The Poincaré–Hopf Theorem for line fields revisited

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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 2k4. 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.

MSC

primary
57R22
secondary
57R25
55M25
53C80
76A15

Keywords

Poincaré–Hopf Theorem
Line fields
Topological defects
Condensed matter physics

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