Flat cone

A cone is intrinsically flat away from its vertex but has nontrivial angular holonomy.

The total angle around the tip of a Euclidean cone. For a translation surface zero of order $m$ it equals $2\pi (m+1)$, while for a holomorphic quadratic differential zero of order $m$ it equals $(m+2)\pi$. An angle $2\pi$ is a regular point.

A cone point on a metric surface has total angular measure $\theta$ about its tip. The flat model is $d{r}^2+(\theta /2\pi {)}^2{r}^{2}d{\phi }^2$, with $0\le \phi <2\pi$; the curvature-minus-one model replaces $r$ by $\sinh r$. In these models the tip is smooth when $\theta =2\pi$. For integer $m>1$, angle $2\pi /m$ gives a cyclic orbifold point of order $m$. Under a degree-$q$ local branched lift, the total angle is multiplied by $q$, and the residual orbifold order, when defined, is $m/q$.

The punctured circular coneis geodesically incomplete because a generator reaches the missing vertex in finite time. It is an inextendible embedded surface: any proper connected embedded extension would have to add the vertex, but its tangent planes have different limits as $\theta$ varies. Nevertheless its Gaussian curvature of a cone away from its vertex is identically zero.

A developing map locally unfolds a flat surface into the Euclidean plane.

Cone isometries preserve its angular identification; those fixing a point are the identity or an axial-plane reflection.