Two Morse functions and singularities of the product map (Q303546)

Two distances \(D(F,G)\) and \(d(F,G)\) between Morse functions \(F,G:X\to\mathbb{R}\), where \(X\) is a closed smooth manifold, are defined and studied. These distances are described in terms of the number of births, deaths of critical points and passing of critical values of all generic homotopies between \(F\) and \(G\). In the paper the author gives some upper bounds for the distances \(D(F,G)\) and \(d(F,G)\) in terms of singularities of the product map \(\varphi=(F,G): X\to\mathbb{R}^2\) under assumption that \(\dim X\geq 2\) and \(\varphi\) is stable. The main result provides a very detailed estimates of both distances by the use of the number of negative slope inflection points, negative double tangent lines, negative slope cusp and double points of negative slope subarcs of \(\varphi\) and Morse indices of \(F\) and \(G\).