Question about the geometric structure of Newtonian mechanics

My point here is about the mathematical structure of Classical pre-relativistic physics and general relativity (GR). It became more clearly, after GR, about the fact that pseudo-riemannian are a nice mathematical model to structure physics (at least basic classical mechanics i.e. without sympletic theory expositions, and general relativity). The standard structure is then assume that the physics occurs in some "place" and not the space or the time are the "stages" but, in fact, the structure called spacetime.

So mathematically a spacetime is a triple:

$$\Big(\mathcal{M},\mathbf{g},\nabla\Big) \tag{1}$$

Where $$\mathcal{M}$$ is a differentiable manifold, $$\mathbf{g}$$ is the metric tensor field and $$\nabla$$ is some connection (mainly the levi-civita connection). It can be said that $$(1)$$ is the model of a spacetime for GR.

But, and here lies my doubt, what about the structure of Newtonian picture?

My guess:

The spacetime of newtonian mechanics is:

$$\Big(\mathbb{R^{3}},\mathbf{\delta},\nabla\Big) \equiv \mathbb{E^{3}} \tag{2}$$ Where $$\mathbb{E^{3}}$$ is the euclidean spacetime. $$\mathbb{R^{3}}$$ is a differentiable manifold, $$\mathbf{\delta}$$ is the euclidean metric tensor field and $$\nabla$$ is the levi-civita connection.

• Where is the time in this picture? – Alex Trounev Feb 22 at 18:05
• That is the space, not the spacetime. – DanielC Feb 22 at 18:06
• Newtonian space and time don't mix, so you need two metrics for the spacetime in this case: one for space intervals, and another one for time intervals. – Cuspy Code Feb 22 at 19:05
• Your question is answered here: physics.stackexchange.com/questions/372843/… – safesphere Feb 23 at 3:46