If $X$ is a metric space and $x$, $yin X$, a segment from $x$ to $y$ is a subset $Ssubseteq X$ such that $x$, $yin S$ and $S$ is isometric to $[0,d(x,y)]$.
Let now $nin{1,2,3}$ 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.]
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 gives a small history of the problem and references to earlier work.
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