A Theorem on Local Isometries

A mapping 0 of a G-space R (Busemann [l ]) on itself is a locally isometric mapping if for each x£P there is a number r)x>0 such that (x), vx). The problem we are concerned with is that of determining conditions on a G-space R under which every locally isometric mapping of R on itself is an isometry. Several such conditions have recently been given by Busemann [l, §27], [2],Szenthe [4], [5], and the author [3]. In this paper we are concerned with the more general of the conditions given by Szenthe [5]. For a fixed point PER, consider the collection G(p) of all geodesic curves which begin and end at p, and which do not contain subarcs traversed more than once. For hEG(p), let 1(h) denote the length of h. Let \i(p) and \s(p) equal, respectively, inf 1(h) and sup 1(h) for all hEG(p). Put \i(p) = oo and \s(p) =0 if G(p) is empty. Let