Teardrop orbifold

A teardrop orbifold is an orbifold whose underlying topological space is a two-sphere and which has exactly one cone point of an orbifold of order $m>1$. A neighborhood of that point is modeled on a disk modulo rotations by multiples of $2\pi /m$; every other point has trivial local group.

Topologically, a disk modulo this finite rotation group is still a disk. The local stabilizer is therefore extra orbifold information invisible in the underlying sphere. The singularity order distinguishes teardrops that have the same underlying topological space.

The teardrop is a standard example of a bad orbifold: it has no manifold orbifold covering. Cutting it into an ordinary disk and a disk with one order-$m$ cone point exposes incompatible nontrivial covering behavior along their common boundary. Thus being locally a finite quotient does not imply being globally covered by a manifold.^[ref]