On locally isometric mappings of a $G$-space on itself

In his study [2] of locally isometric mappings of a G-space W on a G-space R, Busemann considers the following question: Under what conditions is every locally isometric mapping of a G-space R on itself a motion? He proves that such is the case if either (i) the fundamental group of R is not isomorphic to a proper subgroup of itself, or (ii) R is compact. Busemann suggests [2, p. 405] that conditions (other than (i)) be sought which apply to noncompact spaces, in particular, conditions which apply to an ordinary cylinder. Szenthe replies to this in a recent paper [3 ] in which he finds conditions in terms of certain bounds on the lengths of nonoverlapping geodesic curves which begin and end at the same point. In [I ] Busemann shows that under appropriate hypotheses on the order of magnitude of volumes of spheres a locally isometric mapping of a noncompact G-space on itself is a motion. Our paper provides another condition. We first show that if a locally isometric mapping of R on itself has a fixed point, then it is a motion. From this it readily follows that if the motions of R form a transitive group, then every locally isometric mapping of R on itself is a motion. Let 4 denote a locally isometric mapping of a G-space R on a Gspace R. The terminology we use and the following properties of q are found in Busemann [2, ?27]. (1) If ;c(r), ao 0 such that if 4fQil) =)(fi2) =P, f1#l P2, then lfP2? 2p(p). (4) The number of points of R which lie over a given point of R is countable and is the same for different points of R. (5) If 4 is 1-1 then 4 is an isometry. Since each two points of a G-space are joined by a metric segment of the space, the following is an immediate consequence of (1) and (2).