The circular Morse-Smale characteristic of closed surfaces

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 S-1 valued functions on a surface Sigma subset of R-3, 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 g, embedded in a certain way in the first Heisenberg group, with respect to the horizontal distribution of the Heisenberg group.