In this paper we first compute the circular version of the Morse-Smale characteristic
of all closed surfaces. We also observe that the critical points of the real valued
height functions alongside those of some
valued functions on a surface
, are the characteristic points
with respect to some involutive distributions. We finally study the size of the characteristic set of
the compact orientable surface of genus
, embedded in a certain way in the first Heisenberg group, with respect to the horizontal distribution of the Heisenberg group.