  We give a description of the intrinsic geometry of elastic distortions in
three-dimensional nematic liquid crystals and establish necessary and
sufficient conditions for a set of functions to represent these distortions by
describing how they couple to the curvature tensor. We demonstrate that, in
contrast to the situation in two dimensions, the first-order gradients of the
director alone are not sufficient for full reconstruction of the director field
from its intrinsic geometry: it is necessary to provide additional information
about the second-order director gradients. We describe several different
methods by which the director field may be reconstructed from its intrinsic
geometry. Finally, we discuss the coupling between individual distortions and
curvature from the perspective of Lie algebras and groups and describe
homogeneous spaces on which pure modes of distortion can be realised.
