Metrizations of Projective Spaces

A two-dimensional G-space,1 in which the geodesic through two distinct points is unique, is either homeomorphic to the plane £2 and all geodesies are isometric to a straight line, or it is homeomorphic to the projective plane P2 and all geodesies are isometric to the same circle, see [l, §§10 and 31 ]. Two problems arise in either case: (1) To determine the systems of curves (in E2 or P2) which occur as geodesies. (2) If the geodesies are (or lie on) ordinary straight lines, can the space be imbedded in a higher-dimensional space with the ordinary straight lines as geodesies? The author solved both these problems for E2, [l, Theorems (11.2) and (14.8)], but left both open for P2 [l, Appendix (9) and (10)]. Recently Skornyakov [2] solved the first problem for P2; he modified the author's basic idea through replacing a summation by an integration, and thus eliminated the singularities which the author's procedure would produce in the case of P2. The purpose of this note is to show that a device similar to Skornyakov's can be used to solve Problem (2) for Pn. Our method also provides a much simpler solution of Problem (1). Thus we are going to prove simultaneously: