Do space and time geometry obey the same uniqueness rules at every speed limit?

Weyl proved in 1921 that knowing which paths are straight lines plus the local notion of angle is enough to pin down spacetime geometry completely. This paper asks whether the same holds for Galilean geometry (the non-relativistic limit, where light is infinitely fast) and Carrollian geometry (the ultra-relativistic limit, where light stands still). The answer is nuanced: analogous theorems hold, but the geometry of these exotic spacetimes forces careful redefinition of what 'conformal structure' even means.