Liquid crystal director fields in three-dimensional non-Euclidean geometries

TL;DR

This paper explores how nematic liquid crystal director fields deform in various three-dimensional non-Euclidean geometries, revealing which configurations are compatible with each geometry type and extending prior flat-space results.

Contribution

It extends the analysis of liquid crystal director modes to all Thurston geometries, identifying compatible deformation modes in curved spaces beyond flat space.

Findings

Each deformation mode can fill at least one non-Euclidean geometry.

Double twist fits perfectly in the hypersphere $S^3$, extending previous work.

All four deformation modes are compatible with some Thurston geometries.

Abstract

This paper 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. showed that double twist is frustrated in flat space $R^3$, but can fit perfectly in the hypersphere $S^3$. Here, we extend that work to all four deformation modes (splay, twist, bend, and biaxial splay) and all eight Thurston geometries. 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.