On Metric Arcs of Vanishing Menger Curvature

Abstract

1. Let r be a metric space which is a simple arc, that is the topological image of a closed linear segment. Menger introduced the following purely metric definition of curvature ([6], pp. 480, 481).2 Let q, r, s be any distinct points of r. In virtue of the triangle inequality, their mutual distances qr, rs, qs are equal to the sides of a certain euclidean triangle. Let p(q, r, s) = 1/K(q, r, s) (O < p < (X2, 0 < K < ?? ) denote the radius of the circle circumscribed to that triangle, hence K(q, r, s) is its curvature. In fact