We start this chapter by deriving the most important results about the ideal flat linear spaces, that is, the flat projective planes, R2-planes, and point Möbius strip planes. In Section 2.5 we then describe how these results generalize to general flat linear spaces.

To provide full proofs of all results listed below is beyond the scope of an Encyclopedia volume such as this. Also, Salzmann et al. [1995] is an excellent source for many of these results and proofs. Therefore, it was very tempting to restrict ourselves to just listing results and to referring the reader to that book and the literature about flat linear spaces for proofs. However, to make our book as self-contained as possible and to give the reader a good feel for why flat linear spaces behave in the way they do, we have included sketches of proofs for some of the most important results that illustrate the main arguments used to prove them. In particular, we show that the line set of an ideal flat linear space carries a natural topology, that automorphisms of such a geometry are continuous, and that the automorphism group of such a geometry is a finite-dimensional Lie transformation group. On the other hand, our exposition of the group-dimension classification of the flat linear spaces is purely descriptive and we do refer to the relevant literature for proofs. Highlights include the description of all flat projective planes whose automorphism groups are at least 2-dimensional, and all Möbius strip planes and R2-planes whose automorphism groups are at least 3-dimensional.