Metric Geometry

Yuri Burago () and David Shoenthal ()
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Yuri Burago: Steklov Institute for Mathematics at St.
David Shoenthal: Pennsylvania State University, Department of Mathematics

A chapter in New Analytic and Geometric Methods in Inverse Problems, 2004, pp 3-50 from Springer

Abstract: Abstract Much of one’s mathematical experience with regard to metric spaces begins at the level of the metric. Instead of starting with a metric, in many cases we must begin with the length of paths as the primary notion. From this, we will derive a distance function. More precisely, we can introduce a new distance which is measured along the shortest path between two points in a space (as opposed to simply measuring the Euclidean distance between the two points). One says that a distance function on a metric space is an intrinsic metric if the distance between two points can be realized by paths connecting the points (mathematically, it must be equal to the infimum of lengths of paths between the points—a shortest path may not exist). If the length of paths is to be our primary notion, we must ask for a rigorous definition, from where it may arise, and what the properties are of such structures (which we will call length structures).

Keywords: Short Path; Riemannian Manifold; Sectional Curvature; Cayley Graph; Hausdorff Distance (search for similar items in EconPapers)
Date: 2004
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DOI: 10.1007/978-3-662-08966-8_1

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