A comprehensive approach to qualitative problems in intrinsic differential geometry, this text for upper-level undergraduates and graduate students emphasizes cases in which geodesics possess only local uniqueness propertiesand consequently, the relations to the foundations of geometry are decidedly less relevant, and Finsler spaces become the principal subject.
This direct approach has produced many new results and has materially generalized many known phenomena. Author Herbert Busemann begins with an explanation of the basic concepts, including compact metric spaces, convergence of point sets, motion and isometry, segments, and geodesics. Subsequent topics include Desarguesian spaces, with discussions of Riemann and Finsler spaces and Beltrami's theorem; perpendiculars and parallels, with examinations of higher-dimensional Minkowskian geometry and the Minkowski plane; and covering spaces, including locally isometric space, the universal covering space, fundamental sets, free homotopy and closed geodesics, and transitive geodesics on surfaces of higher genus. Concluding chapters explore the influence of the sign of the curvature on the geodesics, and homogenous spaces, including those with flat bisectors.
Table of Contents
I. The Basic Concepts
II. Desarguesian Spaces
III. Perpendiculars and Parallels
IV. Covering Spaces
V. The Influence of the Sign of the Curvature on the Geodesics
VI. Homogenous Spaces
Notes to the Text