The foundations of geometric formulation of Newton's axioms On Professor Frederic P. Schuller's Lecture about General relativity, where you can access it through this link: https://www.youtube.com/watch?v=IBlCu1zgD4Y, he clamed Newton's axioms can be converted into 3 mathematical axioms. However I don't understand how they are connected.
Definition: A Newtonian spacetime is a quintuple $(M,O,A,∇,t)$ where $(M,O,A,∇)$ is a differentiable manifold with a torsion free connection, and $t\in C^\infty$M is a smooth function such that $(dt)_p\neq 0$ and $\nabla(dt)=0$.
He claimed that 
"There is an absolute space" can be written as $(dt)_p\neq 0$. And "Absolute time flows uniformly" can be written as $\nabla(dt)=0$ everywhere. How should I understand these restriction?
 A: Did you find the answer to your question? 
The second one $\nabla(dt)=0$ implies that the $\Gamma^0_{ab}=0$. Using Newton's second law $$\nabla_{v_x} v_x=\frac{F}{m}$$ and the fact that the force field $F$ is spatial i.e. $dt(F)=0$, leads to a linear equation in absolute time $t$ and the parameter $\lambda$ that parameterizes the worldline. I think this is what he meant by time flows uniformly.
The first one I'm not so sure, $(dt)_p= 0$ implies that the time component of every vector (here you may imagine a vector as velocity at point $p$ on the curve $\gamma(\lambda)$ at $\gamma(\lambda_0)=p$) is zero. "Intuitively" this means that there is a point $p\in M$ such that all the curves passing through $p$ have no velocity component in time direction implying there exists segments on the curve $\gamma$ where time is constant as a function of the parameter $\lambda$ contradicting our previous assumption. I don't know why he refers to this as "There is an absolute space".
Please let me know if you have a better explanation.
A: I think I sort of understand the $\nabla (dt)$ condition.
Suppose we have a room and consider any point in the room, in the Newtonian case, it must be that if you kept a clock at each point, then , say if $x$ seconds passed on one clock, in all other clock that same time interval should pass.
I am not quite sure how one would interpret the equation when talking about how the time one form changes when we take covariant derivative in the time basis direction...
A: $\nabla (dt) = 0$, means that given all clocks are synchronized, there's no time flow gradient going from point to point in system, or in other words :
$$ \frac {\partial (\Delta t)}{\partial x} + \frac {\partial (\Delta t)}{\partial y} + \frac {\partial (\Delta t)}{\partial z} = 0 $$
This means that time flow is absolute in Newtonian axioms,- it does not depend on points relative speed difference (otherwise, special relativity), nor on gravitational field potential difference between the points being analyzed (otherwise, general relativity).
