Intrinsic Geometry and Director Reconstruction for Three-Dimensional Liquid Crystals

TL;DR

This paper explores the intrinsic geometry of elastic distortions in 3D nematic liquid crystals, establishing conditions for representing distortions and methods for director field reconstruction, highlighting differences from 2D cases.

Contribution

It introduces necessary and sufficient conditions for intrinsic distortion representation and discusses director reconstruction methods, emphasizing the role of second-order gradients.

Findings

First-order gradients are insufficient for full director reconstruction in 3D.

Multiple methods for director field reconstruction from intrinsic geometry.

Coupling between distortions and curvature analyzed via Lie groups and homogeneous spaces.

Abstract

We give a description of the intrinsic geometry of elastic distortions in three-dimensional nematic liquid crystals and establish necessary and sufficient conditions for a set of functions to represent these distortions by describing how they couple to the curvature tensor. We demonstrate that, in contrast to the situation in two dimensions, the first-order gradients of the director alone are not sufficient for full reconstruction of the director field from its intrinsic geometry: it is necessary to provide additional information about the second-order director gradients. We describe several different methods by which the director field may be reconstructed from its intrinsic geometry. Finally, we discuss the coupling between individual distortions and curvature from the perspective of Lie algebras and groups and describe homogeneous spaces on which pure modes of distortion can be realised.