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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?

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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 $2\cdot\pi-\epsilon$.
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P.S. The curvature can be understood in the sense of Alexandrov. In two-dimensional case it is a measure $\kappa$ such that for any open geodesic triangle $\triangle$, we have $$\kappa(\triangle)=\alpha+\beta+\gamma-\pi,$$ where $\alpha$, $\beta$ and $\gamma$ denote the angles of $\triangle$. –  Anton Petrunin Nov 18 '11 at 1:57
    
Thanks for those facts. Disregarding my edit above, how do you see that the space is a cone with length $2\pi-\varepsilon$ ? –  Chris Gerig Nov 19 '11 at 1:50
2  
@Chris. One way: to find parallel translation around a broken geodesic you have to add its exterior angles and rotate the tangent plane by this angle. An other way: smooth the cone and calculate the integral of curvature of obtained surface using Gauss--Bonnet theorem. –  Anton Petrunin Nov 19 '11 at 2:52

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.)

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