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 $M \setminus \Sigma$, where $\Sigma$ is a discrete subset of $M$. We call the set $\Sigma$ the set of singularities of the section $s : M \setminus \Sigma \to E$. We assume that the behavior of the section $s$ at the singularities is controlled in the following way: $s(M \setminus \Sigma)$ coincides with the interior part of a surface $S \subset E$ with boundary $\partial S$, and $\partial S$ is $\pi^{-1}(\Sigma)$. For such sections $s$ we define an index of $s$ at a point of $\Sigma$, 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 $\Sigma$ 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