  Uniaxial nematic liquid crystals are modelled in the Oseen-Frank theory
through a unit vector field $n$. This theory has the apparent drawback that it
does not respect the head-to-tail symmetry in which $n$ should be equivalent to
-$n$. This symmetry is preserved in the constrained Landau-de Gennes theory
that works with the tensor $Q=s\big(n\otimes n- \frac{1}{3} Id\big)$.We study
the differences and the overlaps between the two theories. These depend on the
regularity class used as well as on the topology of the underlying domain. We
show that for simply-connected domains and in the natural energy class
$W^{1,2}$ the two theories coincide, but otherwise there can be differences
between the two theories, which we identify. In the case of planar domains we
completely characterise the instances in which the predictions of the
constrained Landau-de Gennes theory differ from those of the Oseen-Frank
theory.
