The Poincaré-Hopf Theorem for line fields revisited

Diarmuid Crowley; Mark Grant

doi:10.1016/j.geomphys.2017.03.011

The Poincaré-Hopf Theorem for line fields revisited

Diarmuid Crowley, Mark Grant

Mathematical Science

University of Melbourne

Research output: Contribution to journal › Article › peer-review

5 Citations (Scopus)

8 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 language	English
Pages (from-to)	187-196
Number of pages	10
Journal	Journal of Geometry and Physics
Volume	117
Early online date	29 Mar 2017
DOIs	https://doi.org/10.1016/j.geomphys.2017.03.011
Publication status	Published - Jul 2017

Keywords

Poincaré–Hopf theorem
Line fields
Topological defects
Condensed matter physics

Access to Document

10.1016/j.geomphys.2017.03.011Licence: Unspecified

Accepted Manuscript
© 2017. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/
Accepted author manuscript, 194 KBLicence: CC BY-NC-ND

https://arxiv.org/abs/1612.04073Licence: Unspecified

Cite this

@article{89b28ae2eea4410ab29c0480a4d00b50,

title = "The Poincar{\'e}-Hopf Theorem for line fields revisited",

abstract = "A Poincar{\'e}–Hopf Theorem for line fields with point singularities on orientable surfaces can be found Hopf{\textquoteright}s 1956 Lecture Notes on Differential Geometry. In 1955 Markus presented such a theorem in all dimensions, but Markus{\textquoteright} statement only holds in even dimensions 2k≥4. In 1984 J{\"a}nich presented a Poincar{\'e}–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 Poincar{\'e}–Hopf Theorem for line fields with point singularities, presenting a careful proof which is valid in all dimensions.",

keywords = "Poincar{\'e}–Hopf theorem, Line fields, Topological defects, Condensed matter physics",

author = "Diarmuid Crowley and Mark Grant",

year = "2017",

month = jul,

doi = "10.1016/j.geomphys.2017.03.011",

language = "English",

volume = "117",

pages = "187--196",

journal = "Journal of Geometry and Physics",

issn = "0393-0440",

publisher = "Elsevier",

}

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

SN - 0393-0440

VL - 117

SP - 187

EP - 196

JO - Journal of Geometry and Physics

JF - Journal of Geometry and Physics

ER -

The Poincaré-Hopf Theorem for line fields revisited

Abstract

Keywords

Access to Document

Fingerprint

Cite this