Generalized positive scalar curvature on \(\text{spin}^c\) manifolds
type: Item
instance of: scholarly article
title: Generalized positive scalar curvature on \(\text{spin}^c\) manifolds
author: B. I. Botvinnik
author: Jonathan M. Rosenberg
published in: Annals of Global Analysis and Geometry
publication date: 2024-11-25T00:00:00Z
review text: The authors consider a compact \(\text{spin}^c\) manifold \((M,g)\) whose auxiliary line bundle has a connection \(A\). They introduce the notion of generalized scalar curvature which is expressed in terms of the scalar curvature of the manifold and the operator norm of Clifford multiplication \(c(\Omega)\), where \(\Omega\) is the curvature of the connection \(A\). They show that the positivity of the generalized scalar curvature is equivalent to the positivity of a twisted scalar curvature that they have introduced in a previous paper [\textit{B. Botvinnik} and \textit{J. Rosenberg}, J. Reine Angew. Math. 803, 103--136 (2023; Zbl 1530.53065)]. Then, they use the trichotomy theorem of Kazdan-Warner concerning the existence of metrics with positive scalar curvature to get a corresponding result concerning the generalized scalar curvature on \(\text{spin}^c\) manifolds.
reviewed by: Georges Habib
Mathematics Subject Classification ID: 53C21
Mathematics Subject Classification ID: 53C27
Mathematics Subject Classification ID: 58J22
Mathematics Subject Classification ID: 55N22
Mathematics Subject Classification ID: 19L41
zbMATH DE Number: 7948989
zbMATH Keywords: positive scalar curvature
zbMATH Keywords: \(\text{spin}^c\) manifolds
zbMATH Keywords: bordisms
zbMATH Keywords: \(K\)-theory
zbMATH Keywords: index difference
MaRDI profile type: Publication
cites work: Q3995493
cites work: Lower eigenvalue estimates for Dirac operators
cites work: \(\text{Spin}^c\)-structures and \(S^1\)-actions
cites work: Positive scalar curvature on manifolds with fibered singularities
cites work: Scalar curvature and conformal deformation of Riemannian structure
cites work: Q4072462
cites work: A direct approach to the determination of Gaussian and scalar curvature functions
cites work: Spin\(^c\) structures and scalar curvature estimates
cites work: The two definitions of the index difference
cites work: Infinite loop spaces and positive scalar curvature
cites work: Infinite loop spaces and positive scalar curvature in the presence of a fundamental group
cites work: A conformal lower bound for the smallest eigenvalue of the Dirac operator and Killing spinors
cites work: On Spaces Having the Homotopy Type of a CW-Complex
cites work: Generalized Killing spinors and conformal eigenvalue estimates for \(\text{Spin}^c\) manifolds
cites work: Lower bounds for the eigenvalues of the Dirac operator on Spin\(^c\) manifolds
cites work: Existence and conformal deformation of metrics with prescribed Gaussian and scalar curvatures
cites work: Parallel and Killing spinors on \(\text{Spin}^ c\) manifolds
cites work: Almost complex structures on connected sums of complex projective spaces
cites work: Index theory in spaces of manifolds
cites work: The index of elliptic operators. I
DOI: 10.1007/S10455-024-09977-6
description: scientific article; zbMATH DE number 7948989