Concordance and isotopy of metrics with positive scalar curvature (Q359549)

The paper discusses the relation between two notions for Riemannian metrics on compact manifolds: positive scalar curvature (psc) concordant metrics and psc-isotopic. Theorem A claims that for manifolds of dimension greater or equal to 3 for any two psc-concordant metrics \(g_0\) and \(g_1\) there exists a pseudo-isotopy \(\Phi\) such that the psc-metrics \(g_0\) and \(\Phi^{*}g_1\) are psc-isotopic. The second main claim is a corollary of the first one based on the Cerf's result for simply connected compact manifolds of dimension greater than 4. In this case the author claims that two psc-metrics are psc-isotopic if they are psc-concordant. In view of [\textit{B. Botvinnik}, Geom. Funct. Anal. 24, No. 3, 1037 (2014; Zbl 1297.53030)] these statements should be considered as conjectures.