A characterization of the dependence of the Riemannian metric on the curvature tensor by Young symmetrizers
type: Item
instance of: scholarly article
title: A characterization of the dependence of the Riemannian metric on the curvature tensor by Young symmetrizers
zbMATH Open document ID: 0904.53030
DOI: 10.4171/ZAA/813
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 1999-01-27T00:00:00Z
review text: The first method for the determination of a Riemannian metric from the components of the curvature tensor in normal coordinates was given by Herglotz. The so-called Herglotz relations are nonlinear differential equations and give (at least in the analytic case) the power series expansion of the metric. The related coefficients are determined by the symmetrized partial derivatives of the curvature tensor. Later on, P. Günther found a power series expansion of the metric with explicit coefficients which depend on covariant derivatives of the curvature tensor symmetrized in a certain way. In the paper under review, the author studies the difference between both kind of symmetrizations. For this reason the representation theory of the symmetric group \({\mathcal S}_r\) is used. The partial derivatives \(\partial_{i_1}\dots\partial_{i_r} R_{ijkl}\) induce group ring elements belonging to a direct sum \(J_{(r)}\oplus\widehat J_{(r)} \oplus \breve J_{(r)}\) of three minimal left ideals of the group ring \(\mathbb{C} [{\mathcal S}_{r+4}]\). Then the symmetrized partial derivatives \(\partial_{(i_1}\dots\partial_{i_r} R_{| a| i_{r+1} i_{r+2})b}\) correspond to a linear mapping of the direct sum into \(J_{ (r)} \cdot \varepsilon\), \(\varepsilon\in \mathbb{C} [{\mathcal S}_{r+4}]\) which maps \(\widehat J_{(r)} \oplus \breve J_{(r)}\) to 0. For the symmetrized covariant derivatives \(\nabla_{(i_1} \dots \nabla_{i_r)} R_{ijkl}\) and \(\nabla_{(i_1}\dots\nabla_{i_r} R_{| a| i_{r+1} i_{r+2})b}\) only the ideals \(J_{(r)}\) and \(J_{(r)} \cdot \varepsilon\) do appear. So, the inverse mapping \(J_{(r)} \cdot \varepsilon \to J_{(r)}\) gives a relation between both kinds of symmetrized covariant derivatives. This shows \({\mathcal R}= {\mathcal R}^s\), where \({\mathcal R}\) is the algebra of tensor polynomials generated by \(\nabla_{(i_1} \dots \nabla_{i_r)} R_{ijkl}\) and \({\mathcal R}^s\) is the algebra generated by \(\nabla_{(i_1} \dots \nabla_{i_r} R_{| k| i_{r+1} i_{r+2})l}\)…
Mathematics Subject Classification ID: 53C21
Mathematics Subject Classification ID: 05E10
Mathematics Subject Classification ID: 53A55
zbMATH DE Number: 1129194
zbMATH Keywords: algebras of tensor polynomials
zbMATH Keywords: symmetrization
zbMATH Keywords: Herglotz relations
zbMATH Keywords: partial derivatives of the curvature tensor
Wikidata QID: Q115211680
author: Bernd Fiedler
describes a project that uses: PERMS
describes a project that uses: MathTensor
MaRDI profile type: Publication
cites work: Q5719436
cites work: A use of ideal decomposition in the computer algebra of tensor expressions
cites work: Normal forms for tensor polynomials. I. The Riemann tensor
cites work: Spinorkalkül und Normalkoordinaten
cites work: Q4040088
cites work: Q3952291
cites work: Representations of permutation groups. Part I
cites work: Q4887322
cites work: Q5797085
cites work: Q3947818
cites work: Q4273962
cites work: Q5772226
full work available at URL: https://doi.org/10.4171/zaa/813
OpenAlex ID: W2012242895
description: scientific article; zbMATH DE number 1129194