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

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.