Length Structures: Path Metric Spaces
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 TxX of X. We will study the associated notions of length and distance for their own sake.
KeywordsRiemannian Manifold Homotopy Class Geodesic Segment Alexandrov Space Length Structure
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