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)=12π∫MKdσ, where Z(V) is the set of zeros of V, indz(V) is the index of V at z∈Z(V), and K is the curvature of g. We consider a locally trivial fiber bundle π:E→M over a compact oriented two-dimensional manifold M, and a section s of this bundle defined over M∖Σ, where Σ is a discrete subset of M called the set of singularities of the section. We assume that the behavior of the section s at the singularities is controlled in the following way: s(M∖Σ) coincides with the interior part of a surface S⊂E with boundary ∂S, and ∂S is π−1(Σ). For such sections s we define an index of s at a point of Σ, which generalizes in the natural way the index of zero of a vector field, and then prove that the sum of these indices at the points of Σ can be expressed as an integral over S of a 2-form constructed via a connection in E, thus we obtain a generalization of the GBHP formula. Also we consider branched sections with singularities, define an index of a branched section at a singular point, and find a generalization of the GBHP formula for the branched sections.