Poincaré-Hopf Theorem for Filippov vector fields on 2-dimensional compact manifolds

Abstract: The Poincar\'e-Hopf Theorem relates the Euler characteristic of a 2-dimensional compact manifold to the local behavior of smooth vector fields defined on it. However, despite the importance of Filippov vector fields, concerning both their theoretical and applied aspects, until now, it was not known whether this theorem extends to Filippov vector fields. In this paper, we demonstrate that the Poincar\'e-Hopf Theorem applies to Filippov vector fields defined on 2-dimensional compact manifolds with smooth switching manifolds. As a result, we establish a variant of the Hairy Ball Theorem, asserting that "any Filippov vector field on a sphere with smooth switching manifolds must have at least one singularity (in the Filippov sense) with positive index". This extension is achieved by introducing a new index definition that includes the singularities of Filippov vector fields, such as pseudo-equilibria and tangential singularities. Our work extends the classical index definition for singularities of smooth vector fields to encompass those of Filippov vector fields with smooth switching manifolds. This extension is based on an invariance property under a regularization process, allowing us to establish all classical index properties. We also compute the indices of all generic $\Sigma$-singularities and some codimension-1 $\Sigma$-singularities, including fold-fold tangential singularities, regular-cusp tangential singularities, and saddle-node pseudo-equilibria.