On the Covering of a Complete Space by the Geodesics Through a Point

Abstract

where F is an analytic function of x and dx satisfying further conditions given in ?2. In ?3 it is shown that these, together with a completeness condition for the space as a whole, are sufficient to ensure that each geodesic can be produced indefinitely in both directions' and that any ordered pair of points are joined by at least one oriented extremal of minimum length. Because of the properties described above a normal coordinate system at a goiven point 0 can be extended to define an analytic transformation y P. of the entire tangent Minkowski space at 0, which we call Mn, into F,, the space Mn covering the whole of Fn. The points in Mn at which y -+ P fails to be locally (1.1) are those which are conjugate to 0 with respect to suitable geodesics. They constitute an analytic complex which we call the conjugate complex. Another important locus in Mn is defined by analogy with a locus introduced