The Poincaré-Hopf Theorem for line fields revisited

Diarmuid Crowley, Mark Grant

Research output: Contribution to journalArticle

2 Citations (Scopus)
6 Downloads (Pure)

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.
Original languageEnglish
Pages (from-to)187-196
Number of pages10
JournalJournal of Geometry and Physics
Volume117
Early online date29 Mar 2017
DOIs
Publication statusPublished - Jul 2017

Keywords

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

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