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 Lebesgue 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?
Best Answer
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:
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.