Sharp estimates and the Dirac operator
type: Item
instance of: scholarly article
title: Sharp estimates and the Dirac operator
zbMATH Open document ID: 0895.53037
author: Marcelo Llarull
published in: Mathematische Annalen
publication date: 1998-02-01T00:00:00Z
review text: The following statement is proved: assume that \(M\) is a compact Riemannian spin manifold of dimension \(n\) and assume that there is a distance decreasing map \(f: (M,g) \rightarrow (S^n, g_0)\) of non-zero degree onto the standard sphere of normalized scalar curvature 1. Then either there is a point on \(M\) with scalar curvature \(< 1\) or \(f\) is an isometry of the standard sphere. In the case that \(f\) is the identity map of the sphere this settles a conjecture of M. Gromov. The idea of the proof is to pull back a suitable spinor bundle on the sphere to \(M\) and to use it to twist a similar spinor bundle on \(M\). Then a Bochner type argument and the Atiyah-Singer index theorem are used.
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 53C21
zbMATH DE Number: 1111626
zbMATH Keywords: Dirac operator
zbMATH Keywords: scalar curvature
zbMATH Keywords: distance decreasing map
zbMATH Keywords: twisted spinor bundle
zbMATH Keywords: Atiyah-Singer index theorem
reviewed by: Hans-Bert Rademacher
MaRDI profile type: Publication
full work available at URL: s002080050136
OpenAlex ID: W2036523184
DOI: 10.1007/S002080050136
description: scientific article; zbMATH DE number 1111626