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.
Highlights
To motivate our constructions, we first review the compatibility condition and reconstruction formula for a director in two dimensions, given by Niv and Efrati [13]
In this paper we have given the connection between the distortions of a liquid crystal and the metric connection, which allows us to derive a set of compatibility conditions between those distortions
We describe several different methods for reconstructing a director from its gradients, which can be applied to a variety of different geometric problems
Summary
We first review the compatibility condition and reconstruction formula for a director in two dimensions, given by Niv and Efrati [13]. The reconstruction problem in two dimensions is to find a director field given (generic) splay and bend functions, s and κ. The compatibility condition gives the component of the director along e1; the component along e2 follows from normalisation, |n| = 1 This frame is undefined when both the splay and bend are constant; in this case a uniform frame with the desired functions is constructed, as described in reference [13]. In three dimensions there are nine structure functions (equivalently components of the connection) so that more information than just the basic director distortions of splay, twist and bend is needed to specify them all and facilitate a geometric reconstruction
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