Curvature \(h\)-principles
type: Item
instance of: scholarly article
title: Curvature \(h\)-principles
zbMATH Open document ID: 0909.58005
DOI: 10.2307/2118552
author: Joachim Lohkamp
published in: Annals of Mathematics. Second Series
publication date: 1999-03-21T00:00:00Z
review text: This paper derives some properties of the spaces of all metrics on a differentiable manifold \(M\) which satisfy curvature bounds. In particular, the spaces \(\text{Ric}^{<\alpha}(M)\) and \(S^{<\alpha}(M)\) of metrics on \(M\) with, respectively, Ricci curvature less than \(\alpha\) and scalar curvature less than \(\alpha\) for \(\alpha\in\mathbb{R}\) are investigated. Theorem A says that these spaces are contractible, while Theorem B is an approximation result that these spaces are dense in the space of all metrics in the \(C^0\) and the Hausdorff topologies. Thus, any metric is \(C^0\) close to a metric of arbitrarily negative Ricci curvature. Theorems C and D are local versions of these results.
Mathematics Subject Classification ID: 58D17
Mathematics Subject Classification ID: 53C21
zbMATH DE Number: 837029
zbMATH Keywords: space of metrics
zbMATH Keywords: differentiable manifold
zbMATH Keywords: curvature bounds
reviewed by: Peter W. Michor
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2307/2118552
OpenAlex ID: W2123908708
description: scientific article; zbMATH DE number 837029