### 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≥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.

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.

Original language | English |
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Pages (from-to) | 187-196 |

Number of pages | 10 |

Journal | Journal of Geometry and Physics |

Volume | 117 |

Early online date | 29 Mar 2017 |

DOIs | |

Publication status | Published - Jul 2017 |

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### Keywords

- Poincaré–Hopf theorem
- Line fields
- Topological defects
- Condensed matter physics

### Cite this

Crowley, D., & Grant, M. (2017). The Poincaré-Hopf Theorem for line fields revisited.

*Journal of Geometry and Physics*,*117*, 187-196. https://doi.org/10.1016/j.geomphys.2017.03.011