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