On the uniqueness of the Gauss-Bonnet-Chern formula (after Gilkey-Park-Sekigawa)

Abstract

On an oriented Riemannian manifold, the Gauss-Bonnet-Chern formula asserts that the Pfaffian of the metric represents, in de Rham cohomology, the Euler class of the tangent bundle. Hence, if the underlying manifold is compact, the integral of the Pfaffian is a topological invariant; namely, the Euler characteristic of the manifold. In this paper we refine a result originally due to Gilkey that characterizes this formula as the only (non-trivial) integral of a differential invariant that is independent of the underlying metric.