Riemannian surface

A Riemannian surface is a two-dimensional smooth manifold equipped with a smoothly varying inner product on each tangent plane.

A surface metric has a cone singularity of total angle $\Theta >0$ if a punctured neighbourhood has the metric obtained by joining the radial boundaries of a sector of angle $\Theta$. For constant Gaussian curvature $-1$ away from the vertex, its polar expression is $d{r}^2+(\Theta /2\pi {)}^2{\sinh }^{2}rd{\theta }^2$, where $\theta$ has period $2\pi$. A circle of small radius $r$ has length asymptotic to $\Theta r$. A smooth Riemannian surface instead has circumference asymptotic to $2\pi r$, so the metric extends smoothly at the vertex exactly when $\Theta =2\pi$.