The Poincaré–Hopf Theorem for line fields revisited

Abstract

A Poincaré–Hopf Theorem for line fields with point singularities on orientable surfaces can be found in 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 2 k ≥ 4 . In 1984 Jänich presented a Poincaré–Hopf theorem for line fields with more complicated singularities and focussed on the complexities arising in the generalized setting. In this expository note we review the Poincaré–Hopf Theorem for line fields with point singularities, presenting a careful proof which is valid in all dimensions.