I wonder what are some statements that although they can be formulated for pseudoriemannian manifolds of arbitrary signature they turn out to be true only in the lorentzian case.

I admit things like: "no analog to cauchy hypersuface can be defined in arbitrary pseudoriemanniann manifolds, just in lorentzian ones". I don't know if this last statement is true, it's just to exemplify.

Another one could be: the equivalent topological condition for a given compact manifold to admit a pseudoriemannian metric of a given signature is to have 0 euler characteristic. I think this is true only for lorentzian manifolds.

Thanks in advance!