On Evolution of Invariant Riemannian Metrics on Generalized Wallach Spaces Under the Normalized Ricci Flow

Abstract

The aim of this paper is to discuss some results of [2, 3] relating to the study of the evolution of invariant Riemannian metrics on generalized Wallach spaces with \(a_1=a_2=a_3=a\), where \(a\in (0,1/2)\). We proved that for the Wallach spaces \(SU(3)/T_{\max }\), \(Sp(3)/Sp(1)\times Sp(1)\times Sp(1)\), and \(F_4/Spin(8)\), the normalized Ricci flow evolves all generic invariant Riemannian metrics with positive sectional curvature into metrics with mixed sectional curvature . Moreover, we obtained general results concerning the evolution of invariant Riemannian metrics on generalized Wallach spaces with \(a\in (0,1/2)\setminus \{1/4\}\) under the normalized Ricci flow . The very special case \(a=1/4\) is also considered.