# Clarification on a statement Bernevig's textbook on Topological insulators

On page 11 of the aforementioned book Bernevig claims after doing some calculation that the integral of Berry curvature over a sphere containing a monopole is $$2\pi$$. Now my question is the following: the berry curvature is defined as the curl of the berry connection and hence its integral over a closed surface should vanish by the divergence theorem. How is this possible.

It hinges on the fact that the berry connection is not globally defined on the sphere. For Stokes' theorem(or Divergence theorem as you stated) to work, you need a vector field $${A}$$ defined on the whole manifold.