Moving frames and compatibility conditions for three-dimensional director fields

The geometry and topology of the region in which a director field is embedded impose limitations on kind supported orientational order. These manifest as compatibility conditions that relate quantities describing to embedding space. For example, two dimensions (2D) splay bend fields suffice determine uniquely (up rigid motions) must comply with one relation linear Gaussian curvature manifold. In 3D there are additional local director, i.e. available observer residing within material, number distinct ways yield geometric frustration. So far it was unknown how many such required describe field, nor what relations they satisfy. this work, we address these questions directly. We employ method moving frames show fully determined by five fields. shown be related each other space through six differential relations. As an application our method, characterize all uniform distortion fields, i.e., directors for characterizing constant space, manifolds curvature. classification phases has been recently provided Euclidean where textures correspond foliations parallel congruent helices. non-vanishing curvature, pure twist phase only solution positively curved while hyperbolic (non-necessarily parallel) Further analysis expected allow also construct new non-uniform