A generalization of the Gauss–Bonnet–Hopf–Poincaré formula for sections and branched sections of bundles

For a two-dimensional compact oriented Riemannian manifold (M,g), and a vector field V on M, the Hopf–Poincaré theorem combined with the Gauss–Bonnet theorem gives the Gauss–Bonnet–Hopf–Poincaré (GBHP) formula: ∑z∈Z(V)indz(V)= [Formula presented] ∫MKdσ, where Z(V) is the set of zeros of V, indz(V) i...