Length Structures: Path Metric Spaces

Abstract

In classical Riemannian geometry, one begins with a C^∞ manifold X and then studies smooth, positive-definite sections g of the bundle S²T*X. In order to introduce the fundamental notions of covariant derivative and curvature (cf. [Grl-Kl-Mey] or [Milnor], Ch. 2), use is made only of the differentiability of g and not of its positivity, as illustrated by Lorentzian geometry in general relativity. By contrast, the concepts of the length of curves in X and of the geodesic distance associated with the metric g rely only on the fact that g gives rise to a family of continuous norms on the tangent spaces T_xX of X. We will study the associated notions of length and distance for their own sake.