A classic theorem of Weyl (1921) states that a Weyl metric -- a natural generalisation of a pseudo-Riemannian metric -- is uniquely determined by its conformal and projective structures (i.e. by its conformal structure and its set of unparametrised geodesics). An equivalent formulation of Weyl's result is that a torsion-free linear connection compatible with a pseudo-Riemannian conformal structure is uniquely determined by its projective structure. We discuss analogous results for suitably defined notions of conformal structure for Galilei and Carroll geometry, i.e. for spacetime geometries arising as the `non-relativistic' and `ultra-relativistic' limits of Lorentzian geometry.
Schwartz et al. (Sat,) studied this question.