Intrinsic geometry and director reconstruction for three-dimensional liquid crystals

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.

Automated summary

This article gives a description of the intrinsic geometry of elastic distortions in three-dimensional nematic liquid crystals and establishes necessary and sufficient conditions for a set of functions to represent these distortions by describing how they couple to the curvature tensor. It is shown that, unlike 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 and additional information about the second-order director gradients is necessary. Furthermore, various methods for reconstructing the director field from its intrinsic geometry are described, and the coupling between individual distortions and curvature is discussed from the perspective of Lie algebras and groups, along with the description of homogeneous spaces where pure modes of distortion can be realized.