Congruence and superposability in elliptic space
Leonard M. Blumenthal · Transactions of the American Mathematical Society · 1947
Introduction.Two subsets A, B of a metric space M are congruent provided a mapping f(A) =B exists such that p, q(EA implies pg = dist.(p, q) = dist.(f(p), f(q)) =f(p)f(q).Congruence of A and B is denoted by A **B or by A »/Z? if one wishes to indicate a particular mapping/ which transforms A congruently onto B.Two subsets of a metric space M are superposable provided a congruent mapping of M onto itself exists which maps one of the subsets onto the other.It follows that two subsets A, B are superposable if and only if a congruence / exists between the sets which can be extended to the whole space.It should be observed, however, that superposability of two sets does not imply that each congruent mapping of one onto the other can be extended to the whole space.Thus, for example, if M is the line segment seg.(p, q'), with middle-point labelled both p' and q, then the two pairs of points (p, q), (p', q') are superposable since the congruent mapping f(p) =q', f(q) =p' can be extended to the segment by a reflection in q, but the congruence g(p) =p', g(q) =q' can obviously not be extended to the space.Moreover, a given congruence between two subsets of a metric space does not imply either that that congruence or any other between the subsets can be extended to a congruence of the whole space with itself.It is well known that euclidean, spherical, and hyperbolic spaces possess the strong property (a) any two congruent subsets are superposable, and the even stronger property (b) any congruence between any two subsets can be extended to a congruence of the whole space with itself.In a space with property (b) each two congruent subsets may be called freely superposable, and (as G. Birkhoff [l](') has recently shown) among all metric spaces in which each two points are joined by a segment, locally unique, only in the three spaces named above are congruent subsets freely superposable.In this paper congruence and superposability in »-dimensional elliptic space E",r are studied.Since congruent but not superposable subsets exist in every elliptic space of dimension greater than 1 ( §3) the En,r (»>1) does not have property (a).Moreover, it will be seen ( §6) that not every congruence between superposable subsets of En,r is necessarily extendible to the whole space.These circumstances give rise to two problems: (1) to find neces-