The Atiyah Singer index theorem and Chern Weil forms
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.
scientific article; zbMATH DE number 5913491