Is there any idea of evaluating integer topological invariant that is given by the integral form, say the Gauss-Bonnet theorem, by “not doing the integral”?
In other words, is there any way to evaluate the global quantity by only looking at local quantities?
I want to calculate the Chern number of physical system, especially the Chern insulator. Conventionally we calculate the Chern number by integrating the Berry curvature in the full Brillouin zone. But my question is that whether there is any method of calculating the Chern number without any integration.