Novel Results in the Geometry of Geodesics

Abstract

This work is devoted to results on the G-spaces of Busemann found after the appearance of his book "The Geometry of Geodesics" (Academic Press, New York, 1955); and denoted here as G. The axioms for G-spaces are found on p. 37 of G. They include all complete Riemann spaces and many others which the letter G denotes as Finsler spaces, which is a gross misnomer. Finsler was the first who investigated non-Riemannian spaces (under strong differentiability hypotheses) guided by Carathéodory, whose methods in the calculus of variations form the basis of Finsler′s thesis, which has no relation to our G-spaces (but is still an active field of research). The misnomer was caused by the fact that Finsler′s thesis (1918) was inaccessible until 1951, when Birkhäuser reissued it unchanged. Nevertheless, we denote here as Finsler spaces all non-Riemannian G-spaces, because we refer frequently to G and this greatly facilitates references.