The bifurcation theory of the Gauss-Bonnet-Chern topological current and Morse function

We detail the bifurcation of the GBC topological current and find different directions of the bifurcation. The GBC topological particles are found to split at the degenerate point of the field function φ, but the total charge of the GBC topological particles remains unchanged. Further, we find the GBC topological current of the gradient field of arbitrary function f bifurcating at its critical points until f evaluates into the Morse function; the topological charge of each topological particle is then ±1. (orig.)