Champs de vecteurs unités tangents: invariants homotopiques non abelien et énergie de Dirichlet

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, Ɛ(H) , for continuous tangent maps of arbitrary homotopy type H. The expression for Ɛ(H) involves a topological invariant – the spelling length – associated with the (nonabelian) fundamental group of the n-times punctured two-sphere, pi1(s2 - {s1,...,sn}, *). These results have applications for the theoretical modelling of nematic liquid crystal devices.