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Sorry if the title of this question is not formatted in a way that stackexchange likes, though my question is actually much more specific.

I was reading the book A Course in Metric Geometry, by Burago, Burago, and Ivanov. So I understand the basic idea behind length spaces and length structures seems to be an abstraction of distance. That is, that different metrics may exist on some space and we can group a subset of these metrics intro a length structure. Then a length space applies a metric over the set of metrics in the length structure. Please correct me if I am getting anything wrong.

However the book does not really explain what the mathematical motivation behind metric geometry is? Like what set of problems or questions was metric geometry designed to answer?

I am looking for some sort of statement like Tao gives on measure theory, where he describes the origin or measure theory in trying to understand the area or volume of a set. And then the development Lebesque measure as arising from the need to develop a definition of measure that went beyond simple elementary sets, as in the Riemann or Darboux integral.

So did metric geometry arise out of problems in measure theory, or from problems arising out of differential topology and riemann geometry, etc?

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  • $\begingroup$ this isn't an answer as I'm not that advanced ( especially in this area) but the taxicab metric in theory could have real world applications: youtube.com/watch?v=ineO1tIyPfM and in theory if a taxi can only move in certain directions in a city, you can figure out the minimum distance a taxi, would need to drive to get from place to place ( taxicab metric, idealized city) though I've also seen that talked about in graph theory terms. $\endgroup$ – user451844 Jul 20 '17 at 1:50
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    $\begingroup$ It looks like the later chapters look at subjects like spaces of bounded curvature, spaces with curvature bounded above/below, large scale geometry, and the space of metric spaces. Perhaps those things are sufficient motivation? Also: I don't think it takes a whole lot of motivation to justify getting from metrics and geometry to metric geometry, just my opinion. $\endgroup$ – Zach Teitler Jul 20 '17 at 1:52
  • $\begingroup$ @ZachTeitler Yeah, see this is where the question starts to arise. If I am looking at the properties of a set of metrics over the same space, then each metric could represent a deformation of the space. So then that gets into the territory of algebraic topology. But if you are looking at properties that persist over all metrics, like differentiability, then that is along the lines of differential geometry or riemann geometry. So I was just trying to understand where the ideas from metric geometry are coming from so I can basically orient my viewpoint as I read. $\endgroup$ – krishnab Jul 20 '17 at 1:55
  • $\begingroup$ en.wikipedia.org/wiki/Category:Metric_geometry might help ? $\endgroup$ – user451844 Jul 20 '17 at 2:03
  • $\begingroup$ @RoddyMacPhee thanks Roddy. This material is covered in the book too. But I was just hoping to get a clear statement of some of the motivations behind these concepts. $\endgroup$ – krishnab Jul 20 '17 at 2:07
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In one way it's similar to how we don't always stay within $C^k$ function spaces, but introduce larger function spaces (Lebesgue, Sobolev, etc). For example, $C^1$ is not complete in the norm $\int (|f|+|f'|)$, so when minimizing an integral functional of that kind we would find it hard to prove the existence of a minimizer - a sequence of functions that achieve better and better values of the functional may fail to have a limit.

Similar things happen with manifolds. The hyperboloids $x^2+y^2=z^2+1/n$ are nice manifolds, but their limit as $n\to\infty$ is a cone, which has a non-manifold point. So when we consider a geometric variational problem, like finding a surface of minimal area with given boundary, one has to worry about losing smoothness in the limit. Bringing non-manifold spaces into the discussion of surface area, curvature, etc may be necessary to prove the existence of a minimizer.

For example, consider Gromov's compactness theorem:

The set of compact Riemannian manifolds of a given dimension, with Ricci curvature $\ge c$ and diameter $\le D$ is precompact in the Gromov–Hausdorff metric.

That's great, but one is immediately led to ask: what is the compactification? I.e., what kind of spaces do we get as limits of sequences of such manifolds in the Gromov-Hausdorff metric? Those aren't necessarily manifolds, but they are metric spaces in which one can still detect that Ricci curvature bound using the methods of metric geometry.

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The preface answers your question (as I’ve understood it). Metric geometry is intended to be, or at least began as, a project to see how many differential geometric concepts and results could be defined and obtained without recourse to calculus. The example they give of two ways of looking at convex functions is very enlightening from a bird’s eye perspective. I’m nothing like an expert on metric geometry myself having been reading it for a couple of weeks now.

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