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

TL;DR

This paper refines a result characterizing the Gauss-Bonnet-Chern formula as the unique metric-independent integral of a differential invariant on oriented Riemannian manifolds, emphasizing its topological significance.

Contribution

It provides a refined proof that the Gauss-Bonnet-Chern formula is the only non-trivial metric-independent integral of a differential invariant.

Findings

The Gauss-Bonnet-Chern formula uniquely characterizes the Euler characteristic.

Metric independence of the integral implies it is a topological invariant.

The result extends Gilkey's characterization of the formula.

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.