On invariant operations on a manifold with connection or metric
type: Item
instance of: scholarly article
title: On invariant operations on a manifold with connection or metric
zbMATH Open document ID: 0765.53019
author: Jan Slovák
published in: Journal of Differential Geometry
publication date: 1993-04-01T00:00:00Z
review text: The paper starts with a readable survey of the classical reduction theorems for some natural operators on connections and Riemannian metrics and of the nonlinear Peetre theorem, which implies the finiteness of the order in several interesting cases. Let \(Q_ \tau P^ 1\), \(S_ +^ 2 T^*\), \(T^{(0,r)}\) or \(\Lambda T^*\) denote the natural bundle of symmetric linear connections, of Riemannian metrics, of \(r\)-times contravariant tensor fields or of exterior forms, respectively. The author first deduces that all natural operators \(Q_ \tau P^ 1\times T^{(0,r)}\to\Lambda T^*\), \(r>0\), are generated by the Chern forms, the alternation and the exterior differentiation. Then it is proved that there is no natural operator \(S_ +^ 2 T^*\times T^{(0,r)}\to\Lambda T^*\), \(r>0\), with positive weight and all conformal operators of this type are generated by the Pontryagin forms, the alternation and the exterior differentiation. The cases \(r=0\) are discussed separately.
Mathematics Subject Classification ID: 53C05
Mathematics Subject Classification ID: 58A20
zbMATH DE Number: 146254
zbMATH Keywords: nonlinear Peetre theorem
zbMATH Keywords: natural operators
zbMATH Keywords: Chern forms
zbMATH Keywords: exterior differentiation
zbMATH Keywords: Pontryagin forms
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214453183
OpenAlex ID: W1506807058
Wikidata QID: Q115180441
DOI: 10.4310/JDG/1214453183
reviewed by: Ivan Kolář
description: scientific article; zbMATH DE number 146254