Flat space metrics This question concerns the metric of a flat space:
$$ds^2=dr^2+cr^2\,\,d\theta^2$$ where $c$ is a constant. Why is it necessary to set $c=1$ to avoid singularities and to restrict $r\ge 0$?
Thanks.
 A: The metric, in the form that you give, has no problems until you make the restriction $0 \leq \theta < 2\pi$, or allow $c < 0$.  In either of these cases, you can run into problems when regularizing the metric.  As has been stated, this metric is not invertible when $r=0$.  Normally, this doesn't matter, because we're always free to make the substitution $x = r\cos \theta$ and $y=r \sin \theta$, and then you have transformed this metric into ordinary minkowski space.
If you have $0\leq \theta < 2\pi$ and $c \neq 1$, however, this transformation becomes $x = r \cos (\sqrt{c}\theta)$ and $y= r \sin (\sqrt{c}\theta)$, and then you're mapping more (or fewer $if 0<c<1$) points in your $r,\theta$ domain than there are in your $x,y$ domain, and the singularity is not resolved.  In particular, if $0< c < 1$, you just get a sliver of the plane.  You can remove badness at the boundary by identifying the point $(r,\theta_{max})$ with the point $(r,0)$, but when you do this, the space basically defines a cone, and the manifold is not smooth at $r=0$  Hence why these singularities are called conical singularities.
