type: Item
instance of: scholarly article
title: One component of the curvature tensor of a Lorentzian manifold
zbMATH Open document ID: 1203.53043
author: Anton Galaev
published in: Journal of Geometry and Physics
publication date: 2010-06-02T00:00:00Z
full work available at URL: https://arxiv.org/abs/1001.4441
review text: The basic local invariant of a (pseudo)-Riemannian manifold \((M^n,g)\) is the Riemannian curvature tensor \(R\). The values of \(R\) at a point are contained in the holonomy algebra \(\mathfrak{g}\) at that point. In Riemannian signature the algebraic spaces \(\mathcal{R}(\mathfrak{g})\) of curvature tensors of type \(\mathfrak{g}\) are known for any \(\mathfrak{g}\). The present article aims to compute the \(\mathcal{R}(\mathfrak{g})\) for the holonomy algebras \(\mathfrak{g}\) of a Lorentzian manifold. In the Lorentzian case \(\mathcal{R}(\mathfrak{g})\) is derived from \(\mathcal{R}(\mathfrak{h})\) and \(\mathcal{P}(\mathfrak{h})\), where \(\mathfrak{h}\subset\mathfrak{so}(n)\) is the projection of \(\mathfrak{g}\subset\mathfrak{so}(1,n+1)\) to \(\mathfrak{so}(n)\). This again is known to be a holonomy algebra of a Riemannian manifold, i.e., a Berger algebra. Hence, \(\mathcal{R}(\mathfrak{h})\) is known and it remains to compute the component \[ \mathcal{P}(\mathfrak{h})\;:=\{P\in(\mathbb{R}^n)^*\otimes \mathfrak{h}\;|\;(P(u)v,w)+(P(v)w,u)+(P(w)u,v)=0\;\forall u,v,w\in\mathbb{R}^n\}. \] Going through the list of irreducible Berger subalgebras \(\mathfrak{h}\) of \(\mathfrak{so}(n,\mathbb{C})\), the author calculates the spaces \(\mathcal{P}(\mathfrak{h})\) explicitly as \(\mathfrak{h}\)-modules. In particular, he obtaines the \(\mathcal{P}(\mathfrak{h})\) for the symmetric Berger algebras and the exceptional algebras \(\mathfrak{spin}(7)\) and \(\mathfrak{g}_2\). The case \(\mathfrak{h}\subset\mathfrak{u}(n/2)\) was known before already.
reviewed by: Felipe Leitner
Mathematics Subject Classification ID: 53C29
Mathematics Subject Classification ID: 53C50
Mathematics Subject Classification ID: 53B30
zbMATH DE Number: 5716413
zbMATH Keywords: Lorentzian manifolds
zbMATH Keywords: holonomy algebra
zbMATH Keywords: curvature tensor
zbMATH Keywords: Einstein manifolds
describes a project that uses: LiE
MaRDI profile type: Publication
OpenAlex ID: W2091149855
arXiv ID: 1001.4441
cites work: Q3142327
cites work: On the classification of Lorentzian holonomy groups
cites work: METRICS THAT REALIZE ALL LORENTZIAN HOLONOMY ALGEBRAS
cites work: Q3517575
cites work: Supersymmetry, holonomy and Kundt spacetimes
cites work: Metrics with vanishing quantum corrections
cites work: SUPERGRAVITY SOLUTIONS WITH CONSTANT SCALAR INVARIANTS
cites work: Kundt spacetimes
cites work: Breaking the sanserifM-waves
cites work: Time-dependent multi-centre solutions from new metrics with holonomy Sim( n − 2)
cites work: Holonomy Old and New
cites work: Holonomy groups and spacetimes
cites work: The spaces of curvature tensors for holonomy algebras of Lorentzian manifolds
cites work: Q5563881
cites work: Q4720334
cites work: Q3426908
cites work: Holonomy of supermanifolds
cites work: Irreducible complex skew-Berger algebras
cites work: Classification of irreducible holonomies of torsion-free affine connections
cites work: Connections with irreducible holonomy representations
cites work: Ricci-flat holonomy: a classification
cites work: On the de Rham decomposition theorem
cites work: Sur les groupes d'holonomie homogènes de variétés à connexion affine et des variétés riemanniennes
cites work: Q4358569
cites work: Lectures on exceptional Lie groups. Ed. by Zafer Mahmud and Mamoru Mimura
Wikidata QID: Q125338010
DOI: 10.1016/J.GEOMPHYS.2010.03.002
Recommended article: The Riemann-Lovelock curvature tensor
Recommended article: A type of conformal curvature tensor
Recommended article: CURVATURE TENSORS IN A LORENTZIAN PARA SASAKIAN MANIFOLD
Recommended article: Q4234701
Recommended article: Curvature tensor for the spacetime of general relativity
Recommended article: Lorentzian manifolds and scalar curvature invariants
Recommended article: Lorentzian Manifolds: A characterization with semiconformal curvature tensor
Recommended article: Lorentzian Manifolds
Recommended article: Q4298860
Recommended article: Curvature tensor under the Ricci flow
description: scientific article