  Three-dimensional orientational order in systems whose ground states possess
non-zero, chiral gradients typically exhibits line-like structures or defects:
$\lambda$ 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 generalisation of the umbilic points
of a surface. We characterise these lines in terms of the natural vector
bundles that the order defines and show that they give a way to localise and
identify Skyrmion distortions in chiral materials -- in particular that they
supply a natural representative of the Poincar\'{e} 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.
