The Atiyah Singer index theorem and Chern Weil forms
type: Item
instance of: scholarly article
title: The Atiyah Singer index theorem and Chern Weil forms
zbMATH Open document ID: 1222.58015
author: James Simons
author: Dennis Sullivan
published in: Pure and Applied Mathematics Quarterly
publication date: 2011-06-27T00:00:00Z
review text: A real even total cohomology class \(C\) on a smooth manifold \(M\) satisfies the Atiyah Singer Index Property if \([f^*C\cdot T(V), V]\) is an integer where \(f:V\to M\) is a mapping from a closed stably almost complex manifold \(V\) into \(M\), \(T(V)\) is the Todd genus of \(V\) and \([,]\) is the pairing between homology and cohomology. This article characterizes closed forms occurring as Chern Weil forms of connections on complex vector bundles with compact or arbitrary holonomy over \(M\) by using the Atiyah Singer Index Property. A real even total closed form is the total Chern Weil form of a complex bundle with a connection having compact holonomy if and only if its cohomology class satisfies the Atiyah Singer Index Property. A complex even total closed form is the total Chern Weil form of a complex bundle with a connection if and only if its cohomology class is real and satisfies the Atiyah Singer Index Property.
reviewed by: Ren Guo
Mathematics Subject Classification ID: 58J20
zbMATH DE Number: 5913491
zbMATH Keywords: Chern Weil form
zbMATH Keywords: Atiyah Singer index
zbMATH Keywords: holonomy
MaRDI profile type: Publication
DOI: 10.4310/PAMQ.2010.V6.N2.A14
description: scientific article; zbMATH DE number 5913491