  Topological quantum states have been proposed and investigated on
two-dimensional flat surfaces or lattices with different geometries like the
plane, cylinder and torus. Here, we study quantum anomalous Hall (QAH) or Chern
insulator (CI) states on two-dimensional singular surfaces (such as conical and
helicoid-like surfaces). Such singular geometries can be constructed based on
the disk geometry and a defined unit sector with $n$-fold rotational symmetry.
The singular geometry induces novel and intriguing features of CI/QAH states,
such as in-gap and in-band core states, charge fractionalization, and multiple
branches of edge excitations.
