It's well known that a connected Riemannian manifold induces a metric space: the distance between two points ia measured as the infimum of the length of curves joining the points. The inverse: given a metric space, there is a connected Riemannian manifold that induces it?
I post it as a curiosity. All what I know is the metric space would be a path metric space, so, for example, it has to be locally path connected.
no. Take the curve $$ y = \sqrt {|x|} $$ in the plane and then revolve it around the $y$ axis, so now there is a surface in $\mathbb R^3$ which is nice except for a cusp at the origin. Take a circle at geodesic radius $\rho$ from the cusp. As $\rho \rightarrow 0,$ the geodesic length of that circle goes to $0$ faster than $\rho.$ That is, the arc length of a geodesic circle around the cusp grows more slowly than linear in $\rho.$
For an actual Riemannian manifold of dimension $2,$ the circumference of a geodesic circle must have linear growth in the radius when that is small.
