On topological structure of some sets related to the normalized Ricci flow on generalized Wallach spaces

Abstract

We study topological structures of the sets $(0,1/2)^3 \cap \Omega$ and $(0,1/2)^3 \setminus \Omega$, where~$\Omega$ is one special algebraic surface defined by a symmetric polynomial in variables $a_1,a_2,a_3$ of degree~$12$. These problems arise in studying of general properties of degenerate singular points of dynamical systems obtained from the normalized Ricci flow on generalized Wallach spaces. Our main goal is to prove the connectedness of $(0,1/2)^3 \cap \Omega$ and to determine the number of connected components of $(0,1/2)^3 \setminus \Omega$. Key words and phrases: Riemannian metric, generalized Wallach space, normalized Ricci flow, dynamical system, degenerate singular point of dynamical system, real algebraic surface, singular point of real algebraic surface.