On properties of the sets of positively curved Riemannian metrics on generalized Wallach spaces

Abstract

Sets related to positively curved invariant Riemannian metrics on generalized Wallach spaces are considered. The problem arises in studying of the evolution of such metrics under the normalized Ricci flow equation. For Riemannian metrics of the Wallach spaces $\operatorname{SU}(3)/T_{\max}$, $\operatorname{Sp(3)}/ \left(\operatorname{Sp(1)}\right)^3$ and $F_4/\operatorname{Spin(8)}$ which admit positive sectional curvature and belong to a given invariant surface $Σ$ of the normalized Ricci flow we established that they form a set bounded by three connected and pairwise disjoint regular space curves such that each of them approaches two others asymptotically at infinity. Analogously, for all generalized Wallach spaces the set of Riemannian metrics which belong to $Σ$ and admit positive Ricci curvature is bounded by three curves each consisting of two connected components as regular curves. Intersections and asymptotical behaviors of these components were studied as well.

Figures (3)

• Figure 1: The left panel: the curves $s_1, s_2, s_3$; The right panel: the curves $r_1, r_2, r_3$, $l_1, l_2, l_3$ and singular points $\bold{o}_0, \bold{o}_1, \bold{o}_2,\bold{o}_3$ corresponding to $a=1/6$.
• Figure 2: The left panel: the cones $\Gamma_1,\Gamma_2, \Gamma_3$, $\Lambda_1, \Lambda_2, \Lambda_3$ and the planes $x_k=x_i+x_j$ for $\{i,j,k\}=\{1,2,3\}$; The right panel: crossing $r_2$ and $r_3$ by $r_1$.
• Figure 3: The separatrices $l_1, l_2, l_3$ (in cyan color), $I_1, I_2, I_3$ (in yellow color) and trajectories of \ref{['three_equat']} (in black color).

Theorems & Definitions (7)

• Theorem 1
• Theorem 2
• Lemma 1: Ab24
• Lemma 2
• Remark 1
• Remark 2
• Remark 3