  The Weyl principle is extended from the Riemannian to the pseudo-Riemannian
setting, and subsequently to manifolds equipped with generic symmetric
$(0,2)$-tensors. More precisely, we construct a family of generalized curvature
measures attached to such manifolds, extending the Riemannian Lipschitz-Killing
curvature measures introduced by Federer. We then show that they behave
naturally under isometric immersions, in particular they do not depend on the
ambient signature. Consequently, we extend Theorema Egregium to surfaces
equipped with a generic metric of changing signature, and more generally,
establish the existence as distributions of intrinsically defined
Lipschitz-Killing curvatures for such manifolds of arbitrary dimension. This
includes in particular the scalar curvature and the Chern-Gauss-Bonnet
integrand. Finally, we deduce a Chern-Gauss-Bonnet theorem for
pseudo-Riemannian manifolds with generic boundary.
