Deformations in the general position of the optimal functions on oriented surfaces with boundary

Abstract

UDC 516.91
It is considered simple functions with non-degenerated singularities on smooth compact oriented surfaces with the boundary. Authors describe a connection between optimality and polarity of Morse functions, $m$-functions and $mm$-functions on smooth compact oriented connected surfaces. The concept of an equipped Kronrod – Reeb graph is used to define a deformation in general position. Also, it is obtained the whole list of deformations of simple functions of one of abovedescribed class on torus, 2-dimensional disc with the boundary and on connected sum of two toruses.