A proof of the Chern-Gauss-Bonnet theorem for indefinite signature   metrics using analytic continuation
  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. 
P. Gilkey and J. H. Park
math.DG