A Gauss-Bonnet formula for metrics with varying signature

A Gauss-Bonnet formula for compact orientable connected Riemannian or Lorentzian 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.