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?



epsilon-Manifold with curvature at one point
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 $2\cdot\pi-\epsilon$.


