Flat space in polar coordinates takes the form
$$ds^2=dr^2+r^2d\phi^2$$
To avoid a conical singularity at the origin, we must impose that $\phi$ is periodic with period $2\pi$. 


Now consider the following generalization:
$$ds^2=dr^2+r^2d\phi^2+r^2d\psi^2$$
My basic question is how to interpret such a metric geometrically, in particular at the point where $r\rightarrow 0$? Has these geometries been studied before, and if so what are they called?


I am mostly interested in if there is some natural regularity condition we could impose that is analogous to the absence of a conical singularity in the polar coordinate metric. For example, it is tempting to just impose $2\pi$ periodicity in both angular coordinates, which would make the polar coordinate metric, obtained by ignoring one of the angular coordinates, regular. But it is unclear if this is the right thing to do for the geometry to be well behaved at the point $r\rightarrow0$.



Regularity of a generalized polar coordinate metric with two angles
This is the cone $x^2+y^2=u^2+v^2$ with metric induced from $\mathbb{R}^4$ with coordinates $(x,y,u,v)$. It is clearly not smooth. You just take usual polar coordinates on $(x,y)$ and $(u,v)$ separately and then set the magnitudes equal.


