In $n$ dimensions, the Ricci scalar curvature $R$ at a point measures how much the volume $V$ of a small $n$-dimensional ball of radius $\epsilon$ around that point differs from the Euclidean value $V_E$:
I am not aware of similar formulas for higher-order invariants, but that doesn’t mean they ...
Provided all observers agree when it is zero, they will all agree on the same speed of light.
As far as I can see, the universal speed of light per se imposes no other requirement.
But the universal speed of light is not the only input. We also require there to be a single underlying reality behind the different observations made in the different frames. ...
The physical implication would be that different inertial observers would find different values for the speed of light. It is the assumption that c is constant for all observers that leads to that particular equation for the interval.