Geometries on the torus, complex numbers and Lobachevsky geometry
The description of locally Euclidean geometries, which made up Chapter II, contained one gap, and we now fill this. It was proved that every such geometry can be constructed as a geometry ΣΓ for a certain uniformly discontinuous group Γ of motions of the plane. It would seem that the classification of all such groups given in Chapter II, §8 then solves the problem. However, this is not quite the case: what we have done is to present a list of geometries such that every geometry of the type we are interested in is identical to one in the list; however, we have not determined when two geometries from our list are identical (that is, can be superposed on one another), and when they are different. Hence the question as to how many distinct geometries there are in existence has not yet received a definitive answer. We can make some assertions in this direction at once; for example, geometries belonging to the different Types I, II.a, II.b, III.a and III.b are different, since they are distinguished by properties such as the existence of closed curves, boundedness, and whether right and left are distinguishable. But it remains unclear whether the geometries within each type are distinct or are the same: for example, are the two geometries defined by groups Γ and Γ’ of Type II.a, when Γ and Γ’ are generated by the pairs of vectors illustrated in Figure 13.1, the same or distinct?
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