Umbilic Lines in Orientational Order

Three-dimensional orientational order in systems whose ground states possess nonzero gradients typically exhibits linelike structures or defects:. lines in cholesterics or Skyrmion tubes in ferromagnets, for example. Here, we show that such lines can be identified as a set of natural geometric singularities in a unit vector field, the generalization of the umbilic points of a surface. We characterize these lines in terms of the natural vector bundles that the order defines and show that they give a way to localize and identify Skyrmion distortions in chiral materials-in particular, that they supply a natural representative of the Poincare dual of the cocycle describing the topology. Their global structure leads to the definition of a self-linking number and helicity integral which relates the linking of umbilic lines to the Hopf invariant of the texture.