Surface Nematic Quasi-Uniformity

TL;DR

This paper introduces the concept of quasi-uniform nematic line fields on surfaces, characterizing them as parallel transported along geodesics, which broadens understanding of nematic order patterns beyond uniform fields.

Contribution

It generalizes the notion of uniform nematic fields to quasi-uniform fields on arbitrary surfaces and proves their characterization via Levi-Civita parallel transport.

Findings

Quasi-uniform fields are characterized by Levi-Civita parallel transport.

Construction methods for quasi-uniform fields on various surfaces are provided.

The notion of quasi-uniformity extends the understanding of nematic patterns on complex surfaces.

Abstract

Line fields on surfaces are a means to describe the nematic order that may pattern them. The least distorted nematic fields are called uniform, but they can only exist on surfaces with negative constant Gaussian curvature. To identify the least distorted nematic fields on a generic surface, we relax the notion of uniformity into that of quasi-uniformity and prove that all such fields are parallel transported (in Levi-Civita's sense) by the geodesics of the surface. Both global and local constructions of quasi-uniform fields are presented to illustrate both richness and significance of the proposed notion.