Umbilics: Geometry and Topology of Orientational Order Seminar

A variety of physical systems are characterised by points of degeneracy or singularity known as umbilics. Classical examples include points of degeneracy in the principal curvatures of surfaces, of shear fields as in gravitational lensing, and points (or lines) of circular polarisation, known as C lines, in generic optical or electromagnetic fields. Precisely analogous lines also characterise the geometrical properties of general vector fields, such as the magnetisation in Skyrmion textures of chiral magnets and the director field in cholesteric liquid crystals, where they are known as lambda defects. I shall describe the properties of umbilic lines as degeneracies in the shape operator, how they encode the topology of three-dimensional vector fields in terms of an integer homology class and the characteristic classes of associated vector bundles, and how they relate to classical invariants such as the hedgehog charge or Skyrmion number in applications to cholesteric liquid crystals and chiral magnets.