The scalar curvature on totally geodesic fiberings
type: Item
instance of: scholarly article
title: The scalar curvature on totally geodesic fiberings
zbMATH Open document ID: 0981.53020
DOI: 10.1023/A:1006644823883
author: Yanyan Li
published in: Annals of Global Analysis and Geometry
publication date: 2002-03-11T00:00:00Z
review text: A compact Riemannian manifold \(N\) with scalar curvature \(\kappa _n\) is said to satisfy a comparison theorem for the scalar curvature iff for any other compact Riemannian manifold \(M\) (\(\dim M = \dim N\)) the inequality \(\kappa _M(x)\leq\kappa _N(f(x))\) holds at some \(x\in M\) whenever \(f:M\to N\) is a vector contracting spin map of non-zero degree. Rigidity holds if \(\kappa_M(x)\geq\kappa _N(f(x))\) for all \(x\) implies that \(f\) is an isometry. The main result reads as follows: If \(\pi :\tilde N\to N\) is a Riemannian submersion with totally geodesic fibres and \(\tilde N\) satisfies the above comparison theorem with rigidity, then \(N\) itself satisfies these conditions.
Mathematics Subject Classification ID: 53C20
zbMATH DE Number: 1563296
zbMATH Keywords: Riemannian submersion
zbMATH Keywords: scalar curvature
zbMATH Keywords: rigidity
reviewed by: Paweł Walczak
MaRDI profile type: Publication
full work available at URL: a:1006644823883
OpenAlex ID: W1970675086
description: scientific article; zbMATH DE number 1563296