A note on the structure of algebraic curvature tensors
type: Item
instance of: scholarly article
title: A note on the structure of algebraic curvature tensors
zbMATH Open document ID: 1056.53014
author: Eduardo García-Río
author: José Carlos Díaz-Ramos
published in: Linear Algebra and its Applications
publication date: 2004-08-06T00:00:00Z
review text: A \((0,4)\)-tensor on an \(n\)-dimensional vector space~\(V^n\) is an algebraic curvature tensor if it satisfies the algebraic identities of a Riemannian curvature tensor. It is known [\textit{P. B. Gilkey}, Geometric properties of natural operators defined by the Riemann curvature tensor. Singapore: World Scientific (2001; Zbl 1007.53001)] that any algebraic curvature tensor on~\(V^n\) can be written as a linear combination of algebraic curvature tensors built from some symmetric bilinear form on~\(V\). Using the Nash embedding theorem and the Gauss equation, the authors show that a maximum number of \(n(n+1)/2\) symmetric forms suffices, but this estimate is not optimal. As is shown in the paper, the optimal value for \(n=2\) is~\(1\) and for \(n=3\) is~\(2\).
reviewed by: Eric Boeckx
Mathematics Subject Classification ID: 53B20
zbMATH DE Number: 2081915
zbMATH Keywords: algebraic curvature tensor
zbMATH Keywords: normal coordinates
zbMATH Keywords: Nash embedding theorem
zbMATH Keywords: Gauss equation
MaRDI profile type: Publication
full work available at URL: j.laa.2003.12.044
OpenAlex ID: W2097562314
cites work: Determination of the structure of algebraic curvature tensors by means of Young symmetrizers
cites work: Geometric properties of natural operators defined by the Riemann curvature tensor
cites work: Q3024378
cites work: Isometric embeddings
cites work: The imbedding problem for Riemannian manifolds
DOI: 10.1016/J.LAA.2003.12.044
description: scientific article; zbMATH DE number 2081915