This work investigates nematic liquid crystals in three-dimensional curved space, and determines which director deformation modes are compatible with each possible type of non-Euclidean geometry. Previous work by Sethna et al. [1] showed that double twist is frustrated in flat space $\mathbb{R}^3$, but can fit perfectly in the hypersphere $\mathbb{S}^3$. Here, we extend that work to all four deformation modes (splay, twist, bend, and biaxial splay) and all eight Thurston geometries [2]. Each pure mode of director deformation can fill space perfectly, for at least one type of geometry. This analysis shows the ideal structure of each deformation mode in curved space, which is frustrated by the requirements of flat space.

1. Sethna J. P., Wright D. C. and Mermin N. D., 1983 Phys. Rev. Lett. 51 467–70.
2. J.-F. Sadoc, R. Mosseri and J. Selinger, New Journal of Physics 22 (2020) 093036.