Parallel transport and defects on nematic shells

Nematic shells are thin films of nematic liquid crystal deposited on the boundary of colloidal particles,
where liquid crystal molecules may freely glide, while remaining tangent to the surface substrate. The
surface nematic order is described here by an appropriate tensor field Q, which vanishes wherever a defect
occurs in themolecular order.We show howthe classical concept of parallel transport on a manifold introduced
by Levi-Civita can be adapted to this setting to define the topological charge m of a defect. We arrive at a
simple formula to compute m from a generic representation of Q. In a number of separate appendices, we
revisit in a unified language several, apparently disparate applications of Levi-Civita’s parallel transport.