Topological structure of Gauss–Bonnet–Chern density and its topological current

Abstract

The decomposition of spin connection on the sphere bundle of a compact and oriented even-dimensional Riemannian manifold is detailed. It is shown that the Gauss–Bonnet–Chern form can be expressed in terms of a smooth vector field Φ and taken the form of the δ function δ(Φ). The topological current of the Gauss–Bonnet–Chern theorem and its topological structure are discussed.