Surface nematic uniformity

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.