I remember briefly hearing about this notion (stated in the title), of a manifold where there is a nonzero curvature at precisely one point (a delta-function distribution), and such that there is a holonomy around that point with a vector being rotated by $\varepsilon$ (depending on what that curvature constant is).

I have googled all over, with not even a hint about what I am referring to. How is this curvature distribution even defined?

**Edit/Addendum:** It turns out that I am speaking about the "$\varepsilon$-cone" in 2D, which is generalized by simply taking the cartesian product with $\mathbb{R}^n$. It is defined in Regge's *General Relativity without Coordinates*. It is a polyhedron with one vertex, described by taking the metric $ds^2=d\rho^2+\rho^2d\theta^2$ in the Euclidean plane, but instead of identifying multiples of $2\pi$ on the $\theta$-coordinate, you identify multiples of $2\pi-\varepsilon$.

Are there other applications of this concept beyond Regge Calculus?