A proof of the Chern-Gauss-Bonnet theorem for indefinite signature metrics using analytic continuation

TL;DR

This paper extends the Chern-Gauss-Bonnet theorem to indefinite signature metrics by employing analytic continuation from the Riemannian case, focusing on manifolds with boundary and complex tangent space metrics.

Contribution

It introduces a novel proof of the theorem for pseudo-Riemannian manifolds using analytic continuation of the Pfaffian, bridging Riemannian and indefinite signature geometries.

Findings

Established the theorem for manifolds with boundary in pseudo-Riemannian geometry

Connected Riemannian and indefinite signature cases through analytic continuation

Provided a new perspective on characteristic classes in pseudo-Riemannian contexts

Abstract

We derive the Chern-Gauss-Bonnet Theorem for manifolds with smooth non-degenerate boundary in the pseudo-Riemannian context from the corresponding result in the Riemannian setting by examining the Euler-Lagrange equations associated to the Pfaffian of a complex "metric" on the tangent space and then applying analytic continuation.