The space of negative scalar curvature metrics
type: Item
instance of: scholarly article
title: The space of negative scalar curvature metrics
zbMATH Open document ID: 0771.58006
DOI: 10.1007/BF01231339
author: Joachim Lohkamp
published in: Inventiones Mathematicae
publication date: 1993-05-16T00:00:00Z
full work available at URL: https://eudml.org/doc/144057
review text: Let \(M\) be a closed \(C^ \infty\)-manifold, let \(S^ -(M)\) be the space of Riemannian metrics on \(M\) whose scalar curvature is strictly negative, equipped with the \(C^ \infty\)-Fréchet topology. The main result of the paper is Theorem 1. \(\pi_ i(S^ -(M))=0\) for \(i=0,1,2,\dots\). Combining Theorem 1 with a general result on infinite dimensional topology due to Palais and Whitehead one gets Theorem 2. \(S^ -(M)\) is contractible. Let \(S_{-1}(M)\) be the space of Riemannian metrics whose scalar curvature \(\equiv -1\). After constructing a continuous map \(p:S^ - (M)\to S_{-1}(M)\) with \(p|_{S_{-1}(M)}=id\) one concludes Corollary. \(S_{-1}(M)\) is also contractible.
Mathematics Subject Classification ID: 58D17
zbMATH DE Number: 167401
zbMATH Keywords: negative scalar curvature
zbMATH Keywords: conformal deformation
zbMATH Keywords: conformal Laplacian
reviewed by: Christian Bär
MaRDI profile type: Publication
cites work: Métriques riemanniennes et courbure
cites work: Scalar curvature and conformal deformation of Riemannian structure
cites work: Q3995493
cites work: Q4340768
cites work: Homotopy theory of infinite dimensional manifolds
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description: scientific article; zbMATH DE number 167401