Height functions on products of spheres and associated Reeb digraphs and level sets

Abstract

Height functions are fundamental and important objects and tools in mathematics, especially in geometry such as differential topology and differential geometry and some related singularity theory of differentiable maps. Our interest, especially interest of the author, lies in obtaining explicit lists of such functions. Recently, as information on so-called higher degrees, he is also interested in their naturally defined 1st derivatives. We consider natural maps on products of spheres which are variants of specific cases of so-called moment maps on toric symplectic manifolds. We also generalize cases of the canonical projections of the unit spheres. This is a further result on related previous study of the author. We use Reeb graphs, graphs being natural quotient spaces of manifolds of the domains of nice functions such as Morse-Bott functions, and consisting of connected components of level sets. They are fundamental tools and objects since the last century.