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Surface nematic uniformity

Andrea Pedrini* and Epifanio G. Virga†

  • *Contact author: andrea.pedrini@unipv.it
  • †Contact author: eg.virga@unipv.it
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Phys. Rev. E 113, 015422 – Published 30 January, 2026

DOI: https://doi.org/10.1103/xpvv-zv23

Abstract

An antlike observer confined to a two-dimensional surface traversed by stripes would wonder whether such a 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 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 evoke 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). The proved existence of surface nematic uniform fields suggests the definition of a generalized intrinsic elastic energy for fluid membranes with nematic order, which is but one of the many possible applications of our geometric result.

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Phys. Rev. E 113, 015422– Published 30 January, 2026
  • Received 9 May 2025
  • Revised 13 October 2025
  • Accepted 9 January 2026
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