Ricci curvature and normalized Ricci flow on generalized Wallach spaces

TL;DR

This paper investigates how the normalized Ricci flow affects Ricci curvature positivity on generalized Wallach spaces, identifying conditions under which positivity is preserved or lost, with specific results for classical Lie group quotients.

Contribution

The authors establish new criteria for Ricci curvature positivity preservation under normalized Ricci flow on generalized Wallach spaces, expanding understanding of geometric evolution in these spaces.

Findings

Normalized Ricci flow does not preserve Ricci positivity for certain spaces with a_1+a_2+a_3≤1/2.

Positivity is preserved for spaces with a_1+a_2+a_3>1/2 under specific inequalities.

Identified conditions for classical Lie group quotients with up to 11 dimensions to maintain Ricci positivity.

Abstract

We proved that the normalized Ricci flow does not preserve the positivity of Ricci curvature of Riemannian metrics on every generalized Wallach space with $a_1+a_2+a_3\le 1/2$, in particular on the spaces $\operatorname{SU}(k+l+m)/\operatorname{SU}(k)\times \operatorname{SU}(l) \times \operatorname{SU}(m)$ and $\operatorname{Sp}(k+l+m)/\operatorname{Sp}(k)\times \operatorname{Sp}(l) \times \operatorname{Sp}(m)$ independently on $k,l$ and $m$. The positivity of Ricci curvature is preserved for all original metrics with $\operatorname{Ric}>0$ on generalized Wallach spaces $a_1+a_2+a_3> 1/2$ if the conditions $4\left(a_j+a_k\right)^2\ge (1-2a_i)(1+2a_i)^{-1}$ hold for all $\{i,j,k\}=\{1,2,3\}$. We also established that the spaces $\operatorname{SO}(k+l+m)/\operatorname{SO}(k)\times \operatorname{SO}(l)\times \operatorname{SO}(m)$ satisfy the above conditions for $\max\{k,l,m\}\le 11$, moreover, additional conditions were found to keep $\operatorname{Ric}>0$ in cases when $\max\{k,l,m\}\le 11$ is violated. Similar questions have also been studied for all other generalized Wallach spaces given in the classification of Yuri\u\i\ Nikonorov.