The Ricci curvature and the normalized Ricci flow on the Stiefel manifolds $\mathrm{SO}(n)/\mathrm{SO}(n-2)$

Abstract

We proved that on every Stiefel manifold $V_2{\mathbb{R}}^n\cong \mathrm{SO}(n)/\mathrm{SO}(n-2)$ with $n\ge 3$ the normalized Ricci flow preserves the positivity of the Ricci curvature of invariant Riemannian metrics with positive Ricci curvature. Moreover, the normalized Ricci flow evolves all metrics with mixed Ricci curvature into metrics with positive Ricci curvature in finite time. From the point of view of the theory of dynamical systems we proved that for every invariant set~$\Sigma$ of the normalized Ricci flow on~$V_2{\mathbb{R}}^n$ defined as $x_1^{n-2}{x}_2^{n-2}{x}_3=c$, $c>0$, there exists a smaller invariant set $\Sigma \cap {\mathcal{R}}_{+}$ for every $n\ge 3$, where~${\mathcal{R}}_{+}$ is the domain in ${\mathbb{R}}_{+}^3$ responsible for parameters $x_1,x_2,x_3>0$ of invariant Riemannian metrics on~$V_2{\mathbb{R}}^n$ admitting positive Ricci curvature.

15 pages, 4 figures