A generalisation of the Poincar\'e-Hopf Theorem

TL;DR

This paper extends the Poincaré-Hopf Theorem to encompass real-analytic vector fields on surfaces with piecewise smooth boundaries, broadening its applicability beyond smooth boundary cases.

Contribution

It generalizes the classical Poincaré-Hopf Theorem to include vector fields tangential to surfaces with piecewise smooth boundaries, not necessarily aligned with outward normals.

Findings

Extended the theorem to piecewise smooth boundaries

Provided mathematical framework for non-normal tangential vector fields

Broadened the theorem's applicability to more complex surfaces

Abstract

The Poincar\'e-Hopf Theorem is a conservation law for real-analytic vector fields, which are tangential to a closed surface (such as a torus or a sphere). The theorem also governs real-analytic vector fields, which are tangential to surfaces with smooth boundaries; in these cases, the vector field must be pointing in the outward normal direction along the boundary. In this paper, I will generalise the Poincar\'e-Hopf Theorem for real-analytic vector fields that are tangential to surfaces with piecewise smooth boundaries, and not parallel to the outward normal of the boundary.