Surface Nematic Uniformity

TL;DR

This work addresses the problem of identifying uniform nematic fields tangent to smooth surfaces, showing that such fields can exist only on pseudospherical surfaces with Gaussian curvature $K<0$ and must be parallel transported along a special class of geodesics called uniform geodesics. The authors develop a moving-frame formalism to express uniformity via constant distortion components and derive a precise condition $K=-(b_{\perp}^2+S^2)$, enabling a full classification of uniform geodesics on Beltrami's pseudosphere and, by isometry, on all pseudospherical surfaces. They provide an explicit construction of all uniform fields on the pseudosphere, revealing a dual right/left pairing of geodesic bundles and illustrating how the fields can possess or avoid defects depending on the chosen system. The results establish a principled, intrinsic approach to surface nematic uniformity, with potential implications for planar handedness and geometric frustration on curved substrates, and they suggest pathways to extend the framework to more general manifolds via isometries.

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).

Figures (7)

• Figure 1: Beltrami's pseudosphere represented in the parametrization \ref{['eq:pseudosphere:r']}. The coordinate frame $(\bm{e}_u,\bm{e}_v,\bm{\nu})$ in \ref{['eq:pseudosphere:frame_e']} glides over it in accordance with \ref{['eq:pseudosphere:gliding_laws']}.
• Figure 2: Pre-images of geodesics in the $(u,v)$ plane. The meridians of $\mathscr{S}$ are straight lines parallel to the $v$-axis. All other geodesics are families of circles with centers on the $u$-axis. The particular family represented here is obtained by choosing in \ref{['eq:pseudosphere_geodesic_coordinates']}$u_0(\varphi)$ as a constant and $\rho(\varphi)$ as a monotonic function of $\varphi$, so as to make \ref{['eq:pseudosphere_invertibility']} satisfied. A single circle in this family is drawn for a fixed value of $\varphi$ and $\vartheta$ ranging in the admissible interval \ref{['eq:pseudosphere_admissible_interval']}.
• Figure 3: Meridians are uniform geodesics of the pseudosphere. They convey a uniform field $\bm{n}$ (represented by headless directors) whose (red) field lines are loxodromes.
• Figure 4: Pre-images of uniform geodesics shown in the $(u,v)$ plane for different values of the parameter $m$ designating the separating meridian (marked in red). Right uniform geodesics are blue, while left uniform geodesics are green. The (exceptionally) black geodesic in panels (a) and (c) is precisely the same curve, belonging to two separate systems of right and left uniform geodesics.
• Figure 5: Hybrid uniform geodesics.