epsilon-Manifold with curvature at one point 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?
 A: I did note hear about this notion. 
But here some facts:


*

*If your space is homeomorphic to a manifold and it has zero curvature at all points but one then it is either flat manifold or the dimension is $\le 2$.

*In case dimension is $=2$ the space looks like a cone over a circle with length $2\cdot\pi-\epsilon$.

A: This kind of geometry, with $\varepsilon$ a negative multiple of $2\pi$, appears in particular in translation surfaces. These are surfaces with a flat metric, singular at some points, with trivial holonomy. They are obtained by taking a set of polygons whoses edges can be identified pairwise using translations (for example, a regular octogon with opposite edges identified has genus 2 and one singular point).
Also, Einstein manifolds with conical singularities (concentrated on codimension $2$ submanifolds) have been studied (a motivation is that it is difficult to produce examples of regular Einstein manifold, so enabling some singularities can produce examples, giving some insight.)
