I just start to read Mathematical Methods of Classical Mechanics by Arnold. And I am sort of very puzzled by all the notion. Firstly, if the universe is seen as a 4D affine space, why is time a mapping from [itex]R^4→R[/itex]? I mean this kind of 4D contains time, right? Secondly, I thought the kernel of such a mapping t should be the set of events simultaneous with a given event (affine apace), yet it saids kernel is a 3D linear subspace of a vector space [itex]R^4[/itex]. Thirdly, I never formally took a class on group theory, and google did not exactly answer me. But what is the dimension of the galilean group (or any other group). Thanks in advance!