A Gauss-Bonnet Formula for Metrics with Varying Signature

Abstract

A Gauss–Bonnet formula for compact orientable connected Riemannian or Lorentz-ian 2-manifolds is well-known. We investigate singular metrics on 2-manifolds with varying signature. Such metrics are necessarily degenerate at some points of $M$ where most of the usual definitions for geometric quantities break down. We prove that under some additional assumptions there is a Gauss–Bonnet formula for compact orientable connected 2-manifolds with a singular metric. Some examples are given.