A generalization of the Gauss-Bonnet and Hopf-Poincar\'e theorems

We consider a locally trivial fiber bundle $\pi :E\to M$ over a compact oriented two-dimensional manifold $M$, and a section $s$ of this bundle defined over $M\setminus \mathrm{\Sigma }$, where $\mathrm{\Sigma }$ is a discrete subset of $M$. We call the set $\mathrm{\Sigma }$ the set of singularities of the section $s : M \setminus \Sigma \to E$\mathrm{.}Weassumethatthebehaviorofthesection$s$ at the singularities is controlled in the following way: $s(M\setminus \mathrm{\Sigma })$ coincides with the interior part of a surface $S\subset E$ with boundary $\mathrm{\partial }S$, and $\mathrm{\partial }S$ is ${\pi }^{-1}(\mathrm{\Sigma })$. For such sections $s$ we define an index of $s$ at a point of $\mathrm{\Sigma }$, which generalizes in the natural way the index of zero of a vector field, and then prove that the sum of this indices at the points of $\mathrm{\Sigma }$ can be expressed as integral over $S$ of a $2$-form constructed via a connection in $E$. Then we show that the classical Hopf-Poincar'e-Gauss-Bonnet formula is a partial case of our result, and consider some other applications. Keywords: singularity of section, index of singular point, curvature, projective bundle, $G$-structure