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


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