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 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 $2k\ge 4$. In 1984 Jänich presented a Poincaré-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 Poincaré-Hopf Theorem for line fields with point singularities, presenting a careful proof which is valid in all dimensions.

13 pages, 3 figures. v2: Final version, to appear in Journal of Geometry and Physics

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