The Poincar\'e-Hopf Theorem for line fields revisited

TL;DR

This paper revisits the Poincaré-Hopf theorem for line fields with point singularities, providing a comprehensive proof valid in all dimensions, clarifying historical developments and extending previous results.

Contribution

It offers a unified, detailed proof of the Poincaré-Hopf theorem for line fields with point singularities applicable to all dimensions.

Findings

The theorem holds in all dimensions for line fields with point singularities.

Clarification of the theorem's historical development and scope.

Extension of previous results to a more general setting.

Abstract

A Poincar\'e-Hopf Theorem for line fields with point singularities on orientable surfaces can be found Hopf's 1956 Lecture Notes on Differential Geometry. In 1955 Markus presented such a theorem in all dimensions, but Markus' statement only holds in even dimensions $2k \geq 4$. In 1984 J\"{a}nich presented a Poincar\'{e}-Hopf theorem for line fields with more complicated singularities and focussed on the complexities arising in the generalised setting. In this expository note we review the Poincar\'e-Hopf Theorem for line fields with point singularities, presenting a careful proof which is valid in all dimensions.