Mathematical Formulation of Classical Spacetime

Prof. Schuller is apparently referring to the Axiomatic formulation of the Newtonian theory of gravity, of which two versions I'm aware, one by Andrzej Trautman, described in:

• Trautman, Andrzej (1963). “Sur la théorie newtonienne de la gravitation”. In: Comptes rendus hebdomadaires des séances de l’ Académie des sciences 257, pp. 617–620,
• – (1967). “Comparison of Newtonian and relativistic theories of space-time”. In: Perspectives in Geometry and Relativity. Essays in honor of V. Hlavat’y. Ed. by B. Hoffman. Bloomington: Indiana University Press, pp. 413–425,
• section 5.5 of Trautman, A., F. A. E. Pirani, and H. Bondi (1965). Lectures on general relativity. Prentice-Hall,

and another introduced by Künzle–Ehlers. Both succeed the Newton-Cartan theory. Its main practical use is to show that such a theory of gravity has many of the problems general relativity has.

According to the development of this theory, the metric of the hypersurfaces of constant $t$ is the part of the degenerate matrix $g^{ab}$ that is non-singular. In order to fix the time coordinate and conclude that $g^{ab}$ is constant, we need to do the calculations using the flat connection $\overset{0}{\Gamma}{}_{bc}^a$, which corresponds to the inertial coordinate system.

One simple way to conclude this, even though it is outside the context of the aforementioned axiomatic formulation, is if you write the lagrangian: $$L = \frac{1}{2} h_{ab}(x) \dot{x}^a \dot{x}^b - V(x),$$ and derive the equations of motion, which will yield the geodesics. Newton's gravitational law will emerge in the familiar form only if the connection vanishes, which implies that the coordinate system is inertial, and leads to the correct interpretation of the metric $h_{ab}$.