On the dynamics of a three-dimensional differential system related to the normalized Ricci flow on generalized Wallach spaces

Abstract

We study the behavior of a three-dimensional dynamical system with respect to some set $S$ given in 3-dimensional euclidian space. Geometrically such a system arises from the normalized Ricci flow on some class of generalized Wallach spaces that can be described by a real parameter $a\in (0,1/2)$, as for~$S$ it represents the set of invariant Riemannian metrics of positive sectional curvature on the Wallach spaces. Establishing that $S$ is bounded by three conic surfaces and regarding the normalized Ricci flow as an abstract dynamical system we find out the character of interrelations between that system and~$S$ for all $a\in (0,1/2)$. These results can cover some well-known results, in particular, they can imply that the normalized Ricci flow evolves all generic invariant Riemannian metrics with positive sectional curvature into metrics with mixed sectional curvature on the Wallach spaces corresponding to the cases $a\in \{1/9, 1/8, 1/6\}$ of generalized Wallach spaces.