Let $(M,g)$ be a globally hyperbolic Lorentzian manifold. It is known that there exists a smooth Cauchy surface $\Sigma\subset M$ and that $M$ is diffeomorphic to $\mathbb{R}\times\Sigma$. I suspect that any other smooth Cauchy surface $\Sigma'\subset M$ is diffeomorphic to $\Sigma$, and there is a natural candidate for such a diffeomorphism:
For every $x\in \Sigma$ the maximal timelike, futurepointing path $\gamma$ starting at $x$ intersects $\Sigma'$ in exactly one point $\phi(x)$, thus defining a bijection $\phi:\Sigma\to\Sigma'$. But is $\phi$ smooth?