  The Lebwohl-Lasher (LL) model is the traditional model used to describe the
nematic-isotropic transition of real liquid crystals. In this paper, we develop
a numerical study of the temperature behaviour and of finite-size scaling of
the two-dimensional (2D) LL-model. We discuss two possible scenarios. In the
first one, the 2D LL-model presents a phase transition similar to the
topological transition appearing in the 2D XY-model. In the second one, the 2D
LL-model does not exhibit any critical transition, but its low temperature
behaviour is rather characterized by a crossover from a disordered phase to an
ordered phase at zero temperature. We realize and discuss various comparisons
with the 2D XY-model and the 2D Heisenberg model. Adding to previous studies of
finite-size scaling behaviour of the order parameter and conformal mapping of
order parameter profile, we analyze the critical scaling of the probability
distribution function, hyperscaling relations and stiffness order parameter and
conclude that the second scenario (no critical transition) is the most
plausible.
