Theorem: Classical chronological reconstruction

Let $(M,g)$ and $(M^{\prime },g^{\prime })$ be smooth time-oriented Lorentzian manifolds of the same dimension at least three, both past and future distinguishing. A bijection preserving chronology in both directions is a smooth conformal diffeomorphism:

for a smooth positive function $\Omega$.

Proof. This is the chronological reconstruction result of Malament, using the causal, differential, and conformal rigidity results associated with Hawking, King, and McCarthy Malament,HKM. The distinguishing formulation and its relation to causal bijections are discussed in ParrikarSurya.