### Abstract

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

*Journal of Geometry and Physics*,

*117*, 187-196. https://doi.org/10.1016/j.geomphys.2017.03.011

**The Poincaré-Hopf Theorem for line fields revisited.** / Crowley, Diarmuid; Grant, Mark.

Research output: Contribution to journal › Article

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

}

TY - JOUR

T1 - The Poincaré-Hopf Theorem for line fields revisited

AU - Crowley, Diarmuid

AU - Grant, Mark

PY - 2017/7

Y1 - 2017/7

N2 - 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.

AB - 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.

KW - Poincaré–Hopf theorem

KW - Line fields

KW - Topological defects

KW - Condensed matter physics

U2 - 10.1016/j.geomphys.2017.03.011

DO - 10.1016/j.geomphys.2017.03.011

M3 - Article

VL - 117

SP - 187

EP - 196

JO - Journal of Geometry and Physics

JF - Journal of Geometry and Physics

SN - 0393-0440

ER -