Surface Nematic Uniformity

TL;DR

This paper characterizes all uniform nematic director fields on surfaces with constant negative Gaussian curvature, showing they are related to special geodesic systems and explicitly solving the case of Beltrami's pseudosphere.

Contribution

It provides a complete classification of uniform nematic fields on negatively curved surfaces and links them to geodesic systems, extending known results to all such surfaces.

Findings

Uniform nematic fields exist only on surfaces with constant negative Gaussian curvature.

All uniform fields are parallel transported along special geodesic systems.

Explicit solutions are provided for Beltrami's pseudosphere.

Abstract

An ant-like observer confined to a two-dimensional surface traversed by stripes would wonder whether this striped landscape could be devised in such a way as to appear to be the same wherever they go. Differently stated, this is the problem studied in this paper. In a more technical jargon, we determine all possible uniform nematic fields on a smooth surface. It was already known that for such a field to exist, the surface must have constant negative Gaussian curvature. Here, we show that all uniform nematic fields on such a surface are parallel transported (in Levi-Civita's sense) by special systems of geodesics, which (with scant inventiveness) are termed uniform. We prove that, for every geodesic on the surface, there are two systems of uniform geodesics that include it; they are conventionally called right and left, to allude at a possible intrinsic definition of handedness. We found explicitly all uniform fields for Beltrami's pseudosphere. Since both geodesics and uniformity are preserved under isometries, by a classical theorem of Minding, the solution for the pseudosphere carries over all other admissible surfaces, thus providing a general solution to the problem (at least in principle).