Similarities and differentiability
Herbert Busemann · Tohoku Mathematical Journal · 1957
Introduction.Prof. K. Yano mentioned in a lecture that a Riemann space is euclidean if it possesses a one-parameter group H of non-isometric similarities.With "Minkowskian" replacing "euclidean" the theorem holds also for Finsler spaces, but fails to hold when differentiability hypotheses are altogether omitted by substituting G-spaces 1 ) for Finsler spaces.This remains so, even under very strong supplementary hypotheses: without being Minkowskian the space may, in addition to H, possess groups of motions of a rather high dimension and its geodesies may be the euclidean straight lines.On the other hand, a very mild differentiability suffices for concluding from the existence of H and the axioms of a G-space that the space is Minkowskian.Yet, nothing in the formulation of the original theorem suggests the necessity of smoothness requirements.The author is not aware of any similarly striking example where differentiability assumptions in their usual form conceal strong purely geometric implications.Therefore a systematic analysis of the situation seems justified, We begin 'by discussing similarities in general G-spaces, then convince ourselves by examples'^ that the above mentioned phenomena actually occur.Next, we discuss a simple intrinsic, geometric condition for differentiability.Examples show that this condition is still too weak to deduce the Minkowskian character of the metric from the existence of H, because, in fact, the local metric need not be Minkowskian.However, strengthening the condition slightly into an analogue of continuous differentiability proves sufficient: A G-space which admits a similarity with dilation factor k *1 (a group H of similarities is not needed) and is continuously differentiable at one of the {always existing) fixed points of the similarity, is Minkowskian in the small when k>l, and in the large when k