# Understanding the shape of 2-dimensional orbifold given a metric tensor [closed]

I am trying to grasp some intuition about the shape represented by the following two-dimensional metric tensor:

$$ds^2=dr^2+n^2\cdot r^2d\theta^2$$

where $$r$$ is "radial" coordinate, and $$\theta$$ is the "angular" coordinate for $$n=1$$ the metric represents smooth closed circles on $$\mathbb R^2$$ for a given $$r\in\mathbb R$$ with known periodicity in $$\theta$$ of $$2\pi$$: $$\theta\equiv\theta+2\pi$$ For general $$n$$ the periodicity of the circle is changing by recognizing:

$$n\theta=\theta'$$ so the periodicity is now:

$$\theta'\equiv\theta'+2\pi\Rightarrow\theta\equiv\theta+\frac{2\pi}{n}$$ It looks like a periodicity of a regular polygon with $$n\in\mathbb N$$ sides, but according to the metric this kind of spherical shape with coordinates $$r$$, according to my supervisor this shape is known as an "orbifold" I would like to have some geometrical and graphical expression in the space to this 'orbifold'.

• Jul 12, 2023 at 12:08

I think that intuitively the idea behind the answer is that if you take a circle with a given radius $$r$$ and you cut a sector between zero angles and your periodicity of $$\frac{2\pi}{n}$$ modulus $$2\pi$$ and glue the sides of the sector and you get a cone embedded in $$\mathbb{R}^3$$ (maybe the corresponding quotient group is $$\mathbb{R}^2/\mathbb{Z}$$ or $$\mathbb{R}^2/\mathbb{Z}_n$$ for $$n\in\mathbb{N}$$ )
So actually there is no such "beast" that lives on the plane with radial symmetry and periodicity different than $$2\pi$$ in the angular coordinate. In addition, we notice that the tip of the cone is sharpened so there is no smoothness of the manifold at that point, this is known as "conical singularity" In string theory such spaces appear in the context of "gravitational conical singularities" and are usually a sign of extra degrees of freedom which are located at a locus point in spacetime. In the case of the cone, these degrees of freedom are the twisted states, which are strings "stuck" at the fixed points. When the fields related to these twisted states acquire a non-zero vacuum expectation value, the singularity is deformed, i.e. the metric is changed and becomes regular at this point and around it.