  In this paper, we propose a model of arm reaching movements expressed in
terms of geodesics in a sub-Riemannian space. We will choose a set of kinematic
variables to which motor cortical cells are selective with the purpose of
modelling the specific task of reaching. Minimizing trajectories will be
recovered as suitable geodesics of the geometric spaces arising from the
selective behaviour of M1 neurons. We will then extend this model by including
the direction of arm movement. On this set, we will define a suitable
sub-Riemannian metric able to provide a geometric interpretation of
two-dimensional task-dependent arm reaching movements.
