 This work determines that a 3D director field can be uniquely described by five local fields, which are related to each other and the curvature of the embedding space through six differential relations. The study shows that in positively curved space, the pure twist phase is the only solution, while in hyperbolic space, uniform distortion fields correspond to foliations of space by, not necessarily parallel, congruent helices. Further analysis of the obtained compatibility fields is expected to allow for the construction of new non-uniform director fields. This article was offered by Louis C. B. D. A. Silver and F. E. Frati.