# Definition of Galilean structure in Arnold's book?

I am reading Arnold's Mathematical Methods of Classical Mechanics. He quickly introduces the notion of Galilean structure. The universe is defined as the affine space $A^4$ and time is defined as a linear mapping t from $R^4$ to $R$. And the time interval between two events in the universe is simply $t(b-a)$.

Don't we need an additional requirement that the linear mapping $t$ is essentially a projection on the temporal part of the affine space? Otherwise, transformation of uniform motion would not preserve this time interval invariance.

No, there is no 'temporal part of the universe' until you make your affine space isomorphic to $$\mathbf{R}^3 \times \mathbf{R}$$. What Arnold says is that you take two events in the universe, then this defines a vector in $$\mathbf{R}^4$$, and to that vector you see what the image of the time map $$t:\mathbf{R}^4 \rightarrow \mathbf{R}$$ is.