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.