A theorem on local isometries

A mapping b of a G-space R (Busemann [1]) on itself is a locally isometric mapping if for each xER there is a number 77j>0 such that X maps the spherical neighborhood S(x, qjx) isometrically on S(b(x), 7 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 [1, ?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 L5]. 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 Xi(p) and X,(p) equal, respectively, inf 1(h) and sup 1(h) for all hCG(p). Put Xi(p) = oo and X8(p) =0 if G(p) is empty. Let Xi= inf Xi(p); X8 = supX8(p). pER pER