Tangent unit-vector fields: Nonabelian homotopy invariants and the Dirichlet energy

Apala Majumdar, J.M. Robbins, Maxim Zyskin

Research output: Contribution to journalArticlepeer-review

3 Scopus citations


Let O be a closed geodesic polygon in S2. Maps from O into S2 are said to satisfy tangent boundary conditions if the edges of O are mapped into the geodesics which contain them. Taking O to be an octant of S2, we evaluate the infimum Dirichlet energy, E (H), for continuous tangent maps of arbitrary homotopy type H. The expression for E (H) involves a topological invariant - the spelling length - associated with the (nonabelian) fundamental group of the n-times punctured two-sphere, π1 (S2 - {s1, ..., sn}, *). These results have applications for the theoretical modelling of nematic liquid crystal devices. To cite this article: A. Majumdar et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009). © 2009 Académie des sciences.
Original languageEnglish (US)
Pages (from-to)1159-1164
Number of pages6
JournalComptes Rendus Mathematique
Issue number19-20
StatePublished - Oct 2009
Externally publishedYes

Bibliographical note

KAUST Repository Item: Exported on 2020-10-01
Acknowledged KAUST grant number(s): KUK-C1-013-04
Acknowledgements: A.M. is supported by Award No. KUK-C1-013-04, made by King Abdullah University of Science and Technology (KAUST) to the Oxford Centre for Collaborative Applied Mathematics (OCCAM). We thank Ulrike Tillmann for stimulating discussions and we thank Cameron Hall for help with the French summary.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.


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