Flat Projective Planes

Abstract

In this chapter we concentrate on flat projective planes. Just to reiterate, a flat projective plane is a geometry whose point set is the real projective plane (considered as a topological space only), whose lines are topological circles, and that satisfies the Axiom (of joining) P1 for projective planes (see Section 1.2). Every such geometry automatically also satisfies the other two axioms for projective planes; that is, it is automatically a projective plane. Also, there are essentially two different ways in which a topological circle can be embedded in the real projective plane. In one kind of embedding the curve is embedded such that it separates the surface into two open components; one is homeomorphic to the unit disk, the other one to the Möbius strip. In the other kind of embedding the curve does not separate the surface. All lines in a flat projective plane are embedded like this.