Aspects of Global Analysis of Circle-Valued Mappings

We deal with the minimum number of critical points of circular functions with respect to two different classes of functions. The first one is the whole class of smooth circular functions and, in this case, the minimum number is the so called circular \(\varphi\) -category of the involved manifold. The second class consists of all smooth circular Morse functions, and the minimum number is the so called circular Morse–Smale characteristic of the manifold. The investigations we perform here for the two circular concepts are being studied in relation with their real counterparts. In this respect, we first evaluate the circular \(\varphi\)-category of several particular manifolds. In Sect. 5, of more survey flavor, we deal with the computation of the circular Morse–Smale characteristic of closed surfaces. Section 6 provides an upper bound for the Morse–Smale characteristic in terms of a new characteristic derived from the family of circular Morse functions having both a critical point of index 0 and a critical point of index n. The minimum number of critical points for real or circle valued Morse functions on a closed orientable surface is the minimum characteristic number of suitable embeddings of the surface in \(\mathbb{R}^{3}\) with respect to some involutive distributions. In the last section we obtain a lower and an upper bound for the minimum characteristic number of the embedded closed surfaces in the first Heisenberg group with respect to its noninvolutive horizontal distribution.