This is probably an insanely hard question, but given an abstract metric space, is there some way to determine whether it's a manifold with a Riemannian, or more generally a Finslerian, metric? If that's too hard, one could start off by assuming that the underlying space is a manifold. The example that got me thinking about this was the induced metric on the 2-sphere embedded in $R^3$...the underlying space is obviously a smooth manifold, and the metric should be smooth(the geodesics would even be great circles, as they are for the standard metric on $S^2$,) but I don't see how you could prove the triangle inequality pointwise, as you'd have to to show that it's Finslerian, let alone Riemannian. This example is already incredibly simple, since it's a homogeneous space with a the metric induced by being a subspace of another homogeneous space.



Characterization of Riemannian metrics
If $X$ is a metric space and $x$, $y\in X$, a *segment* from $x$ to $y$ is a subset $S\subseteq X$ such that $x$, $y\in S$ and $S$ is isometric to $[0,d(x,y)]$. 


Let now $n\in\{1,2,3\}$ and let $X$ be a metric space which is locally compact, $n$-dimensional and such that *(i)* every two points are the endpoints of a unique segment, *(ii)* if two segments have an endpoint and one other point in common, then one is contained in the other, and *(iii)* every segment can be extended, at either end, to a larger segment. Then $X$ is homeomorphic to $\mathbb R^n$.


This is a metric characterization of $\mathbb R^n$ for small $n$. Using it locally, you get a characterization of topological manifolds of small dimension. I imagine higher dimensions or smoothness are harder to come by...


The theorem above, along with quite a few other nice results, is proved in [Berg, Gordon O. Metric characterizations of Euclidean spaces. Pacific J. Math. 48 (1973), 11--28.]




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**LATER...**


Of course, the paper [Nikolaev, I. G. A metric characterization of Riemannian spaces. Siberian Adv. Math. 9 (1999), no. 4, 1--58.] is relevant here! The paper gives purely metric characterizations of Riemannian manifolds of all smoothnesses up to $C^\infty$.
The [MR review](http://www.ams.org/mathscinet-getitem?mr=MR1749850) gives a small history of the problem and references to earlier work.


