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

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.